Time-causal and time-recursive wavelets

1 Introduction

Temporal signals can inherently contain different types of structures at different temporal scales. Making such multi-scale relationships in the data explicit and exploring these relationships can often substantially improve the performance of signal processing pipelines. For the purpose of handling such scale variations in not only temporal signals, the notion of wavelet analysis has been developed by the signal processing community to decompose signals into different components at different scales (Ricker, 1944; Morlet et al., 1982a; Morlet et al., 1982b; Grossmann and Morlet, 1984; Hosken, 1988; Mallat, 1989; Mallat, 1999; Heil and Walnut, 1989; Strang, 1989; Daubechies, 1992; Meyer, 1992; Chui, 1992; Ruskai et al., 1992; Donoho and Johnston, 1993; Teolis, 1998; Walker, 1999; Debnath and Shah, 2002; Misiti et al., 2007). Due to their scale-separating properties and their computational efficiency, wavelet representations are used as a powerful generic building block for representing the information contents in image data and time-dependent signals for a very wide range of signal processing methods and applications.

While a few wavelet formulations have respected temporal causality (Szu et al., 1992; Vázquez et al., 2005), traditional wavelet theory is largely non-causal, making use of filters that are often symmetric or antisymmetric with respect to the temporal origin. Such an approach is, however, not feasible in real-time situations, when the future cannot be accessed. Non-causal wavelet theory could, in principle, be applied to real-time signals by introducing explicit temporal buffers with sufficiently long temporal delays, beyond which the wavelets would be truncated to zero. For time-critical applications, where the temporal delays must be minimized, it is, therefore, of interest to develop truly time-causal wavelets that do not make use of any information from the future.

The subject of this study is to show how a principled temporal wavelet theory can be developed based on the notion of time-causal temporal scale-space representation—originally developed for video analysis in Lindeberg (2016) and then dedicated for a pure time-causal temporal domain in Lindeberg (2023). This temporal scale-space theory, which determines canonical filter shapes for the temporal wavelets, is based on convolution with a special temporal smoothing kernel, denoted the time-causal limit kernel, and corresponding to the convolution of an infinite set of truncated exponential kernels in cascade, with specially chosen time constants to obtain temporal scale covariance.

Temporal scale covariance in this respect corresponds to self-similarity over temporal scales so that the temporal scale-space representation computed for a rescaled signal corresponds to a rescaling of the temporal scale-space representation of the original signal, complemented by a shift in the dimension of the temporal scale parameter. By combining this type of time-causal temporal scale-space representation with temporal derivatives, we can construct mother wavelets for time-causal wavelet representations from temporal derivatives of the time-causal limit kernel. In this way, the resulting time-causal wavelets will have the ability to handle temporal structures at different temporal scales in a provably self-similar manner. These filters can, thus, constitute the foundation for defining time-causal and scale-covariant temporal basis functions for use as computational primitives in more composed signal processing or learning-based methods.

Furthermore, by forming the differences between temporal scale-space representations at adjacent temporal scales, we can construct time-causal bandpass wavelet representations, which are here formally shown to up to a scaling factor correspond to the time-causal wavelet representation based on first-order temporal derivatives of the time-causal limit kernel. From such a temporal bandpass representation, reconstruction of the original signal is straightforward by mere addition over scales of the time-causal bandpass representations, thereby opening up possibilities for real-time manipulation of the signal as decomposed into a time-causal wavelet representation. This situation is structurally very closely related to the corresponding property of the Ricker/Mexican hat wavelet, based on second-order derivatives of the non-causal Gaussian kernel, where reconstruction, as will be shown, can be directly performed by integrating the corresponding second-order non-causal wavelet representations over the scale parameter, or the difference-of-Gaussians wavelet, where reconstruction is performed by instead summing up the difference-of-Gaussians wavelet representations over the temporal scales.

Since the time-causal limit kernel constitutes the composition of a set of truncated exponential kernels in cascade, the temporal smoothing operation equivalently corresponds to applying a set of first-order integrators in cascade. In terms of discrete implementation, such an operation can be well approximated by a set of first-order recursive filters coupled in cascade. The resulting implementation is thereby fully time-recursive, meaning that no other temporal memory of the past is needed beyond the information contained in the temporal scale channels themselves. Specifically, this implies straightforward and very efficient implementation schemes on regular signal processing architectures.

By combining such mother wavelets with different types of normalizations with respect to the scale parameter, either in terms of L p-norms or the notion of scale-normalized derivatives, scale-selective signal processing algorithms can then be designed that lead to extrema over scales in the magnitude of the resulting time-causal wavelet representations at temporal scale levels proportional to characteristic durations of different types of temporal structures in the input data (Lindeberg, 2017; Lindeberg, 2018a; Lindeberg, 2018b), with close relationships to the previously studied notion of modulus maxima in non-causal wavelet representations (Mallat and Hwang, 1992).

This will thus demonstrate that significant components in more traditional non-causal wavelet theory can be carried over to a time-causal temporal domain, which opens up for a variety of application areas, where either (i) temporal signals are to be processed in real time, such as for purposes of real-time monitoring, time series analysis, and prediction, including learning-based systems (as detailed further in Section 4), or as components in control-loop systems that are to react to a continuous inflow of information; (ii) analysis of biological or physical signals that needs to be performed in a completely time-causal manner to be biologically or physically realistic, such as in mathematically based models of biological or physical processes.

In this study, we develop the underlying fundamentals of the proposed class of time-causal wavelets, with special emphasis on their theoretical properties, combined with characterizations of scaling properties and proofs-of-concept for different types of temporal signals.

Beyond a very few early approaches, this is the first systematic treatment of time-causal wavelet theory, with structurally close relations to non-causal wavelets based on derivatives of the Gaussian kernel. Specifically, we propose that the time-causal wavelet theory presented here constitutes a canonical way of defining wavelet representations over a time-causal temporal domain, just as non-causal wavelets based on derivatives of the Gaussian kernel, such as the Ricker/Mexican hat wavelets, can be regarded as canonical wavelet representations over a non-causal temporal domain according to the theory for variation-diminishing convolution transformations (Section 2.1.3).

1.1 Structure of the study

The study is organized as follows. Section 2 begins by explaining how continuous time-causal wavelets can be constructed based on temporal derivatives of a special time-causal smoothing kernel, referred to as the “time-causal limit kernel”, with the theoretical results summarized into continuous time-causal wavelet representations in Section 3.2.

Section 3.3 then shows how this continuous kernel can be canonically discretized in terms of a set of first-order recursive filters coupled in cascade, complemented by time-causal difference operators, to compute discrete approximations of temporal derivatives of the time-causal limit kernel, thereby leading to a family of discrete time-causal wavelet representations, as summarized in Section 3.4.

Section 3.5 additionally provides a set of characterizations of scaling properties of the resulting discrete wavelet representations, quantifying how well the ideal scaling properties of the continuous time-causal wavelets carry over to a discrete implementation. This specifically depends on the distribution parameter c of the time-causal limit kernel, which specifies how densely the temporal scale levels are quantized, thus leading to trade-off issues regarding the accuracy of computed features versus the inherent temporal delays that will occur due to the time-causal processing of the input signals.

Section 3.6 complements this with examples of computing results of the proposed time-causal wavelet representations for different types of temporal signals, including experimental verification of the exact reconstruction of the input signal from the proposed subclass of pure time-causal bandpass representations. Finally, Section 4 concludes with a summary and discussion, including relations to related multi-scale signal processing and learning approaches, as well as proposals regarding further application areas of the proposed time-causal wavelets.

2 Methods

2.1 Continuous wavelet representations based on temporal derivatives of the time-causal limit kernel

In this section, we will after (i) a first review of the main concepts regarding regular, not necessarily time-causal, wavelet theory, describe (ii) the time-causal limit kernel and how it is constrained with respect to its shape based on (iii) a very general theoretical result regarding provably variation-diminishing convolution transformations and the desirable property of basing the time-causal wavelet analysis on a (iv) provably scale-covariant convolution kernel. This material then constitutes the theoretical foundation for defining time-causal wavelet representations in Section 3.

2.1.1 Continuous wavelet representation

2.1.1.1 Mother wavelet

The notion of a continuous wavelet representation starts from a mother wavelet χ ( t ), which is assumed to satisfy the admissibility criterion [Mallat, 1999, Equation (4.36)]wheredenotes the Fourier transform of the mother wavelet χ ( t ). Specifically, the admissibility criterion (Equation 1) implies that the integral of the mother wavelet should be zero ¹

C χ = ∫ ω = 0 ∞ | χ ̂ ω | 2 ω d ω < ∞ , (1)

χ ̂ ω = ∫ t ∈ R χ t e − i t ω d t (2)

∫ t ∈ R χ t d t = 0 . (3)

Furthermore, if the mother wavelet decays sufficiently quickly to zero when | t | → ∞ then it holds that C χ < ∞ (Mallat, 1999, p. 75).

∫ t ∈ R 1 + | t | | χ t | d t < ∞ , (4)

2.1.1.2 Wavelet transform

Given that the mother wavelet satisfies the admissibility criterion and is normalized to unit L 2-normthe wavelet representation W χ f of any function f ∈ L 2 ( R ) can then be defined as follows (Heil and Walnut, 1989, p. 629):from which the original function can be recovered as (Heil and Walnut, 1989, p. 629)

∫ t ∈ R | χ t | 2 d t = 1 , (5)

W χ f u , v = ∫ t ∈ R f t e − u / 2 χ e − u t − v ̄ d t , (6)

f t = ∫ u ∈ R ∫ v ∈ R W χ f u , v e − u / 2 χ e − u t − v d u d v . (7)

Basically, in this representation, the variable u constitutes a logarithmic parameterization of the scaling factor a = e u in a scaling transformation of the mother wavelet according tofor some a ∈ R + and b ∈ R.

χ a t = 1 a b χ t a , (8)

The normalization of the mother wavelet χ ( t ) in the forward wavelet transform in Equation 6 and the inverse wavelet transform in Equation 7 corresponds to setting b = 1 / 2 in Equation 8. As will be observed later, other forms of normalizations of filters across scales are more commonly than L₂-normalization (Equation 5) used in the related area of scale-space theory, notably b = 1 to obtain maximum scale invariance, when computing wavelet representations in terms of temporal derivatives of the time-causal limit kernel over multiple temporal scales, as considered in Section 3.1.5.

2.1.2 The time-causal limit kernel

While the above theory applies to very general classes of mother wavelets, this treatment focuses solely on mother wavelets obtained from temporal derivatives of a special type of time-causal kernel, the time-causal limit kernel (Lindeberg, 2016, Section 5; Lindeberg, 2023, Section 3)characterized by having a Fourier transform of the form

h t ; τ = Ψ t ; τ , c , (9)

Ψ ̂ ω ; τ , c = ∏ k = 1 ∞ 1 1 + i c − k c 2 − 1 τ ω . (10)

This time-causal kernel has been derived from criteria regarding the notion of temporal scale-space representation, where any continuous signal f : R → R is to be represented by a temporal scale-space representation L : R × R + → R (see Lindeberg, 2023, Equation 26)over multiple temporal scale levels τ ∈ R + in such a way that no new local extrema or zero-crossings may be created from finer to coarser levels of temporal scales. The parameter c is specifically a distribution parameter with c > 1 that determines the discrete scale levels in units of the variance of the time-causal limit kernel according to ² Equation 18 in Lindeberg (2016):where τ 0 > 0 represents a reference scale and where, for common purposes of implementation, τ 0 = 1 and either c = 2 or c = 2 are often chosen. Alternatively, the time-causal limit kernel can also be parameterized in terms of its standard deviation σ = τ, which is of dimension [ time ].

