Single-shot phase-shifting interferometry: methods, challenges, and future perspectives

1 Introduction

Phase extraction from recorded fringe patterns is a fundamental process in optical interferometric techniques, enabling the quantitative evaluation of physical parameters such as surface deformation, refractive index variations, and wavefront aberrations (Abdelazeem et al., 2025; Barbosa and dos Santos, 2008; Dennis et al., 2001; Flores-Bravo et al., 2021; Kumar et al., 2015; 2016; 2022; Kumar and Matoba, 2021; Kumar and Shakher, 2015; Ma et al., 2024; Miao et al., 2025; Xie et al., 2024). These techniques are widely used in different interferometric configurations, such as digital holography and digital speckle pattern interferometry (DSPI) (Kumar et al., 2015; 2024; Kumar et al., 2023a; Kumar et al., 2023b; Sheridan et al., 2020), where accurate phase retrieval plays a key role in achieving precise measurements of deformation and displacement.

In practical interferometric systems, especially in DSPI and holography, fringe patterns are often affected by speckle noise due to the coherent nature of the illumination. To address this issue, a wide range of speckle denoising techniques have been developed, including spatial filtering methods, transform-based approaches (Kumar et al., 2020; Tounsi et al., 2018; Tounsi et al., 2023; Tounsi et al., 2019; Zada et al., 2019). Efficient denoising is essential for enhancing phase extraction accuracy and ensuring the reliability of subsequent measurements.

Over the past decades, numerous methods have been developed for the retrieval and analysis of optical phase information from interferometric fringe patterns. These techniques can generally be divided into two main categories. The first category includes phase-shifting interferometry (PSI), which requires the acquisition of multiple interferograms with controlled phase shifts between the reference and object beams (Wyant, 1984; Ayubi et al., 2017; Yamaguchi and Zhang, 1997; Deck, 2014; Bai et al., 2023; Servin et al., 2009). The second category encompasses single-frame or single-shot methods, in which the phase distribution is extracted from only one recorded fringe pattern (Rodríguez-Zurita et al., 2020; Tounsi et al., 2019).

PSI offers several advantages, including high measurement accuracy, full-field capability, and non-contact operation. Owing to these merits, PSI has been extensively applied in a wide range of fields, such as surface morphology characterization (Muravsky et al., 2011), thin-film thickness measurement (Kim et al., 2017), wavefront sensing (Zhang et al., 2015), temperature field mapping (Kumon et al., 2023), and beam quality evaluation (Jun-Pu et al., 2006). In PSI, multiple interferograms with known phase shifts are recorded by introducing controlled phase variations in the reference wave. Subsequently, quantitative wavefront information is retrieved using phase-shifting algorithms (Yamaguchi and Zhang, 1997). While the conventional multi-frame PSI has achieved remarkable precision and robustness in static measurements, it suffers from certain drawbacks that limit its applicability in dynamic or unstable environments. Its dependence on sequential phase acquisition makes it particularly sensitive to vibrations, air turbulence, and environmental disturbances, which can introduce significant phase errors and reduce measurement accuracy.

To address these limitations, researchers have proposed various strategies for retrieving phase information from a single interferogram. Broadly, these approaches can be classified into two groups. The first group introduces a high-frequency spatial carrier, producing a modulated fringe pattern that can be demodulated in the frequency domain using Fourier (Green et al., 1988; Takeda et al., 1982), Hilbert, (Yang et al., 2020; Yasir and Ivan, 2016), or wavelet transforms (Ghlaifan et al., 2016; Tounsi et al., 2017; Tounsi et al., 2019). The second group aims to extract the phase without introducing a carrier, relying instead on spatial phase demodulation, Riesz transform, or variational and gradient-based techniques. These single-shot approaches, often referred to as Single-Shot Phase-Shifting Interferometry (SSPSI), have gained increasing attention due to their capability to capture transient phenomena and perform real-time measurements. However, despite significant progress, SSPSI still faces challenges related to phase unwrapping, noise sensitivity, and trade-offs between spatial resolution and phase accuracy.

The purpose of this paper is to provide a comprehensive review of SSPSI, emphasizing the principles behind various methods, their practical implementations, and their current limitations. Furthermore, emerging trends and future research directions, including the integration of artificial intelligence, digital holography, and computational imaging, are discussed to highlight the pathway toward robust, real-time optical metrology systems.

