A Variational Model for Wrapped Phase Denoising

Abstract

This paper presents a variational model for the denoising of wrapped phase images. By enforcing the required Pythagorean trigonometric identity between the real and imaginary components of the signal, this model improves the signal-to-noise ratio of the restored signal. To preserve phase map discontinuities, the model is based on total variation. The existence and uniqueness of the model’s solution are demonstrated using standard techniques. In addition, the convergence of a rapid fixed-point method to determine the numerical solution is demonstrated. Experiments on both synthetic and actual patterns validate the model’s performance.

1. Introduction

Phase map digital images are used in several major technologies around the world. For instance, radar technology uses them for the 3D reconstruction of landscapes; in the manufacturing industry, they are used for quality assurance by estimating an object’s properties, such as shape and deformation, in a contact-free process. The raw signal data from these technologies are usually presented in an image format where each pixel of the image renders two-dimensional information. Because of this, each pixel can be seen as a complex element with real and imaginary parts encoding the magnitude A and phase information $\varphi$ of the data.

In this work, $U=U(x,y)$ represents the signal satisfying the expression

$$\begin{array}{c}U(x,y)=A(x,y)e^{i\varphi (x,y)}=A(x,y)cos\left(\varphi (x,y)\right)+iA(x,y)sin\left(\varphi (x,y)\right)\end{array}$$

where the pair $(U_{real}(x,y),U_{im}(x,y))$ represent the pixel values of the acquired complex signal, with $U_{real}=Acos\varphi$ and $U_{im}=Asin\varphi$, and the dependence on $(x,y)$ has been omitted for brevity.

It is standard practice to encode the information in only one variable known as the wrapped phase, here denoted by $\psi$, using the $arctan2$ function as follows:

$$\psi =arctan2(U_{real},U_{im}).$$

(1)

In optical metrology, the wrapped phase derived from a single or collection of interference fringe pattern images can be unwrapped and used to estimate the physical properties of the objects being measured; see References [1,2,3]. In interferometric synthetic aperture radar (InSAR), the wrapped phase is the absolute phase’s principal value, i.e., a modulo-$2\pi$ observation. In optical metrology, InSAR, or any other similar technology, the pair of signal data elements, i.e., $U_{real}$ and $U_{im}$, and consequently $\psi$, are not immune to noise, which can originate from a variety of sources [4,5,6] and is the primary cause of inaccuracies. Therefore, it is necessary to apply signal denoising to the data. This can be performed directly over $\psi$, but there is a risk of removing pertinent information from the phase map $\psi$. Consequently, it is common practice to denoise the real and imaginary components independently. Denoising one sample at a time is not recommended, even when employing cutting-edge denoise algorithms, because it breaks the Pythagorean identity between the real and imaginary signal components. This relationship must be maintained for the $arctan2$ function to recover the phase map with high precision. If the identity condition is not met, the $\psi$ values derived from this formula cannot be relied upon.

The following is an overview of some phase denoising techniques that have been utilized in the past. In References [7,8], the authors proposed simultaneous phase unwrapping and noise filtering algorithms with a probabilistic approach based on random Markov fields and Kalman filters; their proposals work well for continuous phases, but discontinuous phase maps are not mentioned. Reference [9] proposes a phase filtering method based on a variational decomposition model with total variation regularization; they do not demonstrate any results with discontinuous phase maps either. Reference [10] presents a global filtering process that makes use of the local frequencies of the wrapped phase map and tunes each pixel to its instantaneous frequency; this linear filtering method only works for continuous phases. References [11,12] propose different methods for discontinuous maps; however, they assume the phase to be locally polynomial and the overall process requires quality windows, which adds complexity to the scheme and uncertainty if the size of the window is not chosen appropriately. Modern methods for noise filtering based on deep learning algorithms [13,14,15,16] only perform well for continuous phase maps.