L t ; τ , c = ∫ ξ ∈ R Ψ ξ ; τ , c f t − ξ d ξ (11)

τ k = τ 0 c 2 k , (12)

Figure 1 shows graphs of the time-causal limit kernel for a few values of the temporal scale parameter σ = τ, together with its first- and second-order temporal derivatives.

2.1.3 Characterization of continuous smoothing kernels

A theorem by Schoenberg (1950) characterizes the families of continuous smoothing kernels that are variation-diminishing—in the sense that the number of zero-crossings in a smoothed signal must never exceed the number of zero-crossings in the original signal—to having a bilateral Laplace–Stieltjes transform of the formfor − c < Re ( s ) < c and some c > 0, where C ≠ 0, γ ≥ 0, δ, and a i are real and ∑ i = 1 ∞ a i 2 is convergent.

∫ ξ = − ∞ ∞ e − s ξ h ξ d ξ = C e γ s 2 + δ s ∏ i = 1 ∞ e a i s 1 + a i s (13)

From this result, ³ it follows that any continuous 1-D smoothing kernel can, beyond trivial rescaling operations and shifts, be decomposed into the following classes of primitive smoothing transformations: for some strictly positive | λ |.

• convolution with Gaussian kernels

h ξ = e − γ ξ 2 , (14)

• convolution with truncated exponential functions

h ξ = e − | λ | ξ ξ ≥ 0 , 0 ξ < 0 , h ξ = e | λ | ξ ξ ≤ 0 , 0 ξ > 0 , (15)

With respect to the smoothing of temporal signals over multiple temporal scales, this result has the following important implications. where τ 1 ≥ 0 and τ 2 ≥ 0 are the temporal scale parameters. coupled in cascade (Lindeberg and Fagerström, 1996; Lindeberg, 2023, Section 2.2).

• Over a symmetric non-causal temporal domain, where the future is assumed to be accessible, as it can be for pre-recorded temporal signals, this result singles out the Gaussian kernel (Equation 14) as the canonical semi-group of temporal smoothing kernels (Equation 16) (Karlin, 1968, Theorem 5.2; Lindeberg, 1990, Theorem 5) that obey

h ⋅ ; τ 1 ∗ h ⋅ ; τ 2 = h ⋅ ; τ 1 + τ 2 , (16)

• Over a time-causal temporal domain, where the future cannot be accessed, the only possible temporal smoothing kernels are the truncated exponential kernels (Equation 15)

h exp t ; μ k = 1 μ k e − t / μ k t ≥ 0 , 0 t < 0 , (17)

Thus, a complete characterization of which idealized temporal smoothing kernels to use can be obtained for when to smooth 1-D signals to coarser levels of temporal scales in the cases of either a time-causal or a non-causal temporal domain. Please note, however, that the truncated exponential kernels (Equation 17) do not obey a semi-group property corresponding to Equation 16.

2.1.4 Scale covariance property of the time-causal limit kernel

The definition of the time-causal limit kernel according to Equation 9, originally performed in Section 5 of Lindeberg (2016) and then refined in Section 3 of Lindeberg (2023), starts from the above characterization of time-causal temporal smoothing kernels, as can only be expressed in terms of truncated exponential kernels coupled in cascade, to choose the time constants μ k in an infinite cascade of convolution operationsbased on intermediate temporal scale levels τ k according to Equation 12, in such a way that the choice of the temporal scale levels according toimplies that the resulting time-causal limit kernel obeys temporal scale covariance.

Ψ ⋅ ; τ , c = ∗ k = 1 ∞ h exp ⋅ ; μ k (18)

μ k = c − k c 2 − 1 τ (19)

The notion of temporal scale covariance here means that, if for any temporal scaling factor S t that is an integer power of the distribution parameter c > 1 according towe consider a scaling transformation of the original signal f into a rescaled signal f ′ defined bythen the resulting temporal scale-space representations are equal (Lindeberg, 2016, Equation 47)for matching values of the temporal moments t ′ = S t t and the temporal scale parametersfor the discrete-valued temporal scaling factors S_t according to Equation 20. The time-causal limit kernel is, therefore, self-similar under temporal scaling transformations according to Equation 21, which also, with appropriate complementary temporal scale normalization, carries over to temporal derivatives of the responses to the time-causal limit kernel according toas described in Section 3.1.2.

S t = c j for j ∈ Z , (20)

f ′ t ′ = f t for t ′ = S t t , (21)

L ⋅ ; τ , c = Ψ ⋅ ; τ , c ∗ f ⋅ , (22)

L ′ ⋅ ; τ ′ , c = Ψ ⋅ ; τ ′ , c ∗ f ′ ⋅ , (23)

L ′ t ′ ; τ ′ , c = L t ; τ , c (24)

τ ′ = S t 2 τ (25)

L t n t ; τ = ∂ t n L t ; τ , (26)

Note, however, that since the temporal scale levels are inherently discrete, as implied by the construction of the temporal scale-space representation from multiple truncated exponential kernels coupled in cascade, the temporal scale covariance property only holds exactly when the temporal scaling factor S t is an integer power of the distribution parameter c according to S t = c j for j ∈ Z. For other values of S t, the corresponding result at the transformed scale levels (Equation 25) according to Equation 23 will only be approximate.

2.1.5 Properties of the time-causal limit kernel

2.1.5.1 Characterization of the temporal duration

The variance of the time-causal limit kernel, which characterizes the temporal duration, is given by Equation 35 of Lindeberg (2016):

V Ψ ⋅ ; τ , c = ∑ k = 1 ∞ μ k 2 = τ . (27)

2.1.5.2 Characterizations of the temporal delay

The temporal mean of the time-causal limit kernel, which constitutes one way of characterizing the temporal delay, is as follows (Lindeberg, 2016, Equation 34):

M Ψ ⋅ ; τ , c = ∑ k = 1 ∞ μ k = c + 1 c − 1 τ . (28)

An alternative characterization of the temporal delay is given by the position of the temporal maximum of the time-causal limit kernel. Unfortunately, it is hard to calculate the derivatives of the time-causal limit kernel in compact closed form. The maximum point can, however, be estimated from an approximation of the time-causal limit kernel in terms of Koenderink’s (1988) scale-time model according to Equation 39 in Lindeberg (2023):

t max ≈ c + 1 2 τ 2 2 c − 1 c 3 = δ . (29)

2.1.5.3 Trade-off issue regarding the choice of the distribution parameter c

From these estimates of the temporal delay, in combination with the result of Equation 24 concerning exact scale covariance properties, which hold only for temporal scaling factors S t that are integer powers of the distribution parameter c, there is a trade-off issue, in that:

• A lower value of the distribution parameter c will imply a denser set of temporal scale levels, with numerically better ability to approximate temporal scale covariance for temporal scaling factors S t that are not integer powers of the distribution parameter c;
• A larger value of the distribution parameter c will lead to shorter temporal delays, as estimated according to Equations 28, 29, with better ability to respond rapidly in real-time situations.

2.1.5.4 Differential equation formulation of the time-causal temporal scale-space representation

In terms of differential equations, ⁴ the relationships between adjacent levels of temporal scales are given by Equation 11 in Lindeberg (2023):with the initial condition L ( t ; 0 , c ) = f ( t ).

∂ t L t ; τ k , c = − 1 μ k L t ; τ k , c − L t ; τ k − 1 , c , (30)

An important property of this type of temporal scale-space representation is that it is time-recursive. The temporal scale-space representations L ( t ; τ k , c ) over the temporal scale channels with indices k constitute a sufficient temporal memory of the past to compute the temporal scale-space representation and the next temporal moment, given a new input in the input signal f ( t ).

The form of the first-order integrators in Equation 30 also explicitly implies that local perturbations in the signal are gradually smoothed out from finer to coarser levels of temporal scales because of the negative feedback with respect to the differences L ( t ; τ k , c ) − L ( t ; τ k − 1 , c ) between the adjacent temporal scale channels.

2.1.5.5 Cascade smoothing property of the time-causal limit kernel

Due to the definition of the time-causal limit kernel according to Equation 18, it follows that it obeys the following cascade smoothing property between adjacent levels of temporal scales τ and τ / c 2 (Lindeberg, 2016, Equation 41):

Ψ ⋅ ; τ , c = h exp ⋅ ; c 2 − 1 c τ ∗ Ψ ⋅ ; τ c 2 , c . (31)

This property means that when computing the temporal scale-space representation or a temporal derivative of the temporal scale-space representation, it is not necessary to redo all the computations when computing those representations over multiple temporal scales. Instead, it is sufficient to perform most of the temporal smoothing once and for all and then compute the representations at the coarser levels of temporal scale by applying a usually much smaller number of truncated exponential kernels to the corresponding representations at finer temporal scales.

Between adjacent temporal scale levels of the smoothed temporal scale-space representations, it therefore follows from Equation 31 that and furthermore regarding temporal derivatives from differentiation of Equation 32 that

L ⋅ ; τ , c = h exp ⋅ ; c 2 − 1 c τ ∗ L ⋅ ; τ c 2 , c , (32)

L t n ⋅ ; τ , c = h exp ⋅ ; c 2 − 1 c τ ∗ L t n ⋅ ; τ c 2 , c . (33)

Since each truncated exponential kernel is a simplifying kernel, guaranteeing the non-creation of new local extrema or zero-crossings from finer to coarser temporal scales, the temporal derivatives in Equation 33 therefore obey a simplifying property from finer to coarser levels of temporal scale.

Combined with the variance-based characterization of the time-causal limit kernel in Equation 27, this simplifying property from finer to coarser levels of temporal scale means that the convolution of a signal with the time-causal limit kernel with scale parameter τ can be regarded as effectively suppressing temporal structures at temporal scales finer than σ = τ.

2.1.5.6 Truncating the infinite convolution kernel for implementation purposes

For numerical or physical implementation, the time-causal limit kernel can, because of the rapid convergence property of the temporal scale levels as corresponding to the convergence properties of a geometric series, be truncated for a finite number K of layers of exponential kernels coupled in cascade.

For this purpose, the time constants of the K selected truncated exponential kernels coupled in cascade can be chosen according to Lindeberg (2016), Equations 19, 20: ⁵ where the time constant μ 1 of the first truncated exponential kernel has been determined in such a way that the temporal variance of the resulting composed kernel is exactly τ

μ 1 = c 1 − K τ , (34)

μ k = τ k − τ k − 1 = c k − K − 1 c 2 − 1 τ 2 ≤ k ≤ K , (35)

V h ⋅ ; μ 1 , μ 2 , … , μ K = ∑ k = 1 K μ k 2 = τ . (36)

Thus, the first truncated exponential kernel approximates the temporal smoothing effect of all the truncated exponential kernels that have been truncated away, as quantified in terms of the temporal variances of the temporal smoothing kernels involved.