2 Background on phase-shifting interferometry

PSI is one of the most widely used techniques in optical metrology for high-precision phase measurement and surface profiling. The concept of PSI has its roots in electrical engineering, where it was long used to determine the phase difference between two electrical signals through a technique known as synchronous detection. In the mid-1960s, researchers began to adapt these ideas to interferometry, with one of the earliest references dating back to 1966 (Carré, 1966). Although significant work was carried out during the 1960s and 1970s, relatively few results were published, as much of this research was conducted within defense programs or industrial proprietary projects (Bruning et al., 1974; Greivenkamp, 1984; Morgan, 1982; Wyant, 1984). The widespread adoption of PSI did not occur until the 1980s, driven by the availability of high-quality CCD sensors and compact, powerful computing systems. Today, phase-shifting techniques form the backbone of most high-precision interferometric metrology applications.

The basic principle of PSI lies in the controlled introduction of known phase shifts between the reference and object beams to produce a series of interferograms, each corresponding to a specific phase step. By recording and processing several such interferograms, the optical phase distribution of the measured object can be reconstructed with sub-nanometric accuracy under controlled conditions, including precise phase shifts, high signal-to-noise ratio, and a stable experimental setup.

Mathematically, the intensity at each pixel x , y in an interferogram can be expressed as:where a x , y represents the background illumination, b x , y the fringe contrast or modulation, ϕ x , y the phase to be measured, and δ k the known phase shift introduced during the k t h exposure, and n x , y represent and external noise. By acquiring a minimum of three or more frames with different δ k values, the unknown phase ϕ x , y can be determined using trigonometric relationships or least-squares algorithms. The phase, ϕ x , y at each point in the interferogram can easily be recovered. The required phase shift is introduced by making small modifications to the interferometer configuration between measurements. A widely adopted approach is to attach a piezoelectric actuator to one of the mirrors in the system. Actuators based on lead zirconate titanate (PZT) are commonly used for this purpose, as this material produces very small dimensional changes when driven by an electrical signal. Through the application of appropriate voltages, the mirror position can be displaced with nanometer-level precision. The specific optical component that is moved depends on the interferometer design. By regulating these displacements, predetermined values of δ k are incorporated into the optical path difference, enabling the acquisition of multiple phase-shifted interferograms.

I k x , y = a x , y + b x , y cos ϕ x , y + δ k + n x , y , (1)

Figure 1 presents the four interferograms acquired using the four-step phase-shifting technique. Each image corresponds to a controlled phase shift of 0 , π / 2 , π, and 3 π / 2, respectively. Together, these phase-shifted fringe patterns make it possible to analytically determine the phase value at every pixel.

The phase reconstruction in phase-shifting interferometry is performed by acquiring several interferograms with known phase shifts. In the four-step algorithm, the applied phase shifts are ϕ = 0 , π / 2 , π, and 3 π / 2, leading to the following recorded intensity distributions as expressed by Equation 2.

I x , y ; 0 = a x , y + b x , y cos ⁡ ϕ x , y , I x , y ; π / 2 = a x , y − b x , y sin ⁡ ϕ x , y , I x , y ; π = a x , y − b x , y cos ⁡ ϕ x , y , I x , y ; 3 π / 2 = a x , y + b x , y sin ⁡ ϕ x , y . (2)

By combining these four intensity measurements, the phase distribution can be analytically reconstructed using the arctangent relationship.which effectively removes the background intensity and normalizes the modulation term. To improve robustness with respect to noise and phase-shift errors, extended algorithms have been proposed, such as the five-step algorithm introduced by Schwider et al. (1983) and later refined by Hariharan et al. (1987), and further described in detail in (Malacara and Servín, 2018). In this case, five interferograms are acquired with phase shifts ϕ = 0 , π / 2 , π , 3 π / 2, and 2 π, and the phase is reconstructed the phase is reconstructed using Equation 4.

ϕ x , y = tan ⁡ − 1 I x , y ; 3 π / 2 − I x , y ; π / 2 I x , y ; 0 − I x , y ; π (3)

ϕ x , y = tan ⁡ − 1 2 I x , y ; π / 2 − I x , y ; 3 π / 2 2 I x , y ; π − I x , y ; 0 − I x , y ; 2 π . (4)

These reconstruction algorithms form the basis of most PSI techniques, offering a reliable and accurate estimation of the phase distribution for interferometric metrology applications.