More recently, two variational models with second-order regularizers have been proposed in Reference [17] to enhance the preservation of features such as edges, contrast, and smoothness. Reference [18] proposes a scalable subspace clustering method that combines the learning of a concise dictionary with a robust subspace representation in a unified model, thereby reducing the size of optimization problems. Reference [19] suggests a scalable approximate Bayesian method for image restoration based on TV priors, expectation propagation, and expectation-maximization. In order to preserve image details and minimize vanishing gradients, a novel deep learning method of despeckling SAR images with a dilated residual network (SAR-DRN) and skip connections is proposed in Reference [20]. Reference [21] proposes a hybrid denoising method to eliminate SAR speckle distortions using a convolutional neural network (CNN) and consistent cycle spinning (CCS) in the nonsubsample shearlet transform (NSST) domain. The method achieves superior speckle removal performance and preserves more granular data than existing algorithmic techniques. In this paper, the variational approach is utilized, which has proven effective in many other image processing techniques by preserving discontinuities and noise filtering (see, for example, References [22,23,24] and references therein). This paper proposes and analyzes a variational model for the denoising of wrapped phase maps that preserves not only discontinuities but also the Pythagorean relationship between the real and imaginary portions of the phase map.

The following are this paper’s contributions.

• A variational model for the removal of noise from wrapped phase images while preserving the Pythagorean identity of the signal elements. Total variation regularization is incorporated into the model to preserve signal discontinuities when necessary.
• A mathematical analysis of the model, demonstrating the existence and uniqueness of the model’s solution.
• A fixed-point algorithm for the rapid solution of the model.
• The mathematical proof of the fixed-point algorithm’s convergence.

This work is structured as follows: In Section 2, the proposed variational model is described; Section 3 is devoted to the analysis of the model and the derivation of the Euler–Lagrange equations; and Section 4 explains numerical techniques to solve the model. In particular, a fixed-point method, its proof of convergence, and its computational implementation are presented; in Section 5, experiments illustrating the performance of the model and a comparison with state-of-the-art methods for denoising are presented. In Section 6, conclusions are presented.

2. A Variational Model to Denoise Wrapped Phase Maps

In this section, we present our model for the denoising of wrapped phase images. Consequently, let us start by defining the relevant variables:

• $\widehat{\psi }$, ${\widehat{U}}_{real}=Acos\widehat{\psi }$, and ${\widehat{U}}_{im}=Asin\widehat{\psi }$ are the noisy wrapped phase image and its noisy components;
• $U_{real}$, $U_{im}$ and $\psi$ are the phase components and wrapped phase to be estimated;
• $U_{real}^{\ast }$, $U_{im}^{\ast }$, and ${\psi }^{\ast }$ are the denoised solutions from the proposed model.

The well-known mathematical model of a wrapped phase map $\widehat{\psi }$ contaminated with additive Gaussian noise $\eta$ is

$$\widehat{\psi }=\psi +\eta ,$$

(2)

Possibly the most common technique for the filtering of wrapped phase maps is the one proposed in Reference [25], in which ${\widehat{U}}_{real}$ and ${\widehat{U}}_{im}$ are filtered independently and $\psi$ is then calculated as $\psi =arctan(U_{im}/U_{real})$ with magnitude $A={[{\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2]}^{1/2}$. Due to the periodicity of the sine and cosine functions, it is possible that modulo-$2\pi$ discontinuities will be lost during this process.

Since there is no difficulty in computing the magnitude A for an arbitrary phase map and normalizing it, we consider the case where $A=1$ to simplify ${\widehat{U}}_{real}=cos\widehat{\psi }$ and ${\widehat{U}}_{im}=sin\widehat{\psi }$ without sacrificing generality. Then, we implement the subsequent minimization

$$(U_{real}^{\ast },U_{im}^{\ast })=arg\underset{(U_{real},U_{im})}{min}F(U_{real},U_{im}),$$

where

$$\begin{array}{cc}\hfill F& \equiv \frac{{\lambda }_1}{2}{\int }_{\Omega }{(U_{real}-{\widehat{U}}_{real})}^{2}d\Omega +\frac{{\lambda }_2}{2}{\int }_{\Omega }{(U_{im}-{\widehat{U}}_{im})}^{2}d\Omega \hfill \\ \hfill & +\frac{{\lambda }_3}{2}{\int }_{\Omega }{({\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1)}^{2}d\Omega +{\int }_{\Omega }|\nabla U_{real}|d\Omega +{\int }_{\Omega }|\nabla U_{im}|d\Omega \hfill \end{array}$$

(3)

where $\Omega \subset {\mathbb{R}}^2$ is the domain of integration and ${\lambda }_1,{\lambda }_2$, and ${\lambda }_3$ are the regularization parameters. Following Ströbel’s technique for the filtering of wrapped phase maps, the first two terms of F constitute the fitting function. The third term is a major contribution of the proposed model and forces the Pythagorean identity ${\left(U_{im}\right)}^2+{\left(U_{real}\right)}^2=1$ to be satisfied, thereby enhancing the estimation of $\psi$ using the arctan function. The final two variables serve to regularize the solutions $U_{im},U_{real}$. It is well known that the total variation regularizer permits discontinuities in the solution while simultaneously removing [26].