2.1.5.7 Fulfilment of the admissibility criterion for a mother wavelet

Based on a partial fraction expansion of the Laplace transform of the time-causal limit kernel defined according to Equation 10:with the time constants μ k as functions of τ and c according to Equation 35, the coefficients A k in Equation 37 can be determined according to Lindeberg (2023), Equation 73:This implies that the nth order temporal derivative of the time-causal limit kernel will have the following series representation (Lindeberg, 2023, Equation 75):

H Ψ q ; τ , c = ∏ k = 1 ∞ 1 1 + μ k q = ∑ k = 1 ∞ A k 1 + μ k q , (37)

A k = ∏ i = 1 , i ≠ k ∞ 1 1 − μ i μ k . (38)

∂ t n Ψ t ; τ , c = ∑ k = 1 ∞ − 1 μ k n A k μ k e − t / μ k t ≥ 0 . (39)

Due to the exponential decrease of this expression as a function of t, it thereby with A_k according to Equation 38 follows that (Equation 4)which formally verifies that the time-causal limit kernel fulfills the admissibility condition in Equation 1 for being a mother wavelet.

∫ t ∈ R 1 + | t | | Ψ t ; τ , c | d t < ∞ , (40)

3 Results

Given the above treatment of the special properties of the time-causal limit kernel, this section will start by describing (i) how mother wavelets can be defined from temporal derivatives of the time-causal limit kernel and how such mother wavelets can be used for defining a continuous wavelet transform based on a discrete temporal scale parameter. We will also describe (ii) how the normalization of this kernel over scales relates to scale-normalized derivatives in scale-space theory, including the special role of L 1-normalization across scales and scale selection properties for local characteristic model signals, as well as (iii) how that scale normalization relates to the normalization over scales in more traditional wavelet representations.

Based on these foundations, we will then describe (iv) how the first- and second-order temporal derivatives of the time-causal limit kernel can be regarded as time-causal analogs of the Ricker wavelet, also referred to as the “Mexican hat wavelet”, and as corresponding to the second-order derivative of the Gaussian kernel, as well as analogs of the first-order derivatives of the non-causal Gaussian kernel.

The intention behind this treatment is to lay out the theoretical foundations for using the proposed time-causal wavelets as computational primitives in more composed signal processing and learning approaches. Specifically, since the time-causal limit kernel, used as a fundamental basis for defining the time-causal wavelet representations in terms of the derivatives thereof, does not have any compact closed-form expression, this section will devote special attention to describing its theoretical properties, so as to infer the properties of the derived time-causal wavelet representations and thus mediate a better theoretical understanding.

An effectively more compact description of some of the main concepts and results is provided in Section 3.2, with extensions to discrete temporal wavelets in Sections 3.3–3.4, discrete approximations of scaling properties in Section 3.5, and experimental results of computing the proposed types of time-causal wavelet representations for different types of temporal signals in Section 3.6.

3.1 Time-causal and time-recursive mother wavelets and wavelet representations

Our proposal is that the temporal derivatives of the time-causal limit kernel can with appropriate complementary normalization be used as mother waveletsfor computing time-causal and time-recursive wavelet representations according to Equation 6:

χ t ; c = ∂ t n Ψ t ; 1 , c ‖ ∂ t n Ψ t ; 1 , c ‖ p (41)

W χ f u , v ; c = ∫ t ∈ R f t e − u / 2 χ e − u t − v ; c d t . (42)

However, a conceptual difference between this formulation of a time-causal wavelet transform and the more traditional formulation Equation 6 is that because of the discrete nature of the temporal scale levels τ k = c 2 k according to Equation 12, the logarithmic variable u in the time-causal wavelet transform (Equation 42) will only assume discrete values u j according tothat is of the form

e u j = c j , (43)

u j = j log ⁡ c for j ∈ Z . (44)

According to traditional wavelet theory, L 2-norms are normally used when normalizing the mother wavelet. For the purposes of computing derivatives over multiple scales in scale-space theory, however, other choices of L p-norms are more common for relating image structures over scales (Lindeberg, 1998b; Table 3), as detailed further below.

3.1.1 Relations to scale-normalized derivatives in scale-space theory

For the purposes of scale-space theory, the notion of scale-normalized temporal derivatives can be introduced according to Lindeberg (1998b), Equation 6 and Lindeberg (2017), Equation 6:where γ ∈ R + is a scale normalization parameter.

∂ ζ = τ γ / 2 ∂ t , (45)

3.1.2 Scale covariance property of scale-normalized temporal derivatives

As shown in Lindeberg (2016), Appendix 3, under a temporal scaling transformation (Equation 46) of the formwith the corresponding transformation between corresponding temporal scale levels of the formthe scale-normalized temporal derivatives defined from temporal differentiation of convolutions with the time-causal limit kernel Ψ ( t ; τ , c ) do for any temporal input signal relate according to

f ′ t ′ = f t for t ′ = c j ′ − j t , (46)

τ ′ = c 2 j ′ − j τ , (47)

L ζ ′ n ′ t ′ ; τ ′ , c = c j ′ − j n γ − 1 L ζ n t ; τ , c . (48)

Specifically, local temporal scale estimates τ ̂ and τ ̂ ′, as determined from local extrema over temporal scales in the two temporal domains, will be assumed at corresponding temporal scale levels and will thus be transformed in a scale-covariant way for any temporal scaling transformation of the form t ′ = c j ′ − j t according to Equation 47

τ ̂ ′ = c 2 j ′ − j τ ̂ . (49)

Furthermore, in the special case of γ = 1, the scale-normalized derivatives are equal

L ζ ′ n ′ t ′ ; τ ′ , c = L ζ n t ; τ , c . (50)

Thus, the magnitudes of scale-normalized derivatives between matching temporal moments { t ↔ t ′ } and temporal scales { τ ↔ τ ′ } can be readily compared.

3.1.3 Scale selection properties for characteristic temporal signals

By studying the evolution properties of such temporal scale-normalized derivatives over scales for the non-causal temporal scale-space representation defined from convolution with Gaussian kernels according towhere the non-causal temporal Gaussian kernel is given bythe following properties do specifically hold: the strongest magnitude response of the scale-normalized second-order derivative operator derived from the non-causal temporal scale-space representation in Equation 51 is assumed for the time moment t = 0 and temporal scale level (Lindeberg, 2017, Equation 13)proportional to the temporal scale τ 0 in the input signal. For the specific choice ofit follows from Equation 55 that there a perfect match to the characteristic temporal scale in the input signal according to the strongest maximum response of the scale-normalized first-order derivative operatoris assumed for the time moment t = 0 and the temporal scale level (Lindeberg, 2017, Equation 24)proportional to the temporal scale τ 0 in the input signal. For the specific choice ofit follows from Equation 60 that there is a that perfect match with the characteristic temporal scale in the input signal according to

L ⋅ ; τ , c = g ⋅ ; τ ∗ f ⋅ , (51)

g t ; τ = 1 2 π τ e − t 2 / 2 τ . (52)

• For a 1-D Gaussian blob

f t = g t ; τ 0 , (53)

L ζ ζ t ; τ = τ L t t t ; τ (54)

τ ̂ = 2 γ 3 − 2 γ τ 0 (55)

γ = γ 2 = 3 4 , (56)

τ ̂ = τ 0 . (57)

• For a 1-D Gaussian edge

f t = ∫ ξ = − ∞ t g ξ ; τ 0 d ξ , (58)

L ζ t ; τ = τ 1 / 2 L t t ; τ (59)

τ ̂ = γ 1 − γ τ 0 (60)

γ = γ 1 = 1 2 , (61)

τ ̂ = τ 0 . (62)

Thus, the scale selection mechanism based on local extrema over temporal scales leads to scale estimates τ ̂ that reflect the inherent temporal scales τ in the input signal. Notably, these results bear a close relationship to the matched filter theorem (Woodward, 1953; Turin, 1960), in that the scale selection mechanism will choose filters for detecting the different types of temporal structures in the input signal that as well as possible match their size.

3.1.4 Interpretations in terms of normalization over scale with respect to L p-norms of Gaussian derivative kernels

In relation to normalization of the filters with respect to L p-norms, it can specifically be shown (Lindeberg, 1998b, Section 9.1) that the notion of γ-normalized derivatives corresponds to normalizing the nth-order scale-normalized Gaussian derivative kernels g ζ n ( t ; τ ) over a one-dimensional domain to constant L p-norms over scalewithwhere the perfectly scale invariant casecorresponds to L 1-normalization of the scale-normalized Gaussian derivatives (Equation 63) for all orders n of temporal differentiation. Such default L 1-normalization corresponds to scale-normalized temporal derivatives that are dimensionless in units of [ time ] and will therefore treat temporal structures at all temporal scales in a similar manner.

‖ g ζ n ⋅ ; τ ‖ p = ∫ t ∈ R | g ζ n t ; τ | p d x 1 / p = G n , γ , (63)

p = 1 1 + n 1 − γ , (64)

γ = 1 (65)

This can also be compared with the transformation property of scale-normalized temporal derivatives in the scale covariance property in Equation 48, where the choice of γ = 1 corresponding to p = 1 makes the scale-normalized temporal derivatives equal at corresponding temporal moments { t ↔ t ′ } and temporal scales { τ ↔ τ ′ }.

With regard to the above detection of the scale of a 1-D Gaussian blob from extrema over scale in the magnitude of the scale-normalized second-order derivative using the scale normalization power γ 2 = 3 / 4 or the detection of the scale of a 1-D Gaussian edge from extrema over scale in the magnitude of the scale-normalized first-order derivative using the scale normalization power γ 1 = 1 / 2, it can be noted that it follows from Equation 64 that both of these scale normalization methods correspond to normalizing the corresponding Gaussian derivative kernels to constant L p-norms over scale for the choice of p = 2 / 3.

3.1.5 Normalization over scales in wavelet representations

For a rescaled mother wavelet of the form (Equation 8)where a ∈ R + and b ∈ R, its L p-norm is given byimplying that the L p-norm of the rescaled mother wavelet (Equation 66) is constant over scales if and only if

χ a t = 1 a b χ t a , (66)

‖ χ a ⋅ ‖ p = a 1 / p a b ‖ χ ⋅ ‖ p , (67)

b = 1 p . (68)

In regular wavelet representations, the choice p = 2 corresponding to b = 1 / 2 constitutes the default choice. In scale-space theory, the theoretical results described in Sections 3.1.2, 3.1.4 imply that p = 1 corresponding to b = 1 does, on the other hand, constitute the default choice.

3.1.6 First- and second-order derivatives of the time-causal limit kernel as time-causal analogs of the Ricker/Mexican hat wavelet and the wavelet based on the first-order derivative of the Gaussian kernel

In regular wavelet theory, the Ricker wavelet, or Mexican hat wavelet, corresponding to the second-order derivative of the Gaussian kernelconstitutes a natural choice for defining a mother wavelet. With normalization (Equation 41) to unit L 2-norm over scales forwe obtain the mother waveletand with normalization to unit L 1-norm over scales forwe obtain

g t t t ; τ = t 2 − τ τ 2 g t ; τ = t 2 − τ τ 2 1 2 π τ e − t 2 / 2 τ (69)

‖ g t t ⋅ ; τ ‖ 2 = 3 2 2 π 4 τ 5 / 4 , (70)

χ g t t , L 2 t ; τ = g t t t ; τ ‖ g t t ⋅ ; τ ‖ 2 = 2 t 2 − τ e − t 2 2 τ 3 π 4 τ 5 / 4 , (71)

‖ g t t ⋅ ; τ ‖ 1 = 8 e π 1 τ , (72)

χ g t t , L 1 t ; τ = g t t t ; τ ‖ g t t ⋅ ; τ ‖ 1 = e t 2 − τ e − t 2 2 τ 4 τ 3 / 2 . (73)

For the purpose of wavelet analysis, Equations 70–73 thereby constitute different ways of normalizing the underlying second-order Gaussian derivative kernel (Equation 69).