PSI offers several advantages, including high accuracy, strong noise suppression, and insensitivity to nonuniform illumination. Consequently, it has been extensively applied in surface metrology, optical testing, micro-electromechanical systems (MEMS) characterization, and digital holography. However, its sequential nature constitutes a major limitation. Since multiple frames are required, any disturbance between exposures, such as mechanical vibration, air turbulence, or object motion, leads to phase errors and loss of coherence. This restricts conventional PSI to static measurements performed in well-controlled laboratory conditions.

To overcome these drawbacks, several extensions have been developed, including temporal phase unwrapping, spatial-carrier encoding, and hybrid digital reconstruction techniques. The rapid progress in digital image sensors, high-speed cameras, and computational algorithms has further encouraged the development of single-shot PSI methods capable of capturing all required phase information within a single exposure. These emerging techniques form the core focus of the next section.

3 Single-shot phase-shifting interferometry (SSPSI) methods

The main goal of SSPSI is to eliminate the temporal instability inherent in conventional multi-frame PSI by retrieving the optical phase from a single interferogram. This capability is essential for studying dynamic phenomena, vibration-sensitive systems, and real-time optical metrology applications. Several strategies have been developed to achieve single-shot phase retrieval, relying on either optical multiplexing (spatial, polarization, or spectral encoding) or computational reconstruction through mathematical transforms and artificial intelligence. This paper focuses on the second family by presenting a mathematically based method for phase map extraction from a single interferogram.

3.1 Spatial carrier and frequency-domain approaches

One of the earliest solutions to avoid temporal phase shifting involves the introduction of a spatial carrier frequency into the interferogram. This technique produces fringes modulated by a known spatial frequency, enabling phase retrieval in the frequency domain.

The spatial carrier is introduced by creating a small angle between the reference beam and the object beam in the interferometer. This angular tilt produces a regular fringe pattern across the interferogram, corresponding to a known carrier frequency.

3.1.1 Fourier transform method

The Fourier transform method, pioneered by Takeda and Mutoh (1983), is one of the most influential single-shot phase retrieval techniques in fringe analysis. It is based on a simple yet powerful idea: rather than analyzing the fringe pattern directly in the spatial domain, the information is examined in the frequency domain, where the different components of the signal can be more easily separated. In a fringe pattern, the useful phase information is mainly carried by the fundamental frequency component, whereas the background illumination and low-frequency disturbances are concentrated around the zero-frequency region.

For a two-dimensional fringe pattern, the recorded intensity can be modeled as a superposition of a slowly varying background and a modulated carrier. When a spatial carrier is introduced, typically by tilting the reference beam, the Fourier transform of the interferogram reveals a central spectral peak corresponding to the background term and two symmetrically shifted sidebands that contain the phase information. The Takeda method proceeds by isolating one of these sidebands through appropriate frequency-domain filtering while suppressing the remaining components.

Assuming a spatial carrier is introduced, the interferogram intensity can be expressed as in Equation 5:where, a x , y is the background illumination, b x , y is the fringe contrast, f x , f y is the spatial carrier frequency, and ϕ x , y is the phase to be retrieved.

I x , y = a x , y + b x , y cos ⁡ 2 π f x x + f y y + ϕ x , y (5)

Figure 2 illustrates the fringe patterns: (a) I (x,y), (b) the quadrature signal q (x,y), and (c) the modulated fringe pattern with high spatial frequency.

Applying a 2D Fourier transform to the interferogram as given in Equation 6:

F I x , y = A f , y + C f − f 0 , y + C * f + f 0 , y (6)

The spectrum consists of:

• A central lobe (zero frequency) related to the background term
• Two symmetrical sidebands containing the phase information

Applying the Fourier transform to the interferogram separates its spectral components. As shown in Figure 3, the spectrum is composed of three main components: a central lobe corresponding to the zero-order or background term A f , y, and two symmetric sidebands located at ±f 0, denoted as C f − f 0 , y and its complex conjugate C * f + f 0 , y. These sidebands contain the phase information modulated by the spatial carrier. By applying a band-pass filter, one of the sidebands is isolated and shifted to the origin of the frequency plane, removing the background and conjugate components and yielding the complex spectrum C f , y, from which the wrapped phase can be extracted.