3. Model Analysis

In this section, standard techniques are used to demonstrate that the model’s solution exists and is unique. Let us assume that $\Omega$ is a convex region and define its bounded variation norm as $\left|\right|·{\left|\right|}_{BV}$.

• Existence

Let us start by showing that F is BV-coercive. To this end, note that

$$\begin{array}{c}{\int }_{\Omega }{(U_{im}-{\widehat{U}}_{im})}^{2}d\Omega \ge 0,\end{array}$$

(4)

$$\begin{array}{c}{\int }_{\Omega }{(U_{real}-{\widehat{U}}_{real})}^{2}d\Omega \ge 0,\end{array}$$

(5)

$$\begin{array}{c}{\int }_{\Omega }{({\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1)}^{2}d\Omega \ge 0.\end{array}$$

(6)

Then, it is immediately found that

$$\begin{array}{c}\underset{\left|\right|(U_{real},U_{im}){\left|\right|}_{BV}\to \infty }{lim}F(U_{real},U_{im})=\infty \end{array}$$

(7)

meaning that F is $BV$-coercive.

The weakly lower semi-continuity of (4)–(6) is a consequence of the weakly lower semi-continuity of the norms on Banach spaces. The same property holds true for $\left|\right|U_{real}{\left|\right|}_{BV}$ and $\left|\right|U_{im}{\left|\right|}_{BV}$, as demonstrated by Reference [27]. In conclusion, with F being weakly lower semi-continuous and $BV$-coercive, the existence of the solution in the $BV$ space is guaranteed according to Reference [27].

• Uniqueness

The uniqueness of the solution is due to the convexity of the model. The convexity of (4) and (5), $\left|\right|U_{real}{\left|\right|}_{BV}$, and $\left|\right|U_{im}{\left|\right|}_{BV}$ is demonstrated in Reference [27]. Therefore, it only remains to demonstrate the convexity of (6). Section 4.2 demonstrates that (6) can be viewed as $\left|\right|Ku-{d\left|\right|}_2^2$, where $u=(U_{real};U_{im})$ and K is a bounded operator. It is readily apparent that therefore (6) is convex [28]. Note also that F is bounded below by zero.

Euler–Lagrange Equations

This section shows the derivation of the model’s Euler–Lagrange equations. Define $\phi \in C(\Omega )$ with compact support and $\epsilon$ as a given scalar. The first variations are then set to zero, i.e.,

$$\begin{array}{c}\delta F_{real}\equiv \frac{d}{d\epsilon }F(U_{real}+\epsilon \phi ,U_{im}){|}_{\epsilon =0}=0,\mathrm{and}\\ \delta F_{im}\equiv \frac{d}{d\epsilon }F(U_{real},U_{im}+\epsilon \phi ){|}_{\epsilon =0}=0.\end{array}$$

Here, only the computation for $U_{real}$ is provided because $U_{im}$ is nearly identical.

$$\begin{array}{cc}\hfill \delta F_{real}& =\frac{d}{d\epsilon }\left\{\frac{{\lambda }_1}{2}{\int }_{\Omega }{(U_{real}+\epsilon \phi -{\widehat{U}}_{real})}^{2}d\Omega +\frac{{\lambda }_2}{2}{\int }_{\Omega }{(U_{im}-{\widehat{U}}_{im})}^{2}d\Omega \right.\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & +\frac{{\lambda }_3}{2}{\int }_{\Omega }{({(U_{real}+\epsilon \phi )}^2+{\left(U_{im}\right)}^2-1)}^{2}d\Omega \hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \left.+{\int }_{\Omega }|\nabla (U_{real}+\epsilon \phi )|d\Omega +{\int }_{\Omega }|\nabla U_{im}|d\Omega \right\}{|}_{\epsilon =0}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & ={\lambda }_1{\int }_{\Omega }(U_{real}-{\widehat{U}}_{real})\phi d\Omega +2{\lambda }_3{\int }_{\Omega }\left({\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1\right)U_{real}\phi d\Omega \hfill \end{array}$$