Correspondingly, we can also start from the first-order derivative of the Gaussian kernelto define a mother wavelet. With normalization to unit L 2-norm over scales forwe obtain the mother waveletand with normalization to unit L 1-norm over scales forwe obtain

g t t ; τ = − t τ g t ; τ = − t τ 1 2 π τ e − t 2 / 2 τ (74)

‖ g t ⋅ ; τ ‖ 2 = 1 2 π 4 τ 3 / 4 , (75)

χ g t , L 2 t ; τ = g t t ; τ ‖ g t ⋅ ; τ ‖ 2 = − 2 t e − t 2 2 τ π 4 τ 3 / 4 , (76)

‖ g t ⋅ ; τ ‖ 1 = 2 π 1 τ , (77)

χ g t , L 1 t ; τ = g t t ; τ ‖ g t ⋅ ; τ ‖ 1 = − t e − t 2 2 τ 2 τ . (78)

For the purpose of wavelet analysis, Equations 75–78 thereby constitute different ways of normalizing the underlying second-order Gaussian derivative kernel (Equation 74).

Figure 2 shows graphs of the Gaussian derivative operators underlying these constructions, here visualized in terms of scale-normalized derivatives for the scale normalization power γ = 1 (Equation 65) and corresponding to L 1-normalization over scales.

Notably, given the uniqueness of the Gaussian kernel for constructing a scale-space representation over a symmetric one-dimensional domain, as based on the characterization of 1-D smoothing kernels in Section 2.1.3 and the uniqueness of temporal smoothing with truncated exponential kernels over a time-causal temporal domain as also resulting from the same characterization of 1-D smoothing kernels, with the special role of the time-causal limit kernel leading to temporal scale covariance (Section 2.1.4), the Gaussian kernel can be regarded as the canonical choice of smoothing kernel over a non-causal temporal domain, whereas the time-causal limit kernel can be regarded as a canonical choice of smoothing kernel over a time-causal temporal domain.

Given this theoretical and conceptual background, we propose to consider the first- and second-order temporal derivatives of the time-causal limit kernel according (Equation 9) as the time-casual analogs of the first- and second-order temporal derivatives of the Gaussian kernel.

With complementary normalization, the following first-order time-causal mother wavelet based on L 2-normalization is obtained, as are the following first-order time-causal mother wavelet based on L 1-normalizationthe following second-order time-causal mother wavelet based on L 2-normalizationand the following second-order time-causal mother wavelet based on L 1-normalization

χ Ψ t , L 2 t ; τ , c = Ψ t t ; τ , c ‖ Ψ t ⋅ ; τ , c ‖ 2 , (79)

χ Ψ t , L 1 t ; τ , c = Ψ t t ; τ , c ‖ Ψ t ⋅ ; τ , c ‖ 1 , (80)

χ Ψ t t , L 2 t ; τ , c = Ψ t t t ; τ , c ‖ Ψ t t ⋅ ; τ , c ‖ 2 , (81)

χ Ψ t t , L 1 t ; τ , c = Ψ t t t ; τ , c ‖ Ψ t t ⋅ ; τ , c ‖ 1 . (82)

Correspondingly, time-causal analogs of Gaussian-derivative-based mother wavelets for other orders n of temporal differentiation as well as other exponents p can also be formulated according to

χ g t n , L p t ; τ = g t n t ; τ ‖ g t n ⋅ ; τ ‖ p (83)

χ Ψ t n , L p t ; τ , c = Ψ t n t ; τ , c ‖ Ψ t n ⋅ ; τ , c ‖ p . (84)

Unfortunately, the norms ‖ Ψ t n ( ⋅ ; τ , c ) ‖ p in Equations 79–84 are hard to compute in closed form for the temporal derivatives of the time-causal limit kernel. Therefore, for the purposes of implementation, these norms can either be computed numerically or be moved outside of the details of the actual numerical implementation of the wavelet analysis of the signal, by instead basing the corresponding computations on temporal derivatives of the signal that are not explicitly normalized in magnitude with respect to the L p-norms of the mother wavelets, such as by using scale-normalized derivatives according to Equation 45.

For such purposes, we provide the following numerical values of the L 2-norms and L 1-norms of the temporal derivatives of the time-causal limit kernel up to order 2, computed based on a truncation of the time-causal limit kernel to the eight truncated exponential kernels having the longest time constants μ k for the case of the temporal scale parameter being τ = 1 and the distribution parameter set to c = 2 and for the case of the distribution parameter c = 2

‖ Ψ t ⋅ ; 1,2 ‖ 2 ≈ 0.635 , (85)

‖ Ψ t ⋅ ; 1,2 ‖ 1 ≈ 0.995 , (86)

‖ Ψ t t ⋅ ; 1,2 ‖ 2 ≈ 2.084 , (87)

‖ Ψ t t ⋅ ; 1,2 ‖ 1 ≈ 2.385 , (88)

‖ Ψ t ⋅ ; 1 , 2 ‖ 2 ≈ 0.513 , (89)

‖ Ψ t ⋅ ; 1 , 2 ‖ 1 ≈ 0.924 , (90)

‖ Ψ t t ⋅ ; 1 , 2 ‖ 2 ≈ 0.983 , (91)

‖ Ψ t t ⋅ ; 1 , 2 ‖ 1 ≈ 1.555 . (92)

To compute the values of the L 2-norms and the L 1-norms of the temporal derivatives of the time-causal limit kernels for other values of the temporal scale parameter τ, the recursive relationship (Lindeberg, 2016, Equation 177)can in combination with Equations 85–92 be used to express the L 2-norms and the L 1-norms at the scale levelswhich are related to the base level τ 0 = 1 according to integer powers of the distribution parameter squared c 2 in units of the variance of the time-causal limit kernel. The scaling relationship in Equation 93 is notably closely related to the corresponding scaling relationship for L_p-norms of non-causal Gaussian derivatives (Equation 63) and leading to the coupling to the scale normalization power γ according in Equation 64, while for the time-causal limit kernel restricted to discrete temporal scale levels (Equation 94).

‖ Ψ t n ⋅ ; c 2 j , c ‖ p = c − j n + 1 + j / p ‖ Ψ t n ⋅ ; 1 , c ‖ p (93)

τ j = c 2 j , (94)

3.2 Time-causal wavelet representations for continuous signals

3.2.1 Time-causal wavelet representations based on temporal derivatives of the time-causal limit kernel

Given the theory presented above, for any input signal f : R → R and given any order n > 0 of temporal differentiation of the time-causal limit kernel Ψ ( ⋅ ; τ k , c ), we can for a suitable choice of the scale normalization factor γ define the continuous temporal derivative of the time-causal scale-space representation L ζ n : R × Z → R according tofor the temporal scale levels chosen as (Lindeberg, 2016, Equation 18)

L ζ n ⋅ ; τ k , c = Ψ ζ n ⋅ ; τ k , c ∗ f ⋅ = τ n γ / 2 ∂ t n Ψ ⋅ ; τ k , c ∗ f ⋅ (95)

τ k = τ 0 c 2 k for k ∈ Z . (96)

With this parameterization, the discrete parameter k ∈ Z in Equation 96 plays a role similar to the continuous variable in the parameterization of the scaling group under the exponential map e u (Equations 43, 44) in the formulation of the continuous wavelet transform according to Equation 6, to which the time-causal representation in Equation 95 constitutes a time-causal analog of.

If desired, this representation can be complemented by an additional normalization with respect to a suitable L p-norm of the nth order temporal derivative of the time-causal limit kernel according to the theory above, also taking into account the scaling properties of the scale-normalized temporal derivatives.

In the following analysis, however, the mother wavelets to constant L p-norm will not explicitly be normalized, instead fully relying on the uniform scaling factor normalization over scales in terms of scale-normalized temporal derivatives according to Equation 45.

3.2.2 Time-causal bandpass wavelet representations

If from an input signal f ( t ) a set of successively smoothed temporal representations L ( t ; τ k , c ) is constructed by convolution with the time-causal limit kernel according to Equation 22, then the differences between these representationswith the specifically finest bandpass channel given bywill constitute a bandpass representation of the original signal f ( t ), with very close structural relationships to a difference-of-Gaussians representation in the area of image processing (Burt and Adelson, 1983; Crowley and Stern, 1984).

Δ L DoT t ; τ k , c = L t ; τ k − L t ; τ k − 1 , c for k ∈ 2 , K , (97)

Δ L DoT t ; τ 1 , c = L t ; τ 1 − L t ; τ 0 = L t ; τ 1 − f t , (98)

In terms of convolution kernels, such a representation can equivalently, although less efficiently, be computed with a time-causal convolution kernel of the formwhere the difference-of-the-time-causal-limit-kernels (DoT) kernel Δ Ψ underlying the definition in Equation 99 is given byfor (Lindeberg, 2016, Equation 18)

Δ L DoT ⋅ ; τ k , c = Δ Ψ ⋅ ; τ k , τ k − 1 , c ∗ f ⋅ , (99)

Δ Ψ t ; τ k , τ k − 1 , c = Ψ t ; τ k , c − Ψ t ; τ k − 1 , c (100)

τ k = c 2 τ k − 1 . (101)

Specifically, if such a representation is computed between a finest minimum temporal scale level τ 1, for the specific choice of defining L ( t ; τ 0 , c ) = f ( t ) in the boundary case, and a coarsest maximum temporal scale level τ K, then reconstruction of the original signal or the representation at any other temporal scale level is straightforward from such a representation by merely adding the corresponding bandpass representations over the temporal scale channels for all the temporal scale levelswhich for the special choice of j = 0 yields

L ( t ; τ j , c ) = L t ; τ K , c − ∑ k = j + 1 K Δ L DoT t ; τ k , c , (102)

f t = L t ; τ K , c − ∑ k = 1 K Δ L DoT t ; τ k , c . (103)

3.2.3 Exact reconstruction from the time-causal bandpass representation

Notably, provided that

• the bandpass channels Δ L DoT ( t ; τ k , c ) for k ∈ [ 1 , K ] are computed from adjacent temporal scale channels (Equation 101) according to Equations 97, 98 and that
• the reconstruction is performed from the bandpass channels Δ L DoT ( t ; τ k , c ) according to Equation 103,

the reconstruction will be exact if we can assume that the influence of numerical errors can be disregarded and provided that the temporal scale-space representation L ( t ; τ K , c ) at the coarsest temporal scale τ K is also provided as input to the reconstruction algorithm.

This property follows directly from inserting the expressions in Equations 97, 98 for the bandpass representations into the expression in Equation 103 for the reconstruction, which gives

L t ; τ K , c − ∑ k = 1 K Δ L DoT t ; τ k , c = = L t ; τ K , c − L t ; τ K − L t ; τ K − 1 , c − ⋯ − L t ; τ 2 − L t ; τ 1 , c − L t ; τ 1 − f t = = f t . (104)

Of course, when initiating the first-order temporal integrators for the temporal smoothing operations with truncated exponential kernels coupled in cascade, there will be transition effects in the first-order integrators as originating from the first temporal moment when the time-recursive analysis is initiated. From the way that the bandpass representations are defined from explicit differences between the temporal scale channels, such transition effects will not, however, affect the exactness of the reconstruction. ⁶

3.2.4 Non-causal bandpass representations based on the second-order derivatives of the Gaussian kernel

Notably, there is a close structural relationship between the above notion of bandpass wavelets and the Ricker/Mexican hat wavelet, in that the difference-of-Gaussians operator can be seen as an approximation of the Laplacian operator in the 2-D case, or the second-order derivative of the Gaussian in the 1-D case.