An inverse Fourier transform of the filtered spectrum yields a complex exponential fringe pattern given by Equation 7:where F and F − 1 denote the forward and inverse Fourier transforms, respectively, and the tilde indicates the filtering operation in the frequency domain. The function a x , y represents the amplitude of the filtered fringe pattern, while ϕ x , y corresponds to the wrapped phase distribution.

I x , y = F − 1 F ∼ u , v = a x , y exp j ϕ x , y , (7)

The phase can then be directly retrieved from the real and imaginary parts of the complex signal using Equation 8:

ϕ x , y = tan ⁡ − 1 Im ⁡ I x , y Re ⁡ I x , y . (8)

In practice, the complete Takeda method involves the following steps: recording an interferogram with a spatial carrier, performing a Fourier transform to separate the spectral components, isolating one sideband by filtering, applying an inverse Fourier transform to obtain the complex field, and finally computing and unwrapping the phase.

This approach enables accurate and robust phase reconstruction from a single interferogram, making it particularly well suited for dynamic measurements. However, its performance strongly depends on a clear spectral separation between the carrier and the zero-order term, which requires an adequate carrier frequency and careful filter design.

Later improvements, such as windowed Fourier and wavelet transforms, enhanced spatial localization and adaptability to nonstationary or curved fringes (Ghlaifan et al., 2016; Kemao, 2007; Quan et al., 2010).

3.1.2 Windowed fourier transform method

Without loss of generality, the two-dimensional Windowed Fourier Transform (WFT) (Kemao, 2004) and its inverse transform can be written as shown by Equations 9, 10. where j = − 1 and g x , y denotes the window function, typically chosen as a Gaussian expressed in Equation 11:with σ x and σ y representing the standard deviations along the spatial directions.

S u , v , ξ , η = ∬ − ∞ + ∞ I x , y g x − u , y − v e − j ξ x + η y d x d y (9)

I x , y = 1 4 π 2 ∬ − ∞ + ∞ ∬ − ∞ + ∞ S u , v , ξ , η g x − u , y − v e j ξ x + η y d ξ d η d u d v (10)

g x , y = 1 π σ x σ y exp x 2 2 σ x 2 − y 2 2 σ y 2 (11)

Within the windowed Fourier filtering framework, the filtered fringe pattern is obtained from the local spectrum as expressed by Equation 12:

ϕ x , y = tan − 1 Im I ¯ x , y Re I ¯ x , y (12)

I ¯ x , y = 1 4 π 2 ∬ − ∞ + ∞ ∬ − ∞ + ∞ S u , v , ξ , η g x − u , y − v e j ξ x + η y d ξ d η d u d v ˉ (13)

The fundamental frequency component is extracted from the windowed spectrum according to the Equation 13:where thr is a predefined threshold.

S ¯ u , v , ξ , η = S u , v , ξ , η , if ∣ S u , v , ξ , η ∣ ≥ thr 0 , otherwise (14)

For the WFT ridge-based algorithm, the local frequency and phase are determined by Equations 15, 16:

ω x u , v , ω y u , v = arg ⁡ max ξ , η ∣ S u , v , ξ , η ∣ (15)

ϕ u , v = tan ⁡ − 1 Im ⁡ S f u , v , ω x u , v , ω y u , v Re ⁡ S f u , v , ω x u , v , ω y u , v + ω x u , v u + ω y u , v v (16)

It is important to mention that the classical Fourier method analyzes the image as a whole and assumes that fringe frequency remains constant across the field. In contrast, the WFT works locally, allowing it to adapt to changes in fringe frequency and orientation from one region of the image to another.

3.1.3 Wavelets transform based method

The wavelet transform (WT) is a powerful tool for analyzing non-stationary signals, and it has been widely adopted in many signal-processing applications, including phase retrieval from fringe patterns. Unlike the Fourier approach, where the analysis window has a fixed size, the wavelet transform uses a window that adapts to the frequency content of the signal. This multi-scale capability allows simultaneous localization in both space and frequency, making WT particularly suitable for signals whose characteristics vary across the image.