(11)

$$\begin{array}{cc}& +{\int }_{\Omega }\frac{\nabla U_{real}}{|\nabla U_{real}|}·\nabla \phi d\Omega .\hfill \end{array}$$

(12)

Now, by applying Green’s Theorem in the last term,

$${\int }_{\Omega }\frac{\nabla U_{real}}{|\nabla U_{real}|}·\nabla \phi d\Omega =-{\int }_{\Omega }\nabla ·\left(\frac{\nabla U_{real}}{|\nabla U_{real}|}\right)\phi d\Omega +{\oint }_{\partial \Omega }\left(\frac{\nabla U_{real}}{|\nabla U_{real}|}·\mathbf{n}\right)\phi d(\partial \Omega )$$

(13)

the following nonlinear partial differential equation (PDE) is obtained

$$-\nabla ·\left(\frac{\nabla U_{real}}{|\nabla U_{real}|}\right)+{\lambda }_1(U_{real}-{\widehat{U}}_{real})+2{\lambda }_3\left({\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1\right)U_{real}=0\mathrm{in}\Omega$$

(14)

with Neumann’s boundary condition

$$\begin{array}{c}\frac{\nabla U_{real}}{|\nabla U_{real}|}·\mathbf{n}=0\mathrm{on}\partial \Omega \end{array}$$

(15)

where $\mathbf{n}$ denotes the outward unit normal vector on $\partial \Omega$.

The corresponding PDE for $U_{im}$ is

$$-\nabla ·\left(\frac{\nabla U_{im}}{|\nabla U_{im}|}\right)+{\lambda }_2(U_{im}-{\widehat{U}}_{im})+2{\lambda }_3\left({\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1\right)U_{im}=0\mathrm{in}\Omega$$

(16)

with Neumann’s boundary condition

$$\begin{array}{c}\frac{\nabla U_{im}}{|\nabla U_{im}|}·\mathbf{n}=0\mathrm{on}\partial \Omega .\end{array}$$

(17)

4. Numerical Solution

The nonlinear PDEs in (14) and (16) are anisotropic second-order nonlinear equations; consequently, it is anticipated that conventional numerical methods will struggle to quickly converge to the solution. This is the case with the popular gradient descent method, which, due to stability constraints, may converge to the solution very slowly. In both PDEs, the discrete diffusion coefficient $D=\frac{1}{|\nabla U_{real}|}$ for (14) must be regularized to prevent division by zero by adding a very small scalar $\beta >0$, i.e., $D_{\beta }=\frac{1}{|\nabla U_{real}+\beta |}$. It was demonstrated in Reference [29] that as $\beta \to 0$, the time-step parameter in the gradient descent algorithm must also approach zero in order to ensure the stability of the numerical method. Clearly, the smaller the value of $\beta$, the greater the number of iterations required.

The dual formulations [30,31], alternating minimization methods [32], Newton-based method [33], Bregman iterative algorithms [34], and fixed-point methods [35] are among the other numerical methods that have been developed for similar PDEs but have yet to be tested in the proposed wrapped phase denoising model. This paper demonstrates an effective fixed-point method with rapid convergence, leaving the remainder for future research.

4.1. Fixed-Point Method

This section introduces the fixed-point algorithm for the solution of PDEs (14) and (16). The following fixed-point scheme should initially work for $U_{real}$ with $U_{im}$ fixed.

$$\left(-\nabla ·\left(\frac{\nabla }{|\nabla U_{real}^k|}\right)+{\lambda }_1+2{\lambda }_3\left({\left(U_{real}^k\right)}^2+{\left(U_{im}^k\right)}^2-1\right)\right)U_{real}^{k+1}={\lambda }_1{\widehat{U}}_{real}\mathrm{for}k=1,2,3,\dots$$

(18)

Note, however, that the term ${\left(U_{real}\right)}^2+{\left(U_{im}\right)}^2-1$ may be negative at some iterations, and when $2{\lambda }_3\ge {\lambda }_1$, the resulting matrix is no longer strictly diagonally dominant. Therefore, the convergence of algorithms such as Gauss–Seidel and Jacobi cannot be guaranteed to converge.