Since the non-causal Gaussian representation of a 1-D signal defined by convolution with Gaussian kernelssatisfies the 1-D diffusion equationit follows that the difference between the non-causal Gaussian-based temporal scale-space representations (Equation 105) at the scales τ and τ + Δ τ will constitute an approximation of the derivative of the scale-space representation in the scale direction

L ⋅ ; τ = g ⋅ ; τ ∗ f ⋅ (105)

∂ τ L = 1 2 L t t , (106)

∂ τ L t ; τ ≈ L t ; τ + Δ τ − L t ; τ Δ τ . (107)

This difference in the Gaussian-based temporal scale-space representations can also be approximated by convolving the input image with the second-order derivative of the Gaussian kernel

L t ; τ + Δ τ − L t ; τ Δ τ ≈ 1 2 g t t ⋅ ; τ ∗ f ⋅ t ; τ . (108)

Thus, up to a scalar normalization factor and a normalization with respect to the scale parameter, the Ricker/Mexican hat wavelet approximately encodes differences in a Gaussian-based scale-space representation between adjacent temporal scales.

The analogous finite-difference-based relationship to Equation 108 becomes exact if we replace the second derivative of the Gaussian kernel with the corresponding difference-of-Gaussian kernelleading to the following relationship for differences-of-Gaussians bandpass representations,and corresponding to following relationships between the difference-of-Gaussians kernel and the second-derivative of the Gaussian kernel according towith the approximation becoming gradually better for decreasing values of the scale difference Δ τ. This follows since Equation 111 is with the DoG kernel according to Equation 109 convolved with any signal corresponds to Equation 107.

DoG t ; τ , Δ τ = g t ; τ + Δ τ − g t ; τ , (109)

Δ L DoG t ; τ = L t ; τ + Δ τ − L t ; τ = DoG ⋅ ; τ , Δ τ ∗ f ⋅ t ; τ , (110)

DoG t ; τ , Δ τ Δ t ≈ 1 2 g t t t ; τ , (111)

3.2.5 Exact reconstruction of the original signal from non-causal Laplacian or differences-of-Gaussians responses

Regarding the second-order derivative of the non-causal Gaussian kernel, it follows from the diffusion equation interpretation of the non-causal 1-D scale-space representation in Equation 106 that, provided that second-order Gaussian-derivative-based responses are computed over a continuum of scale levels over some range [ t 1 , t 2 ], then the signal at the finer scale level t 1 can be reconstructed from the signal at the coarser scale level t 2 in combination with the second-order derivative responses over this scale range according toThis implies the following explicit relationship regarding non-causal reconstruction from a continuum of Laplacian-of-Gaussian responses:with the following limit case for τ 1 = 0 corresponding to L ( t ; 0 ) = f ( t ):

L t ; τ 2 − L t ; τ 1 = ∫ t = t 1 t 2 ∂ τ L t ; τ d τ = 1 2 ∫ t = t 1 t 2 L t t t ; τ d τ = 1 2 ∫ t = t 1 t 2 g t t ⋅ ; τ ∗ f ⋅ t ; τ d τ . (112)

L t ; τ 1 = L t ; τ 2 − 1 2 ∫ t = t 1 t 2 g t t ⋅ ; τ ∗ f ⋅ t ; τ d τ , (113)

f t = L t ; τ 2 − 1 2 ∫ t = 0 t 2 g t t ⋅ ; τ ∗ f ⋅ t ; τ d τ . (114)

Concerning the non-causal reconstruction of the original signal from a discrete set of differences-of-Gaussians bandpass responses Δ L DoG ( ⋅ ; τ k ), we correspondingly havewith the differences-of-Gaussians response L DoG ( ⋅ ; τ 0 ) at the finest scale level τ 0 = 0 defined according toand with the differences-of-Gaussians responses L DoG ( ⋅ ; τ k ) at the coarser scales τ k defined as

f t = L t ; τ K − ∑ k = 0 K − 1 Δ L DoG t ; τ k (115)

Δ L DoG t ; τ 0 = L t ; τ 1 − f t (116)

Δ L DoG t ; τ k = L t ; τ k + 1 − L t ; τ k . (117)

The expressions in Equations 113–117 can be considered as non-causal analogs underlying the corresponding exact reconstruction (Equation 102) from a time-causal bandpass representation, which again demonstrates a close structural relationship to regard the temporal derivatives of the time-causal limit kernel as time-causal analogs of the derivatives of the regular Gaussian kernel or differences thereof.

3.2.6 Conceptual difference between bandpass representation based on the time-causal limit kernel and bandpass representation based on the non-causal Gaussian kernel

Regarding the difference-of-the-time-causal-limit kernel (Equation 100) used in the temporal bandpass wavelets representation in Section 3.2.2, it follows from the differential equation interpretation (Equation 30) in terms of first-order integrators coupled in cascade that this difference kernel does instead represent the first-order temporal derivative of the time-causal limit kernel according towith the notable difference that the difference between the temporal scale levels τ k and τ k − 1 now has to be of the macroscopic form

Δ Ψ t ; τ k , τ k − 1 , c = − μ k Ψ t t ; τ k , c , (118)

τ k = c 2 τ k − 1 . (119)

Since the time constant μ k for the time-causal limit kernel is according to Equation 19 given by μ k = c − k c 2 − 1 τ, the multiplication of the first-order temporal derivative of the time-causal limit kernel Ψ t ( t ; τ , c ) by the time constant μ k on the right hand side in Equation 118 implies that the first-order temporal derivative of the time-causal limit kernel Ψ t ( t ; τ , c ) is multiplied by the temporal scale parameter τ raised to the power of 1 / 2 and is therefore up to the non-essential additional scaling factor c − k c 2 − 1 expressed in terms of scale-normalized derivatives for the scale normalization power γ = 1. Thereby, the definition of the time-causal bandpass representation from adjacent temporal scale levels according to Equation 119 does automatically normalize the components in the decomposition over multiple temporal scales in terms of scale-normalized derivatives.

3.3 Discrete approximations of temporal derivatives of the time-causal limit kernel

To transfer the above theory over a continuous time-causal temporal domain to a corresponding discrete time-causal domain, there is indeed a corresponding discrete temporal scale-space theory to build upon, which can be used for constructing theoretically well-founded and numerically well-conditioned discrete implementations.

3.3.1 Discretization of the first-order integrators

A basic result is that the canonical discretization of each layer in the set of first-order integrators coupled in cascade (Equation 30)is a first-order recursive filter of the formwhere it is assumed that the discrete signals in Equation 121 are sampled with a grid spacing Δ t relative to the continuous signals in Equation 120. Then, it can be shown (Lindeberg, 2025a, Appendix B.3) that the discrete recurrence relation in Equation 121 constitutes a true discretization of the continuous differential equation in Equation 120, provided that

∂ t f out t = − 1 μ k , cont f out t − f in t (120)

f out t − f out t − 1 = − 1 1 + μ k , disc f out t − 1 − f in t , (121)

μ k , disc = μ k , cont Δ t . (122)

3.3.2 Classification of discrete temporal smoothing kernels

The result that the time-causal smoothing operation should be discretized on the form (Equation 148) also builds upon a classification of discrete variation-diminishing convolution transformations by Schoenberg (1948), which states that a discrete kernel guarantees non-creation of new zero-crossings in the convolved signal if and only if its generating function is of the formwhere C > 0, k ∈ Z, q − 1 , q 1 , α i , β i , γ i , δ i ≥ 0, and ∑ i = 1 ∞ ( α i + β i + γ i + δ i ) < ∞, from which we can recognize the z-transform of the discrete recurrence relation in Equation 121 as one of the members of the class of possible functions, as occurring in the denominator of the product expression in Equation 123:

φ z = C z k e q − 1 z − 1 + q 1 z ∏ i = 1 ∞ 1 + α i z 1 + δ i z − 1 1 − β i z 1 − γ i z − 1 , (123)

f out x = f in x + β i f out x − 1 0 ≤ β i < 1 . (124)

Thus, with the relationship between the corresponding continuous and discrete time constants according to Equation 122, we can by comparison with Equation 124 regard the discrete recurrence relation in Equation 121 as a discrete analog of the continuous smoothing process in Equation 120.

3.3.3 Discrete analog of the time-causal limit kernel

In formulating a discrete analog of the time-causal limit kernel, one special fact to consider is that while the temporal mean M of the first-order recursive filter is structurally similar as for the truncated exponential filter M = μ k, the expression for the variance of the discrete recursive filter is structurally different from the corresponding continuous expression (Equation 36):

Δ τ k = μ k 2 + μ k . (125)

Thus, to determine the time constants μ k from the desired scale increments Δ τ k between adjacent temporal scale levels according towhen k > 1, for some suitably sufficiently low ⁷ selected first temporal level τ k > 0, with the first amount of incremental smoothing emulating a discrete approximation of all the truncated exponential kernels that have been truncated away according tothe following time-constants should be used in the recursive filters that implement the discrete analog of the time-causal limit kernel (Lindeberg 2023, Equation 55):

Δ τ k = τ k − τ k − 1 = τ 0 c 2 k − c 2 k − 1 , (126)

Δ τ 1 = τ 0 c 2 , (127)

μ k = 1 + 4 Δ τ k − 1 2 . (128)

This relationship has been derived to fulfil the discrete amounts of smoothing according to Equation 125 combined with the boundary condition in Equation 127.

3.3.4 Discrete approximations of temporal derivatives of the time-causal limit kernel

Given that the input signal has thus been smoothed to a set of temporal scales, discrete approximations of temporal derivatives can then be computed with temporal difference operators of the formand scale-normalized derivatives according to Equation 45 in turn be computed by multiplying the regular discrete derivative approximations by an appropriate power of the scale parameter τ in units of the variance of the underlying the equivalent discrete temporal smoothing kernel.

δ t = − 1 , + 1 , δ t t = 1 , − 2,1 , (129)

Figure 3 shows examples of such equivalent discrete derivative approximation kernels, obtained by filtering a discrete delta function with the proposed methodology for computing the discrete approximation of temporal derivatives at multiple temporal scales. As can be seen from these graphs, the discrete nature of the smoothing kernels and their discrete derivative approximations is significant for lower temporal scale levels, while the shapes of the kernels approach their continuous counterparts in Figure 1 with increasing values of the temporal scale parameter τ.

Algorithm 1 Pseudocode for combining discrete time-causal temporal filtering with the computation of bandpass wavelet representations and the computation of first- and second-order temporal derivatives, based on a set of first-order recursive filters of the form (Equation 148) coupled in cascade, which implements convolution with a discrete approximation of the time-causal limit kernel Ψ ( t ; τ , c ) according to (Equation 9). Here, it is assumed that the time constants μ k > 0 have been already computed according to (Equation 128). The variable level_prev represents a memory from the previous frame, necessary to compute the temporal differences that drive the recursive filters, also serving as a temporal buffer for computing first-order temporal derivatives. The variable level_prev2 represents a memory from two time steps ago, used for computing second-order temporal derivatives. In this respect, the algorithm is strictly time-recursive, since it only makes use of information from the present moment and a very short-term memory from the previous 1 or 2 frames.