For the one-dimensional wavelet transform (1-D WT), the phase can be extracted directly from the complex wavelet coefficients as shown by Equation 1 where a is the scale (dilation) parameter, which determines the width of the wavelet and is related to the ridge point, while b represents the translation parameter indicating the position of analysis. The 1-D continuous wavelet transform of a signal f x using a Morlet wavelet is given by Equation 8:

ϕ b = tan − 1 Im W f a , b Re W f a , b (17)

W f a , b = 1 a ∫ − ∞ + ∞ f x ψ x − b a d x (18)

The Morlet mother wavelet used here is defined by Equation 9:where γ = 2 ⁡ ln ⁡ 2. This wavelet combines a Gaussian envelope with a complex sinusoidal carrier, providing good localization in both domains.

ψ x = 1 π 1 / 4 2 π γ 1 / 2 exp − 2 π x 2 2 γ 2 + j 2 π x (19)

For the two-dimensional case (2-D WT), which is required for fringe pattern analysis, the phase ϕ can be estimated from the complex wavelet coefficients as

ϕ b = tan − 1 Im W f b , θ , a Re W f b , θ , a − 2 π m exp − π 2 m (20)

In Equation 20, a controls the scale (window width), θ represents the orientation angle of the wavelet, and b denotes the spatial position corresponding to the ridge of the wavelet transform.

The 2-D wavelet transform of f x with a modified wavelet is expressed as in Equation 21:where the modified mother wavelet is given in Equation 22:

W f b , θ , a = 1 a 2 ∫ − ∞ + ∞ f x ψ * a − 1 r − θ x − b d 2 x (21)

ψ x = exp − m ∣ x ∣ 2 + j 2 π x (22)

Here, x = x , y represents the spatial coordinates of a pixel in the fringe pattern, r − θ is the conventional 2 × 2 rotation matrix used to align the wavelet with local fringe orientation, and m is a parameter that controls the width of the Gaussian window. Thanks to its multi-resolution capability and its ability to adapt locally to signal variations, the wavelet transform has become a popular tool for fringe analysis and phase extraction (Afifi et al., 2002; Barj et al., 2005; Barj et al., 2006; Ghlaifan et al., 2016).

To clearly illustrate the differences between the main frequency-based phase retrieval techniques, Table 1 provides a comparison of the FT), WFT, and WT. Although these methods are all based on spectral analysis, they differ in how they handle spatial variations in fringe patterns, their resistance to noise, and their computational demands. The classical Fourier method offers a global frequency representation and works well for regular fringe patterns, whereas WFT and WT introduce spatial localization and multi-resolution analysis, making them better suited for distorted or non-stationary fringes. The table summarizes their underlying principles, advantages, and limitations.

TABLE 1. Comparative analysis of FT, WFT and WT for phase retrieval.

Despite their simplicity and efficiency, spatial-carrier techniques face several challenges. The carrier frequency must be carefully chosen to avoid overlapping between the central and sideband spectra. Additionally, high-frequency modulation can reduce spatial resolution and amplify noise sensitivity, especially in low-contrast or speckle-contaminated interferograms. Nevertheless, these methods remain popular for their robustness and ease of implementation in digital holography and surface metrology.

In the same context, Afifi et al. proposed a novel digital technique for introducing a high-frequency spatial carrier directly into an interferogram in a numerical manner, (Barj et al., 2005), rather than through experimental adjustment. This approach enables the application of a wavelet-based phase extraction algorithm, allowing the phase distribution to be retrieved directly from the fringe pattern without requiring a phase unwrapping step.

After removing the background intensity term I 0 x , y from the interferogram using a high-pass filter, two orthogonal fringe patterns are obtained: one in cosine form and the other in sine form, expressed by Equations 23, 24:and

I x , y = b x , y cos ⁡ ϕ x , y (23)

q x , y = b x , y sin ⁡ ϕ x , y . (24)

These two patterns correspond respectively to the original fringe pattern and its π/2 phase-shifted version. To digitally introduce a high-frequency spatial carrier, the two patterns are numerically combined using a modulation ratio m and trigonometric terms cos m y and sin ⁡ m y as expressed in Equation 25:

I x , y = cos ⁡ m y I x , y − sin ⁡ m y q x , y = B x , y cos ⁡ m y + ϕ x , y . (25)

This operation effectively adds a phase-modulated carrier m y to the fringe pattern, resulting in a digitally modulated interferogram whose phase varies as ϕ x , y + m y.