To resolve this issue, one element is moved to the right, resulting in

$$\left(-\nabla ·\left(\frac{\nabla }{|\nabla U_{real}^k|}\right)+{\lambda }_1+2{\lambda }_3\left({\left(U_{real}^k\right)}^2+{\left(U_{im}^k\right)}^2\right)\right)U_{real}^{k+1}={\lambda }_1{\widehat{U}}_{real}+2{\lambda }_3{U}_{real}^k\mathrm{in}\Omega .$$

(19)

Now, ${\lambda }_1+2{\lambda }_3\left({\left(U_{real}^k\right)}^2+{\left(U_{im}^k\right)}^2\right)$, which is always positive, is added to the main diagonal, ensuring strict diagonal dominance.

Following the same steps, a fixed-point scheme for $U_{im}$ is obtained with $U_{real}$ fixed.

$$\left(-\nabla ·\left(\frac{\nabla }{|\nabla U_{im}^k|}\right)+{\lambda }_2+2{\lambda }_3\left({\left(U_{real}^k\right)}^2+{\left(U_{im}^k\right)}^2\right)\right)U_{im}^{k+1}={\lambda }_2{\widehat{U}}_{im}+2{\lambda }_3{U}_{im}^k\mathrm{in}\Omega .$$

(20)

In shortened form, (19) and (20) can be written as

$$\begin{array}{c}H\left(U_{real}^k\right)U_{real}^{k+1}=f\left(U_{real}^k\right),\end{array}$$

(21)

$$\begin{array}{c}H\left(U_{im}^k\right)U_{im}^{k+1}=f\left(U_{im}^k\right).\end{array}$$

(22)

where f is the right-hand side and H the differential operator multiplying $U_{real}$ and $U_{im}$ of (19) and (20), respectively. This system can be solved initially for $U_{real}$ with $U_{im}$ fixed, and then solved in the opposite direction. However, an improvement is feasible by creating the system

$$\begin{array}{c}L(u^k)u^{k+1}=c^k\end{array}$$

(23)

where

$$\begin{array}{c}L(u^k)=\left(\begin{array}{cc}H\left(U_{real}^k\right)& 0\\ 0& H\left(U_{im}^k\right)\end{array}\right),\\ u^{k+1}=\left(\begin{array}{c}{U}_{real}^{k+1}\\ U_{im}^{k+1}\end{array}\right),c^k=\left(\begin{array}{c}f\left(U_{real}^k\right)\\ f\left(U_{im}^k\right)\end{array}\right).\end{array}$$

The matrix $L\left(u\right)$ inherits the same satisfactory properties of H, i.e., L is symmetric, positive definite, and strictly diagonal dominant.

4.2. Fixed-Point Algorithm Convergence

To prove the convergence of the proposed fixed-point scheme (23), the Weiszfeld method will be used; see References [36,37,38]. Here, a lexicographic ordering on $U_{real}$ and $U_{im}$ is considered, both in ${\mathbb{R}}^N$, to define $u\in {\mathbb{R}}^{2N}$ as the column vector $u=(U_{real},U_{im})$, and $K\in {\mathbb{R}}^{2N\times 2N}$ where $N=m\times n$. Further, we define the set $\{d_1,d_2,d_3\}$ with elements in ${\mathbb{R}}^{2N}$, and $K^{\ast }$ is the adjoint of the bounded operator K as follows:

$$\begin{array}{c}K=\left(\begin{array}{cc}{U}_{real}^k& U_{im}^k\\ -U_{im}^k& U_{real}^k\end{array}\right),d_1=\left(\begin{array}{c}{\widehat{U}}_{real}\\ 0\end{array}\right),d_2=\left(\begin{array}{c}0\\ {\widehat{U}}_{im}\end{array}\right),d_3=\left(\begin{array}{c}1\\ 0\end{array}\right),\end{array}$$

(24)

with 1 the vector with all entries equal to one and 0 the null vector.

Then, the model in (3) can be written as follows:

$$\begin{array}{c}arg\underset{u}{min}F\left(u\right)\end{array}$$

(25)

where

$$\begin{array}{c}F\left(u\right)=\sum _l|A_l^T{u|}_{\beta }+\frac{{\lambda }_1}{2}\left|\right|u-d_1{\left|\right|}_2^2+\frac{{\lambda }_2}{2}\left|\right|u-d_2{\left|\right|}_2^2+\frac{{\lambda }_3}{2}\left|\right|Ku-d_3{\left|\right|}_2^2,\end{array}$$