Algorithm 2 Pseudocode for reconstructing a signal from its bandpass wavelet representation b combined with the time-causal temporal scale-space representation l at the coarsest temporal scale.

3.3.5 Streaming pipelines for the encoding and decoding stages

Algorithm 1 summarizes the different steps involved in implementing a discrete approximation of convolution with the time-causal limit kernel, as well as the computation of temporal bandpass representations and the computation of first- and second-order temporal derivatives. ⁸ Algorithm 2 gives corresponding pseudocode for reconstructing the original temporal signal from its bandpass representation, given complementary access to the time-causal scale-space representation at the coarsest temporal scale.

3.4 Time-causal wavelet representations for discrete signals

Given that the scheme in Algorithm 1 can be used for smoothing a discrete signal stream f : Z → R with an approximation of the discrete analog of the time-causal limit kernel Ψ disc ( t ; τ , c ), and then computing discrete approximations of temporal derivatives using the time-causal discrete temporal derivative operators δ t and δ t t according to Equation 129, complemented with scale normalization according tofor some suitably selected value of the scale normalization parameter γ > 0, we thereby obtain numerical approximations of the discrete analog L ζ n : Z × Z → R of the scale-normalized temporal derivative of order n of the time-causal scale-space representation of the signal f according tofor the temporal scale levels chosen as

δ ζ n = δ t n , norm = τ n γ / 2 δ t n (130)

L ζ n ⋅ ; τ k , c = Ψ ζ n , disc ⋅ ; τ k , c ∗ f ⋅ = τ n γ / 2 δ t n Ψ disc ⋅ ; τ k , c ∗ f ⋅ (131)

τ k = τ 0 c 2 k for k ∈ Z . (132)

With this parameterization, the discrete parameter k ∈ Z again has a similar role to the continuous variable u in the parameterization of the scaling group under the exponential map e u in the formulation of the continuous wavelet transform according to Equation 6.

Alternatively, given the raw undifferentiated temporal scale-space representation equivalently obtained by convolution with the discrete analog Ψ disc ( t ; τ k , c ) of the time-causal limit kernel according toa bandpass wavelet representation can be computed from the temporal scale channels in Equation 133 which is computationally equivalent to convolving the input signal f with the bandpass wavelet filterbetween adjacent levels of temporal scales

L ⋅ ; τ k , c = Ψ disc ⋅ ; τ k , c ∗ f ⋅ , (133)

Δ L DoT t ; τ k , c = L t ; τ k − L t ; τ k − 1 , c , (134)

Δ Ψ disc t ; τ k , τ k − 1 , c = Ψ disc t ; τ k − Ψ disc t ; τ k − 1 , c (135)

τ k = c 2 τ k − 1 . (136)

A technical difference compared to the continuous case is that, whereas for the continuous time-causal wavelets the bandpass wavelet representation is directly related to the first-order temporal derivative of the time-causal scale-space representation according to Equation 30 for the discrete time-causal wavelets, the bandpass wavelet representation is only related to the discrete approximation of the first-order temporal derivative of the time-causal scale-space representation if a temporal shift is additionally included when defining the bandpass representation according to the recurrence relation in Equation 121:

Δ L DoT t ; τ k , c = L t ; τ k , c − L t ; τ k − 1 , c = − μ k ∂ t L t ; τ k , c , (137)

L t − 1 ; τ k , c − L t ; τ k − 1 , c = − 1 + μ k L t ; τ k , c − L t − 1 ; τ k , c = − 1 + μ k δ t L t ; τ k , c . (138)

In this respect, there is a notable structural difference between the continuous expression in Equation 137 and the corresponding discrete expression in Equation 138.

Therefore, if a main purpose is to perform reconstruction from the time-causal wavelet responses, it is more straightforward to make use of the explicit bandpass representation as opposed to using wavelet responses in terms of discrete approximations of temporal derivatives obtained from temporal difference operators. Notably, in practice, the differences-of-time-causal-limit-kernels (Equation 135) are never used for practical implementations. Instead, the bandpass representations (Equation 134) are always computed between adjacent scale levels (Equation 136).

3.5 Characterizations of scaling properties in discretizations of the continuous theory

The theory for scale-covariant time-causal and time-recursive wavelets in Section 2.1 is developed for fully continuous signals. This section characterizes how well the continuous scaling properties carry over to a discrete implementation, as based on the methodology proposed in Sections 3.3, 3.4 and depending on the relationship between the temporal scale levels in relation to the sampling distance in the signal.

3.5.1 Scale dependencies of the discrete l p-norms of the equivalent discrete derivative approximation kernels

Figure 4 shows a first characterization in this respect by showing how the discrete l 1-norms of the equivalent convolution kernels corresponding to discrete approximations of the time-causal limit kernelfor the scale normalization power γ = 1 vary as function of the scale parameter τ in the graphs notably parameterized in terms of the standard deviation σ = τ instead of the variance τ.

‖ Ψ ζ n , disc ⋅ ; τ , c ‖ p = ‖ τ n γ / 2 δ t n Ψ disc ⋅ ; τ , c ‖ p (139)

For reference, the graphs also show the evolution properties of the L 1-norms of the temporal derivatives of the continuous time-causal limit kernel according to Equations 86, 88, 90, 92, which are constant over scales for γ = 1.

As can be seen from these graphs, there are some transient phenomena at finer levels of scales, where the influence of the discretization effects is strongest. Towards coarser levels of scales, however, the discrete l p-norms asymptotically approach constant values.

When the distribution parameter is c = 2, very good matches are obtained between the asymptotic values and the corresponding expressions for the fully continuous time-causal model. When the distribution parameter is c = 2, there are, however, certain deviations of the order of 5% relative to the continuous theory. This quantification thereby demonstrates that the discretization effects due to the discrete nature of the temporal-scale levels may be notable when the distribution parameter c = 2, while there is a much better match between the continuous and discrete theories when the distribution parameter is c = 2.

For interpreting these curves, it should be noted that here the scale parameter is treated as a continuous variable. Given that the temporal scale levels are to be inherently discrete in a temporal scale-space representation, this corresponds to emulating the effect of using possibly different initial scale values τ 0 when initiating the temporal scale levels according to Equation 12. In a practical implementation, a fixed value of the initial scale value τ 0 should, however, always be used, thus implying that the resulting graphs will only assume values for the values of the scale parameter σ = τ that are integer multiples of the distribution parameter c.

There are certain oscillations in these curves because the positive and the negative lobes in the discrete approximations of the derivatives in the time-causal limit kernel interact with the grid spacing, thereby affecting the discrete l 1-norms.

3.5.2 Scale selection properties for local blob-like and edge-like structures in the signal

To investigate the scale selective properties of the proposed methodology for defining time-causal wavelets, our next focus is on the ability of the time-causal wavelets in terms of scale-normalized derivatives of the time-causal limit kernel to reflect the inherent characteristic scales in the input data.

Blob-like and edge-like temporal structures are therefore generated in terms of discrete analog of the Gaussian kernel defined by (Lindeberg, 1990, Equation 19)where I m ( τ ) denotes the modified Bessel functions of integer order (Abramowitz and Stegun, 1964). More, specifically, blob-like temporal structures were generated as (Figure 5, left)and edge-like structures according to (Figure 5, right)

T m ; τ = e − τ I m τ , (140)

f blob t ; τ ref = T t ; τ ref (141)

f edge t ; τ ref = ∑ m = − ∞ t T m ; τ ref . (142)

Then, the discrete time-causal wavelet representations of these signals were computed over the range of temporal scale levels σ = τ ∈ [ 1 / 8,64 ] Figure 5 by computing:

• Discrete approximations of the second-order scale-normalized temporal derivatives for the scale normalization power γ = 3 / 4 of the time-causal wavelet representation of the idealized blob models f blob ( t ; τ ref ) according to Equation 141.
• Discrete approximations of the first-order scale-normalized temporal derivatives for the scale normalization power γ = 1 / 2 of the time-causal wavelet representation of the idealized edge models f edge ( t ; τ ref ) according to (Equation 142).

Then, we determined the time moments and temporal scales at which these discrete approximations of the scale-normalized temporal derivative responses assumed their maximum values over time and temporal scales.

t ̂ blob , τ ̂ blob = argmax t , τ | Ψ ζ ζ , disc ⋅ ; τ , c ∗ f blob ⋅ ; τ ref t | , (143)

t ̂ edge , τ ̂ edge = argmax t , τ | Ψ ζ , disc ⋅ ; τ , c ∗ f edge ⋅ ; τ ref t | . (144)

Such points computed in a temporal scale-space representation are referred to as “scale-space extrema” and constitute a generalization of a corresponding methodology developed for a spatial scale-space representation (Lindeberg, 1998b; Lindeberg, 1998a). Specifically, based on the theory described in Section 3.1.3, in case the time-causal limit kernel is replaced by a non-causal Gaussian kernel, this methodology would in the ideal continuous case determine the inherent temporal duration of the corresponding idealized temporal blob and edge models according to Equations 57, 62.

Figure 6 shows graphs of the dependency of the resulting scale estimates for a range of reference scale levels τ ref when detecting scale-space extrema in this way for the reference blob and edge signals f blob ( t ; τ ref ) and f edge ( t ; τ ref ) according to Equations 141, 142 when using time-causal wavelet representations for a fixed reference scale τ 0 = 1 in Equation 12 using the distribution parameter set to either c = 2 or c = 2. Given the discrete nature of both the temporal scale levels and the temporal sampling, we used local parabolic interpolation around each extremum over temporal scales to increase the accuracy in the resulting estimates of the temporal scales τ ̂.

As observed from the results, the resulting scale estimates largely follow a linear increase in the selected scale levels as a function of the reference scales τ ref of the model signals as can be derived by taking the square root of the ideal expression in Equation 49. Due to the discrete nature of the temporal scale levels and the lack of provable scale covariance properties for temporal scaling factors S t that are not integer powers of the distribution parameter c, the graphs of the selected scale levels cannot be expected to be smooth curves across the discontinuities over the temporal scale levels.

Notably, the quantization effects decrease substantially when decreasing the distribution parameter c from c = 2 to c = 2. In this respect, there is a trade-off issue in that decreasing the value of the distribution parameter c leads to more accurate scale estimates from the underlying signal, whereas increasing the distribution parameter c leads to shorter temporal delays; compare with Equations 28, 29, and see also the following section.

3.5.3 Temporal delays of the discrete approximations of the time-causal limit kernel

To characterize the temporal delays for the discrete approximations of the time-causal limit kernel, corresponding discrete analogs of the temporal delay estimates for the continuous time-causal limit kernel in terms of the temporal mean can be computed according to Equation 28 with the discrete time constants μ k according to Equations 128, 126, and the time value at which the discrete approximation of the time-causal limit kernel assumes its maximum value over scales

M Ψ disc ⋅ ; τ , c = ∑ t = 0 ∞ t Ψ disc t ; τ , c ∑ t = 0 ∞ Ψ disc t ; τ , c = ∑ k = 1 ∞ μ k , (145)

t max = argmax t Ψ disc t ; τ , c . (146)

Figure 7 shows the result of computing these characterizations of the temporal delay for the distribution parameter c ∈ { 2 , 2 } and with the estimate of t max interpolated to sub-quantization resolution using parabolic interpolation. For comparison, the graphs also show the dependencies on the temporal scale levels for the continuous temporal delay estimates according to Equations 28, 29. As can be seen from the graphs, the measures of the temporal delays are significantly shorter for the distribution parameter c = 2 than for the distribution parameter c = 2. We can also note that the measures of the temporal delays are shorter for the discrete approximations of the time-causal limit kernel than for the continuous time-causal limit kernel.