The condition requires that the spatial carrier m exceed the maximum value of the phase derivative with respect to y, ensuring proper phase separation and avoiding spectral overlap during demodulation. The advantage of this approach is that the wavelet phase extraction algorithm can now be applied directly to this modulated signal. The added carrier frequency ensures that the phase can be retrieved without the need for a phase-unwrapping process, providing a robust and purely digital method for phase extraction from only two input fringe patterns (the original and its π/2 shifted version).

3.2 Toward Riesz transform: phase extraction from single fringe pattern and without spatial carrier

In recent years, the Riesz transform has emerged as a powerful tool for fringe pattern analysis, showing outstanding performance in phase extraction and feature characterization.

The Riesz transform is a multidimensional generalization of the Hilbert transform, designed to extend the concept of the analytic signal from 1D to 2D signals. Introduced by Marcel Riesz (1886–1969), it provides a powerful framework for computing the local amplitude, phase, and orientation of image structures. Unlike other extensions of the Hilbert transform, the Riesz transform is isotropic and steerable, meaning it naturally adapts to any orientation in the image. These properties make it particularly effective for image and fringe pattern analysis, offering accurate phase extraction and robust performance under varying conditions.

The Riesz transform is a multidimensional generalization of the Hilbert transform, formulated within the framework of Clifford analysis.

The Riesz operator of order n is defined by Equation 26:where x , y denotes the spatial position, n refers to the order of the Riesz operator, and i is the imaginary unit (i 2 = − 1).

r n x , y = n i n x + i y n 2 π x 2 + y 2 n + 2 / 2 (26)

For the first-order case (n = 1), the Riesz kernels along the x and y directions are expressed by Equation 2:

r x = x 2 π x 2 + y 2 3 / 2 , r y = y 2 π x 2 + y 2 3 / 2 (27)

Given an image intensity distribution I x , y, the first-order Riesz transform is defined by Equation 28:where * denotes the convolution operator.

R 1 I x , y = f 1 = r x * I x , y f 2 = r y * I x , y (28)

These two components can be combined into a complex representation as given in Equation 29:

I R = f 1 + i f 2 = r x * I + i r y * I (29)

This complex representation can be regarded as the 2D analytic signal of the image, providing access to its local amplitude, phase, and orientation. For higher-order Riesz transforms, the operator is applied recursively to capture higher directional derivatives of the image as expressed in Equations 30, 31:

• For n = 2 (second order):

R 2 I x , y = f 3 = r x * f 1 f 4 = r y * f 1 f 5 = r y * f 2 (30)

• For n = 3 (third order):

This recursive formulation allows the construction of higher-order Riesz representations, which capture increasingly complex directional information in an image.

Figure 4 illustrates the frequency responses of the first, second and third order Riesz operator.

As shown in Figure 4, the first-order Riesz transform produces two components, the second order yields three components, and the third order results in four components. In general, an nth order Riesz transforms (n + 1) components.

3.3 The monogenic signal

The monogenic signal (Felsberg and Sommer, 2001) extends the concept of the analytic signal to multidimensional domains. Based on the first-order Riesz transform, it provides a complex representation of an image by combining the original signal with its horizontal and vertical Riesz components. For an input image I x , y, the Riesz transform produces two components, f 1 and f 2, from which the monogenic signal, denoted I m, is expressed by Equation 31:where i and j denote complex units such that i 2 = j 2 = − 1. From a geometric stand point, I m can be viewed as a three-component position vector represented in a Cartesian reference frame (the red vector in the Figure 5).

I m = I x , y + i f 1 + j f 2 (31)

Zada et al. demonstrated that the phase map of the interferogram, denoted as ϕ x , y, is defined by Equation 23 (Zada et al., 2013):

ϕ x , y = sign ⁡ f i · ⁡ atan ⁡ 2 ⁡ f 1 2 + f 2 2 , I (32)

3.4 Digital four step phase shifting method

The Digital Four-Step Phase-Shifting (D4-PS) method is a numerical approach Proposed by (Tounsi, et al., 2019) for phase extraction from a single fringe pattern. Instead of acquiring multiple interferograms, the method digitally generates three additional phase-shifted fringe patterns, each shifted by π/2 radians (90°) with respect to the original one. This allows accurate and stable phase retrieval without requiring multiple measurements or experimental phase-shifting devices.