(26)

and $A_l^T\in {\mathbb{R}}^{2\times 2N}$ is an array that contains the gradient of u that is approximated using finite differences. Here, l defined as

$$\begin{array}{c}l=\left\{\begin{array}{cc}l=(i-1)n+j\hfill & \mathrm{if} (i,j) \mathrm{belongs} \mathrm{to} U_{real}\hfill \\ l=N+(i-1)n+j\hfill & \mathrm{if} (i,j) \mathrm{belongs} \mathrm{to} U_{im}\hfill \end{array}\right.,\end{array}$$

(27)

stands for the pixel coordinates where the gradient is computed and

$$|A_l^T{u|}_{\beta }=\sqrt{{\left({\partial }_x{u}_l\right)}^2+{\left({\partial }_y{u}_l\right)}^2+\beta }$$

where $\beta >0$ is a regularization parameter. Note that $A_l^{T}u$ can be split as $A_l^{T}u=(A_{1l}^T,0)u+(0,A_{2l}^T)u$. For instance, the operator $(A_{1l}^T,0)$ computes the gradient of the first half of u, i.e., $U_{real}$, and pads the vector with zeros. By defining $B_{1l}=(A_{1l}^T,0)$ and $B_{2l}=(0,A_{2l}^T)$, the two regularization terms in (3) can be compactly expressed as ${\sum }_l{\left|A_l^{T}u\right|}_{\beta }$.

Further, the set $\{d_1,d_2,d_3\}$ has elements in ${\mathbb{R}}^{2N}$, and K is a bounded operator with $K^{\ast }$ its adjoint.

$$\begin{array}{c}K=\left(\begin{array}{cc}{U}_{real}^k& U_{im}^k\\ -U_{im}^k& U_{real}^k\end{array}\right),d_1=\left(\begin{array}{c}{\widehat{U}}_{real}\\ 0\end{array}\right),d_2=\left(\begin{array}{c}0\\ {\widehat{U}}_{im}\end{array}\right),d_3=\left(\begin{array}{c}1\\ 0\end{array}\right),\end{array}$$

(28)

with 1 the vector with all entries equal to one and 0 the null vector.

Note that the fixed-point algorithm in (23) then can be written as

$$\sum _l{A}_l\left(\frac{A_l^T{u}^{k+1}}{|A_l^T{u}^k{|}_{\beta }}\right)+{\lambda }_1(u^{k+1}-d_1)+{\lambda }_2(u^{k+1}-d_2)+2{\lambda }_3{K}^{\ast }(K{u}^{k+1}-d_3)=0.$$

(29)

Generalized Weiszfeld Method

The generalized Weiszfeld method consists of choosing a uniformly strictly convex quadratic function $G(w,u)$ that approximates $F\left(u\right)$ under the assumptions of Hypothesis 1 below.

Hypothesis 1.

1. 

$G(w,u)=F\left(u\right)+(w-u,F^{\prime }\left(u\right))+\frac{1}{2}(w-u,C\left(u\right)(w-u))$.

2. 

$C\left(u\right)$ is continuous.

3. 

${\lambda }_{min}\left(C\left(u\right)\right)\ge \mu >0\forall u$.

4. 

$F\left(w\right)\le G(w,u)\forall w$.

Weiszfeld’s generalized method solves (25) via the following iterative scheme:

$$u^{k+1}=\underset{w}{min}G(w,u^k)$$

(30)

where $G(w,u^k)$ is a quadratic approximation of $F\left(u\right)$ at iteration k. Under Hypothesis 1 and for fixed u, $G(w,u)$ is coercive, bounded, and strictly convex. Therefore, the minimum exists and can be computed by

$$0=G_w^{\prime }(u^{k+1},u^k)=F^{\prime }\left(u^k\right)+C\left(u^k\right)(u^{k+1}-u^k)$$

(31)

with global and linear convergence [36,37].

In what follows, it will be shown, with the help of the following Lemma, that (29) and (31) are equivalent.

Lemma 1.

If $C\left(u\right)$ is given by

$$C\left(u\right)=B_{1}diag\left(\frac{1}{|B_{1l}^T{|}_{\beta }}{I}_2\right)B_1^T+B_{2}diag\left(\frac{1}{|B_{2l}^T{|}_{\beta }}{I}_2\right)B_2^T+{\lambda }_1+{\lambda }_2+2{\lambda }_3{K}^{\ast }K$$

(32)

for $B_1=(B_{11},\dots ,B_{1m})$, $I_2$ the $2\times 2$ identity matrix and $B_2$ are defined in a similar way. Then,

$$G(w,u)=F\left(u\right)+(w-u,F^{\prime }\left(u\right))+\frac{1}{2}(w-u,C\left(u\right)(w-u))$$

(33)

defines a generalized Weiszfeld method and (29) is equivalent to (31).