The reason for the latter difference between the continuous and discrete theories is that the time constants μ k are determined in a different manner from the differences Δ τ k in the temporal scale levels for the discrete kernels than for the continuous kernels; compare (Equation 128) in the discrete case with the expression μ k = Δ τ k in the continuous case.

3.6 Experiments

3.6.1 Bandpass decompositions based on time-causal wavelets

To illustrate the properties of the proposed notion of time-causal bandpass representation, Figures 8, 9 show the result of computing the following three types of discrete bandpass representations from two model signals ⁹ using four temporal scale levels according to τ k = τ 0 c 2 k (Equation 12) for τ 0 = 1 and c = 2, with four additional pre-smoothing stages prior to the first scale level used in the time-causal limit kernel:

• Discrete time-causal bandpass representations Δ L DoT ( t ; τ k , c ) according to Equation 134, computed from differences between discrete temporal scale-space representations obtained by smoothing with discrete analogs of the time-causal limit kernel according to Section 3.3.3 and as based on first-order recursive filters coupled in cascade as described in Algorithm 1, with the complementary reconstruction performed as described in Algorithm 2. This approach constitutes the main type of pure time-causal bandpass representation proposed in this study.
• Discrete analogs of the non-causal differences-of-Gaussians bandpass representation Δ L DoG ( t ; τ k ) according to Equation 110, with the underlying continuous convolution operations with continuous Gaussian kernels g ( t ; τ ) replaced by discrete convolutions with the discrete analog of the Gaussian kernel T ( m ; τ ) according to (Equation 140).

This approach can be seen as an improved method over previous forms of difference-of-Gaussians bandpass decompositions (Burt and Adelson, 1983; Crowley and Stern, 1984; Birch et al., 2010; Li et al., 2020; Brown et al., 2024). defined from differences between the temporal scale channels obtained by smoothing the input signal with a set of single first-order integratorswith the time constants μ k defined from the variances τ k according to μ k = ( 1 + 4 τ k − 1 ) / 2 in analogy with Equation 128 and corresponding to the natural discretizations of the time-causal scale-channel representationsobtained from convolution with single truncated exponential kernels h exp ( t ; μ k ) according to Equation 17, although for continuous signals the relationship between the variances τ k and the time constants μ k is instead given by μ k = τ k according to Equation 27.

• Discrete time-causal differences-of-exponentials bandpass representations Δ L DoE ( t ; τ k )

Δ L DoE t ; τ k = L t ; τ k − L t ; τ k − 1 (147)

L t ; τ k − L t − 1 ; τ k = − 1 1 + μ k L t − 1 ; τ k − f t (148)

L t ; τ k = h exp ⋅ ; μ k ∗ f ⋅ (149)

This type of representation can be seen as a natural extension to a time-causal bandpass representation of the Laplace-transform-related temporal memory representations considered by Howard (2024) and Howard et al. (2025). In contrast to the time-causal temporal scale-space representation underlying the differences-of-time-causal-limit-kernels bandpass representation Δ L DoT ( t ; τ k , c ), the relationships between the temporal scale channels underlying this differences-of-exponentials bandpass representation Δ L DoE ( t ; τ k ) according to Equations 147–149 do not, however, obey a similar guaranteed simplifying property from finer to coarser levels of temporal scales, as used as the foundation for the theories of variation-diminishing continuous and discrete smoothing transformations in Sections 2.1.3, 3.3.2.

For this representation, both the bandpass representation and the reconstruction from the bandpass representation are computed in a structurally similar way as for the discrete bandpass representation based on discrete analogs of the time-causal limit kernel. The only difference is that the temporal smoothing is performed using a single recursive filter as opposed to using a cascade of recursive filters. Thereby, the reconstruction from the bandpass representation will also be exact for this discrete time-causal differences-of-exponentials bandpass representation.

As can be seen from the figures, all the three types of bandpass representation lead to intuitively reasonable decompositions of the signals in terms of different types of temporal structures at different temporal scales.

Notably, there are inherent temporal delays for the time-causal bandpass representations. This constitutes an inherent property of any time-causal signal processing operations, since all processing is restricted to what has occurred in the past. For the non-causal differences-of-Gaussians representations, there are, however, no temporal delays because those representations are symmetric relative to the past and future.

From the visualizations of the bandpass representation, we can see that the differences-of-Gaussians representations are somewhat smoother at coarser temporal scales than the differences-of-time-causal-limit-kernel representations. An explanation for this property is that the Gaussian kernel as well as its discrete analog are both infinitely divisible smoothing kernels, thereby implying very strong smoothing effects given any strictly positive scale level. The differences-of-time-causal-limit-kernel representations are, in turn, significantly smoother than the differences-of-exponentials representations, because the former are computed from a set of recursive filters coupled in cascade, whereas the latter are computed from only a single recursive filter for each scale channel.

We can also note that the reconstructions f ̂ of the original signal f from the bandpass representations in combination with the scale-space representation at the coarsest scale fully reconstruct the original signal, and also for the proposed discrete implementation. Numerically, the reconstruction errorsare at the level of machine precision, as shown in Table 1.

ϵ = ‖ f ̂ − f ‖ 2 ‖ f ‖ 2 (150)

TABLE 1. Reconstruction errors according to Equation 150 for the three types of pure bandpass representations considered here, in terms of (i) differences-of-time-causal-limit-kernel (DoT) representations, (ii) differences-of-Gaussians (DoG) representations, and (iii) differences-of-exponentials (DoE) computed for the two model signals shown in Figures 8–9.

3.6.2 Scale-selective property of the time-causal wavelet concept

To demonstrate the ability of the proposed time-causal wavelet representations to adaptively respond to local structures at different temporal scales in a signal, we will next consider the result of performing a time-causal wavelet analysis based on explicit temporal derivatives of the following model signal with exponentially varying frequency for a = 200 and b = 1000, see Figure 10 for an illustration.

f t = sin exp b − t a (151)

Discrete approximations of the first-order and second-order temporal derivatives are thus computed according to Equation 131 over the scale range σ = τ ∈ [ 1 / 8,1024 ] (Equation 132) based on temporal difference operators (Equation 130) for the scale normalization power γ = 1 corresponding to L 1-normalization over scales of the underlying temporal derivatives of the continuous time-causal limit kernel. The motivation for this choice is to treat responses at different scales in a similar manner, so that the magnitudes of the temporal derivative responses can be readily compared across scales, as manifested in the scale covariance property in Equation 50.

To define a feature descriptor that is less sensitive to the local phase in the signal near the preferred scales, as determined by the temporal duration of the locally dominant temporal structures, the following quasi quadrature measure (Lindeberg, 2018a)is also computed for the weighting parameter C = 1 / 2.

Q L = L ζ 2 + C L ζ ζ 2 (152)

As can be seen from the results, the dominant responses in the time-causal wavelet representations across scales move from finer to coarser scales with increasing local wavelengths in the signal, in agreement with the desirable properties of a multi-scale wavelet representation and also in agreement with the scaling properties. Due to the exponential increase of the local wavelengths in the input signal as function of time, the corresponding temporal scale estimates ought also to increase exponentially over time. With the logarithmic parameterization of the temporal scale parameter used in the figures, we experimentally obtain scale dependencies that are well described as linear on the logarithmic scale, and thus with a very good match between the continuous theory and the discrete implementation.

The quantization effects are notably stronger when using a sparser sampling of the temporal scale levels, as induced by setting the distribution parameter to c = 2, while the quantization effect is much lower using a denser sampling of the temporal scale levels, as induced by setting the distribution parameter to c = 2.

In these respects, the results from this experiment agree with the theoretical results presented in the previous sections and demonstrate how the proposed family of time-causal wavelet representations can locally capture and reflect the temporal scales of locally dominant temporal structures in a time-causal analysis of temporal signals.

3.6.3 Time-causal wavelet representations of real-world signals

The effects of applying the time-causal wavelet analysis based on explicit temporal derivatives of orders 1 and 2 with their associated quasi quadrature measure according to Equation 152 to three real-world signals can be illustrated as follows:

Figure 11 shows such results for an electrocardiography signal from the PhysioNet service (http://www.physionet.org) from the MIT-BIH Arrhythmia database, item “NSR/100m(0).mat” from Plawiak (2017). As observed from the results, there are strong values in the wavelet transforms near the transients in the signal. At coarser temporal scales, the temporal delays become longer. The temporal delays are, however, significantly shorter when the distribution parameter is c = 2 compared to using c = 2.

Figure 12 shows such results from a signal with measurements of vibrations from the NASA Bearing Dataset (https://www.kaggle.com/datasets/vinayak123tyagi/bearing-dataset) (first test recorded 2003.11.25.23.39.56). As the results show, there are peaks in the energy of the wavelet representations near the temporal moments where there are also major peaks in signal.

Figure 13 shows corresponding results for an auditory percussion sound of a cymbal (from https://pixabay.com/sound-effects/search/cymbal/ item “Cymbal Crash”). As is evident, the time-causal wavelet transform largely responds over three ranges of temporal scale, with different durations of the responses over these ranges of temporal scale and corresponding to vibrations in the cymbal over different ranges of frequency. In this way, the proposed time-causal wavelet transform leads to an intuitive separation of the temporal variations in the signal over different temporal scales.

4 Summary and discussion

We have presented a theory for how the notions of temporal causality and temporal recursivity can be introduced in wavelet representations.

For this purpose, we have built the time-causal wavelet analysis on a special temporal smoothing kernel, denoted the time-causal limit kernel, and which constitutes the result of coupling an infinite set of truncated exponential kernels in cascade, with specially chosen time constants to obtain temporal scale covariance and self-similarity across scales. Specifically, Sections 2 and 3.2 proposed using temporal derivatives of this kernel as mother wavelets, with complementary normalization in a suitable L p-norm. We have analyzed the continuous scaling properties of this representation and have also from a series expansion (Equation 39) shown that the temporal derivatives of the time-causal limit kernel obey the admissibility condition (Equation 40) for a mother wavelet. We have also argued that the temporal derivatives of the time-causal limit kernel can by theoretical arguments be regarded as canonical time-causal analogs of the derivatives of the regular Gaussian kernel, which, for example, constitute the foundation for the Ricker/Mexican hat wavelet.

Since the time-causal limit kernel or its temporal derivatives do not have any compact closed-form explicit expressions, we have for this purpose specifically focused on what can be stated theoretically about these kernels regarding their recurrence relations across scales, their scaling properties across scales, and regarding bandpass wavelet representations defined from differences between time-causal wavelet representation at adjacent discrete temporal scales.