Mathematically, the four digitally generated fringe patterns I 1 , I 2 , I 3 , I 4 follow the model given in Equation 1. The wrapped phase map is then computed numerically using Equation 3.

This digital formulation provides a fully computational solution for phase extraction, enabling high-accuracy phase recovery from a single interferogram while maintaining robustness to intensity variations and noise.

The detailed procedure of this technique is shown in Figure 6.

The components of the first-, second-, and third-order Riesz transforms provide respectively two, three, and four components, in agreement with the general theory of the Riesz transform, which states that an operator of order n yields n + 1 components. Thus, by applying the three Riesz transform orders R 1, R 2, and R 3 to the fringe image I x , y, a total of nine components is obtained and noted as f i x , y with i = 1 t o 9.

By combining the components of each order, three phase-shifted fringe images are generated, shifted by π / 2, according to the Equation 33 (Tounsi et al., 2019b):where sgn is the sign function.

I x , y , π / 2 = a x , y − b x , y sin ⁡ ϕ x , y = sgn ⁡ f 1 , f 2 f 1 2 + f 2 2 I x , y , π = a x , y − b x , y cos ⁡ ϕ x , y = sgn ⁡ f 3 , f 4 , f 5 f 3 + f 5 I x , y , 3 π / 2 = a x , y + b x , y sin ⁡ ϕ x , y = sgn ⁡ f 6 , f 7 , f 8 , f 9 f 6 2 + f 9 2 (33)

This method makes it possible to generate three digitally phase-shifted fringe patterns from a single input fringe image using the Riesz transform. Its principle is illustrated in Figure 4.3 (Tounsi et al., 2019b).

The footnote variable i plays a crucial role in this technique, as it is directly related to the nature of the fringe pattern, whether the fringes are closed, open, or exhibit peak-like structures. For the first generated fringe pattern I x , y , π / 2, derived from the first-order Riesz components f 1 and f 2, the appropriate choice of the sign function depends on the topology of the fringes. Specifically, for closed fringe patterns, it is recommended to use sgn ⁡ f 2, whereas for open fringes, sgn ⁡ f 1 generally provides better results. The same consideration applies to the generated fringe pattern I x , y , 3 π / 2, where the sign function should involve f i with i = 6 , 7 , 8 , or 9, depending on the fringe configuration.

In general, when the cosine function is transformed into a sine function, the sign function must be carefully selected to eliminate sign ambiguity and ensure correct phase polarity. However, for the generated fringe pattern I x , y , π, this issue does not occur.

For further illustration, a representative fringe pattern is presented in Figure 7a. The first digitally generated π/2 phase-shifted fringe pattern is shown in Figure 7b, where the effect of sign ambiguity can be clearly observed near the center of the image. A similar behavior is noticed for the 3π/2 phase-shifted fringe pattern illustrated in Figure 7c. In contrast, the π phase-shifted fringe pattern, Figure 7d, is generated accurately and exhibits no sign-related artifacts or discontinuities.

To quantitatively evaluate the quality of the generated fringe patterns, the Structural Similarity Index Measure (SSIM) was used. Unlike simple pixel-based metrics such as Mean Squared Error (MSE) or Peak Signal-to-Noise Ratio (PSNR), SSIM provides a perceptually meaningful assessment by comparing the luminance, contrast, and structural similarity between two images. The SSIM between the generated fringe pattern I x , y and the reference pattern I r x , y is defined by Equation 34:where μ I and μ I r represent the mean intensities of the two images, σ I and σ I r their standard deviations, and σ I I r their cross-covariance. The constants C 1 and C 2 are small stabilizing factors to prevent division by zero. SSIM values range between 0 and 1, with values close to 1 indicating a high structural similarity between the generated and reference images. This metric thus provides a reliable quantitative measure of how well the proposed algorithm preserves the fringe structure and phase consistency during reconstruction.