Proof. 

The proof is similar to the one given for Lemma 6.1 in Reference [37]. First, it is shown that Property 1 in Hypothesis 1 is satisfied, i.e., that (29) and (31) are equivalent.

$$\begin{array}{cc}\hfill F^{\prime }(u^k)& +C(u^k)(u^{k+1}-u^k)={\lambda }_1(u^k-d_1)+{\lambda }_2(u^k-d_2)+2{\lambda }_3{K}^{\ast }(K{u}^k-d_3)\hfill \\ \hfill & {\displaystyle +\sum _l{B}_{1l}\left(\frac{B_{1l}^T{u}^k}{|B_{1l}^T{u}^k{|}_{\beta }}\right)+\sum _l{B}_{2l}\left(\frac{B_{2l}^T{u}^k}{|B_{2l}^T{u}^k{|}_{\beta }}\right)}\hfill \\ \hfill & {\displaystyle +\sum _l{B}_{1l}\left(\frac{B_{1l}^T{u}^{k+1}}{|B_{1l}^T{u}^k{|}_{\beta }}\right)+\sum _l{B}_{2l}\left(\frac{B_{2l}^T{u}^{k+1}}{|B_{2l}^T{u}^k{|}_{\beta }}\right)}\hfill \\ \hfill & +{\lambda }_1{u}^{k+1}+{\lambda }_2{u}^{k+1}+2{\lambda }_3{K}^{\ast }K{u}^{k+1}\hfill \\ \hfill & {\displaystyle -\sum _l{B}_{1l}\left(\frac{B_{1l}^T{u}^k}{|B_{1l}^T{u}^k{|}_{\beta }}\right)-\sum _l{B}_{2l}\left(\frac{B_{2l}^T{u}^k}{|B_{2l}^T{u}^k{|}_{\beta }}\right)}\hfill \\ \hfill & -{\lambda }_1{u}^k-{\lambda }_2{u}^k-2{\lambda }_3{K}^{\ast }K{u}^k\hfill \\ \hfill & ={\lambda }_1(u^{k+1}-d_1)+{\lambda }_2(u^{k+1}-d_2)+2{\lambda }_3{K}^{\ast }(K{u}^{k+1}-d_3)\hfill \\ \hfill & {\displaystyle +\sum _l{B}_{1l}\left(\frac{B_{1l}^T{u}^{k+1}}{|B_{1l}^T{u}^k{|}_{\beta }}\right)+\sum _l{B}_{2l}\left(\frac{B_{2l}^T{u}^{k+1}}{|B_{2l}^T{u}^k{|}_{\beta }}\right)}\hfill \\ \hfill & ={\lambda }_1(u^{k+1}-d_1)+{\lambda }_2(u^{k+1}-d_2)+2{\lambda }_3{K}^{\ast }(K{u}^{k+1}-d_3)\hfill \\ \hfill & +\sum _l{A}_l\left(\frac{A_l^T{u}^{k+1}}{|A_l^T{u}^k{|}_{\beta }}\right).\hfill \end{array}$$

(34)

Property 2 holds because the continuity of $C\left(u\right)$ is guaranteed by $|B_{1l}^T{u}^k{|}_{\beta }>0$ and $|B_{2l}^T{u}^k{|}_{\beta }>0$.

Property 3 is fulfilled by noticing that K is a bounded nonzero operator.

Property 4 is straightforwardly obtained from the procedure in the proof of Lemma 6.1 in Reference [37] and will be omitted here. □

4.3. Numerical Realization

Here, the numerical realization of the fixed-point method for the Euler–Lagrange equations is presented. To this end, we define the discrete domain as ${\Omega }_h=[0,m]\times [0,n]$ and, without loss of generalization, set the spatial step size $h=h_x=h_y=1$ over the grid defined as ${\Gamma }_h=\{(x,y):x=i{h}_x,y=j{h}_y,i,j\in \mathbb{Z}\}$.