Computationally, the proposed bandpass wavelet representations specifically allow a much more straightforward reconstruction of the original signal from the wavelet transforms. Whereas a standard (usually non-causal) wavelet representation requires reconstruction from the general inversion formula reproduced in Equation 7, when using the bandpass wavelet representation according to Equation 97, the original signal can instead be recovered from a mere summation of the bandpass wavelet channels according to Equation 102. Notably, this opens up for real-time manipulation of local contents in the signal by restricting the reconstruction to subsets of temporal sub-bands over subsets of the temporal scale channels.

As shown in Equation 118, there is a direct relationship between the underlying computationally equivalent difference-of-time-causal-limit-kernels (DoT) kernel and the first-order temporal derivative of the time-causal limit kernel. This relationship is specifically qualitatively different from the corresponding relationship for non-causal wavelets based on derivatives of the Gaussian kernel, where the corresponding bandpass kernel, according to Equation 112, instead corresponds to the second-order derivative of the non-causal Gaussian kernel.

Section 3.3 turned to the problem of numerically approximating the time-causal limit kernel and its temporal derivatives for discrete signals. In this regard, first-order recursive filters constitute the natural analog of the first-order integrators that equivalently describe temporal smoothing with truncated exponential kernels, as constituting the main computational primitives in the continuous theory. Thereby, we have described a canonical way to compute discrete time-causal wavelet representations in Section 3.4 and also how this representation relates to bandpass representations, as can also be defined from continuous wavelets. Notably, the proposed implementation method is fully time-recursive and therefore lends itself directly to real-time processing on regular signal processing architectures. Specifically, as described in Algorithm 1, temporal smoothing can be implemented in terms of low number (often 4–8) first-order recursive filters coupled in cascade. The wavelet transforms are then obtained by applying compact temporal difference operators according to Equation 129 to the temporally smoothed data, complemented by complementary scale normalization. Thereby, real-time computations can be achieved on very low-capacity computational architectures.

These theoretical properties thus constitute the foundation for treating temporal structures in the input signal at different temporal scales similarly for different types of applications. Sections 3.5 and 3.6 specifically analyzed how well the ideal scaling properties of the continuous theory are transferred to a discrete implementation, with special attention to the ability of the proposed time-causal wavelet representations to reveal locally dominant structures at different temporal scales in the input signals.

In summary, these characterizations clearly demonstrate how the proposed family of time-causal wavelet representations can locally capture and reflect the temporal scales of locally dominant temporal structures based on a fully time-causal analysis of the input signals.

Based on these results, and the above mentioned principled theoretical foundations, we propose this notion of time-causal wavelet analysis as a valuable multi-purpose tool for signal processing tasks, where streams of signals are to be processed and analyzed in real time. Beyond real-time monitoring and time-series analysis and prediction, such applications may concern purposes of real-time feedback for, as an example, control-loop systems, and as time-causal and scale-covariant temporal basis functions in learning-based approaches. Specifically, we argue that the proposed time-causal wavelet representations could be suitable for signals that may contain local variations over a rich span of temporal scales and more generally for analyzing physical or biophysical temporal phenomena, where a fully time-causal analysis is required to be physically realistic, by not accessing data from the future, as otherwise is often done for pre-recorded data.

Concerning further applications of the theory, we have intentionally focused mainly on theoretical and generic properties of the proposed families of time-causal wavelet representations and not on developing more application-oriented experiments beyond the proof-of-concept studies in Sections 3.6 and 3.5.2. The motivation for this is for the results to be as timeless as possible and not dependent on specific properties of the experimental data, which may be different for different application domains and may furthermore warrant the inclusion of additional specialized computational mechanisms, as well as integration with other signal processing approaches to build full-fledged applications.

However, temporal derivatives of the time-causal limit kernel, as used as the mother wavelets in the proposed time-causal wavelet representations have been used for handling the temporal domain in models for spatio-temporal receptive fields for purposes of video analysis in classical computer vision (Jansson and Lindeberg, 2018; Lindeberg, 2018b; Lindeberg, 2018a). Those results clearly demonstrate that the numerical properties of the temporal derivatives of the time-causal limit kernel can be used for computing highly useful results on real-world temporal data.

Temporal derivatives of the time-causal limit kernel have also been used for modeling the temporal component in biological visual receptive fields (Lindeberg, 2021), leading to provable covariance properties under geometric image transformations (Lindeberg, 2025c). A precursor to the time-causal limit kernel, in terms of finite sets of truncated exponential kernels coupled in cascade with logarithmically distributed temporal scale levels, has furthermore been used for modeling the temporal component in auditory receptive fields (Lindeberg and Friberg, 2015). This study instead makes use of the temporal derivatives of the time-casual limit kernel to develop a time-causal wavelet theory and also describes structural relations to non-causal wavelet representations based on derivatives of the Gaussian kernel.

Closely related research based on using single truncated exponential kernels to model neural computations in biological perception, memory processes, and temporal learning has been reported by Jacques et al. (2022), Holt et al. (2022), Howard et al. (2023), Howard (2024), and Howard et al. (2025), with close relations to earlier work on temporal scale-space representations by Lindeberg and Fagerström (1996). Compared to the zero-order truncated exponential functions used as temporal basis functions for learning by Howard and colleagues and Holt and colleagues, the time-causal wavelets in this study obey simplifying cascade properties over temporal scales and are also DC-balanced. For biological support of the logarithmic distribution of the temporal scale levels used both here, according to Equation 12 and in these studies, see Cao et al. (2022).

The raw zero-order (non-differentiated) time-causal limit kernel has additionally been successfully used as a replacement of the non-causal Gaussian window function in time-causal time-frequency analysis. For the corresponding notion of a time-causal and time-recursive analog of Gabor filtering, corresponding to a time-causal analog of the Morlet wavelet (Morlet et al., 1982a; Morlet et al., 1982b), see Lindeberg (2025a). Compared to such time-causal time-frequency analysis, the time-causal wavelets developed here require significantly less computations and lead to shorter temporal delays, with a better ability to handle rapid transients. For more general notions of a non-causal wavelets based on Hermite transforms, which constitute extensions of the wavelets based on first- and second-order derivatives of the Gaussian kernel to higher orders of differentiation, see Markett (1993), Glaeske (2000), and Pandey and Phukan (2020).

In this study, we have solely focused on temporal derivatives of orders 1 and 2, leaving the topic to future researchers of characterizing the properties of time-causal wavelets based on temporal derivatives of the time-causal limit kernel for higher orders of temporal differentiation.

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Footnotes

Edited by: Qiyu Sun, University of Central Florida, United States

Reviewed by: Asst. Prof. Dr. Aws Zuheer Yonis, Ninevah University, Iraq

Pradeep Chaudhary, Mayo Clinic, United States

Footnotes

This can be observed from the fact that the admissibility criterion (Equation 1) implies that by necessity the Fourier transform (Equation 2) at the origin is given by χ ̂ ( 0 ) = 0, which is also the integral (Equation 3) of the mother wavelet.

This logarithmic spacing of the temporal scale levels is similar to the spacing generated by a uniform spacing with respect to the canonical logarithmic parameterization of the scale parameter into an effective scale parameter τ eff according to τ eff = A + B ⁡ log ⁡ τ, as can be derived from axiomatic arguments for a regular spatial scale-space representation (Lindeberg, 1993). Compared to a uniform spacing of the scale levels relative to the scale parameter measured in units of the variance τ of the temporal smoothing kernel, the logarithmic distribution according to (Equation 12) also leads to significantly shorter delays, as further described in Section 4 and Appendix 1 of Lindeberg (2016).

In Equation 13, (i) the factor e γ s 2 corresponds to the Laplace-Stieltjes transform of the Gaussian kernel, (ii) the factor 1 / ( 1 + a i s ) corresponds to the Laplace–Stieltjes transform of a truncated exponential kernel, while (iii) the factors e δ s and e a i s correspond to the Laplace–Stieltjes transforms of shift operations. The overall product form of this expression furthermore corresponds to convolutions between these primitive smoothing transformations.

The fact that the time-causal scale-space representation L ( t ; τ , c ) according to Equation 11 satisfies the recurrence relation in Equation 30 between adjacent levels of temporal scales can be formally verified as follows: We first rewrite the recurrence relation between the input signal f in ( t ) and the output signal f out ( t ) between adjacent levels of temporal scales τ k − 1 and τ k as ∂ t f out ( t ) = − 1 / μ k × ( f out ( t ) − f in ( t ) ). Taking the Laplace transform of this relation then yields s L ( f out ) ( s ) = − 1 / μ k × ( L ( f out ) ( s ) − L ( f in ) ( s ) ). We next assume that the Laplace transforms L ( f out ) ( s ) and L ( f in ) ( s ) of f out ( t ) and f in ( t ), respectively, are related by a transfer function H ( s ) according to L ( f out ) ( s ) = H ( s ) L ( f in ) ( s ). Inserting this expression into the Laplace transform of the recurrence relation gives s H ( s ) L ( f in ) ( s ) = − 1 / μ k × ( H ( s ) L ( f in ) ( s ) − L ( f in ) ( s ) ), from which we can solve for the transfer function as H ( s ) = 1 / ( 1 + μ k s ). From tables of the Laplace transforms of standard functions, the Laplace transform of the truncated exponential kernel e − α t for t > 0, 0 elsewhere, is 1 / ( s + α ). From this, we can conclude that the transfer function H ( s ) corresponds to the Laplace transform of the truncated exponential kernel h exp ( t ; μ k ) = 1 / μ k × e − t / μ k for t > 0, 0 elsewhere, which proves the result.

To derive these relations, we consider that (i) for non-negative convolution kernels, the variances are additive under the convolution operation, and (ii) the variance of truncated exponential kernel h exp ( t ; μ k ) with time constant μ k is V h exp ( ⋅ ; μ k ) = μ k 2. These properties lead to the relationships in Equation 35, given the logarithmic sampling of the temporal scale levels τ k = c 2 k according to Equation 12. Concerning Equation 34, on the other hand, we determine the first time constant μ 1 such that the variance of the first truncated exponential kernel V h exp ( ⋅ ; μ 1 ) = μ 1 2 is equal to the sum of the variances of all the truncated exponential kernels in the infinite convolution operation (Equation 18) that have been truncated away. Consequently, the variance of the resulting composed convolution kernel is equal to the same value τ as would be obtained with the original time-causal limit kernel Ψ ( t ; τ , c ) with temporal scale parameter τ according to Equation 27.

By corresponding arguments, a similar exactness of reconstruction applies also to the corresponding discrete bandpass representations to be described in Section 3.4, as obtained from differences between temporal scale channels obtained from discrete temporal smoothing based on first-order recursive filters in cascade. The algebraic form of the proof for that result is similar to the above result (Equation 104) for proving the result about exact reconstruction from the continuous bandpass representation.

In order to compute an actual discrete approximation of the convolution of a discrete signal at a certain level, sufficient initial scale levels need to be implemented to sufficiently approximate the truncation of the infinite convolution. For our purposes, we often use a minimum of four to eight temporal scale levels.

In an actual implementation, one would typically only make use of either bandpass representations or temporal derivatives, but we here describe both approaches in the same pseudocode to save space.

The model signals “Block” and “Riemann” have been generated with the function “pywt.data.demo_signal()” in the PyWavelets library (Lee et al., 2019) for n = 30 time samples. Such comparably short test signals are chosen in this case to represent the main temporal structures over temporal scales using as few as four bandpass channels, such that the visualizations of all the scale channels fit within a single column for each type of bandpass representation.