S S I M I , I r = 2 μ I μ I r + C 1 2 σ I I r + C 2 μ I 2 + μ I r 2 + C 1 σ I 2 + σ I r 2 + C 2 (34)

The SSIM results shown in Table 2 demonstrate that all digitally generated fringe patterns exhibit high structural similarity with reference images. The π-shifted fringe pattern achieved the highest SSIM value (0.982), confirming that it is the most accurate among the generated patterns. Slightly lower SSIM values for the π/2 and 3π/2 shifted fringes (0.948 and 0.951, respectively) are attributed to local sign ambiguities around the central region of the image, which are inherent to the phase polarity of the Riesz transform. Overall, SSIM values above 0.94 indicate excellent reconstruction quality and validate the robustness of the proposed digital phase-shifting approach.

TABLE 2. Comparison between the reference fringe pattern and the Riesz-generated fringe patterns using SSIM metric.

To assess the performance of the proposed and reference algorithms, three single-shot phase extraction methods were applied to the same fringe pattern: the Phase Modulation Method (PMM), the Monogenic Signal Method (MSM), and the Digital Four-Step Phase-Shifting (D4-PS) technique. The corresponding wrapped phase maps are presented in Figure 8, where Figure 8a shows the reference phase obtained from the simulated interferogram, Figure 8b depicts the phase retrieved using the MSM, Figure 8c displays the phase extracted by the PMM, and Figure 8d illustrates the phase map reconstructed by the D4-PS method.

The Table 3 summarizes the quantitative comparison between the three single-shot phase extraction methods, PMM, MSM, and D4-PS, in terms of Phase Error (PE) and SSIM. These metrics were computed between each reconstructed phase map and the reference phase to provide an objective assessment of accuracy and structural consistency.

TABLE 3. Quantitative comparison of phase extraction performance using SSIM.

Bold values indicate the best results.

The results summarized in the table above demonstrate that all three single-shot phase extraction methods achieved high structural similarity with the reference phase map, as indicated by their SSIM values above 0.90. Among them, the D4-PS method obtained the highest SSIM value of 0.962, confirming its superior accuracy in reconstructing the phase structure and preserving fringe continuity. The MS method also performed well, with an SSIM of 0.945, showing good phase consistency and robustness against noise, though minor variations appear in regions of low contrast. In contrast, the PMM recorded a slightly lower SSIM of 0.921, indicating some structural deviations and localized phase errors. Overall, these results confirm that the D4-PS approach provides the most reliable phase reconstruction among the evaluated methods.

4 Conclusion

In this work, a comprehensive analysis of SSPSI has been presented, focusing on its theoretical principles, algorithmic developments, and performance assessment. Unlike conventional multi-frame approaches, SSPSI enables phase retrieval from a single interferogram, making it highly suitable for dynamic or vibration-sensitive applications.

Among the various techniques discussed, the D4-PS method based on the Riesz transform has demonstrated superior performance, achieving high structural similarity and accurate phase reconstruction when compared to the PMM and MSM methods. The experimental and numerical evaluations confirmed that D4-PS ensures better fringe continuity, higher SSIM values, and reduced phase error.

Beyond these results, research in phase retrieval is currently evolving rapidly, with a growing interest in combining traditional signal processing methods with modern computational techniques. In particular, artificial intelligence and deep learning are increasingly being explored to improve robustness to noise, enhance phase unwrapping, and enable faster, even real-time, processing. These hybrid approaches, which integrate classical transforms such as Fourier, wavelet, or Riesz with data-driven models, are opening new perspectives in the field.

At the same time, advances in digital holography, computational imaging, and optical metrology are broadening the range of applications for SSPSI, especially in situations requiring high precision or operating under dynamic conditions.

Looking ahead, future work will likely focus on improving performance under non-ideal conditions, reducing reliance on carrier fringes, and developing real-time or embedded solutions. In this context, the combination of physics-based models and machine learning techniques appears to be a promising direction for the next-generation of phase retrieval methods.

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Footnotes

Edited by: Rajpal Sirohi, Alabama Agricultural and Mechanical University, United States

Reviewed by: Miguel Mora-Gonzalez, University of Guadalajara, Mexico

Shanta Patil, Indian Institute of Technology Guwahati, India

Rajshekhar Gannavarpu, Indian Institute of Technology Kanpur, India