To approximate the partial derivatives of a given variable w, we use the finite differences method. The norm of the gradient of w at points $\left(i+1,j\right),\left(i-1,j\right),\left(i,j+1\right)$, and $\left(i,j-1\right)$ is computed as follows:

$$\begin{array}{cc}\hfill {|\nabla w|}_{i+1,j}& =\sqrt{\frac{{(w_{i+1,j}-w_{i,j})}^2}{h_x^2}+\frac{{(w_{i,j+1}-w_{i,j})}^2}{h_y^2}},\hfill \end{array}$$

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$$\begin{array}{cc}\hfill {|\nabla w|}_{i-1,j}& =\sqrt{\frac{{(w_{i,j}-w_{i-1,j})}^2}{h_x^2}+\frac{{(w_{i-1,j+1}-w_{i-1,j})}^2}{h_y^2}},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {|\nabla w|}_{i,j+1}& =\sqrt{\frac{{(w_{i+1,j}-w_{i,j})}^2}{h_x^2}+\frac{{(w_{i,j+1}-w_{i,j})}^2}{h_y^2}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {|\nabla w|}_{i,j-1}& =\sqrt{\frac{{(w_{i+1,j-1}-w_{i,j-1})}^2}{h_x^2}+\frac{{(w_{i,j}-w_{i,j-1})}^2}{h_y^2}}.\hfill \end{array}$$

(38)

Then, the curvature term $\nabla ·\frac{\nabla u}{|\nabla u|}$ is approximated at point $\left(i,j\right)$ by

$$\begin{array}{cc}\hfill \nabla ·\frac{\nabla w}{\sqrt{{|\nabla w|}^2+\beta }}& =\frac{(w_{i+1,j}-w_{i,j})/h_x}{\sqrt{{|\nabla w|}_{i+1,j}^2+\beta }}-\frac{(w_{i,j}-w_{i-1,j})/h_x}{\sqrt{{|\nabla w|}_{i-1,j}^2+\beta }}\hfill \\ \hfill & +\frac{(w_{i,j+1}-w_{i,j})/h_y}{\sqrt{{|\nabla w|}_{i,j+1}^2+\beta }}-\frac{(w_{i,j}-w_{i,j-1})/h_y}{\sqrt{{|\nabla w|}_{i,j-1}+\beta }}\hfill \end{array}$$

where $\beta >0$ is a regularization parameter to avoid division by zero. Finally, the boundary conditions are enforced by

$$w_{i,0}=w_{i,1},w_{i,n+1}=w_{i,n},w_{0,j}=w_{1,j},w_{m+1,j}=w_{m,j}.$$

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To realize the fixed-point iteration, we first define

$$\begin{array}{c}\sqrt{|\nabla U_{real}{|}_{i,j}^2+\beta }=G_{i,j},\mathrm{and}\sqrt{|\nabla U_{im}{|}_{i,j}^2+\beta }={\overline{G}}_{i,j}.\end{array}$$

Then, by taking advantage of the fact that the resulting matrix is strictly diagonal dominant, each iteration of the fixed-point method can be solved using the Gauss–Seidel method as follows:

$$\begin{array}{c}{u}_l^{k+1}=\left(\begin{array}{c}{\left(U_{real}^{k+1}\right)}_l\\ {\left(U_{im}^{k+1}\right)}_l\end{array}\right)=\left(\begin{array}{c}{\left({\mathcal{N}}_{real}\right)}_{i,j}/{\left({\mathcal{D}}_{real}\right)}_{i,j}\\ {\left({\mathcal{N}}_{im}\right)}_{i,j}/{\left({\mathcal{D}}_{im}\right)}_{i,j}\end{array}\right)\end{array}$$

(40)

with l as in (27) and

$$\begin{array}{c}{\left({\mathcal{N}}_{real}\right)}_{i,j}=G_{i+1,j}^k{\left(U_{real}^k\right)}_{i+1,j}+G_{i-1,j}^k{\left(U_{real}^{k+1}\right)}_{i-1,j}+G_{i,j+1}^k{\left(U_{real}^k\right)}_{i,j+1}\end{array}$$

$$\begin{array}{c}+G_{i,j-1}^k{\left(U_{real}^{k+1}\right)}_{i,j-1}+{\lambda }_1{\left({\widehat{U}}_{real}\right)}_{i,j}+2{\lambda }_3{\left(U_{real}^k\right)}_{i,j}\end{array}$$

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$$\begin{array}{c}{\left({\mathcal{D}}_{real}\right)}_{i,j}={\lambda }_1+2{\lambda }_3\left({\left(U_{real}^k\right)}_{i,j}^2+{\left(U_{im}^k\right)}_{i,j}^2\right)+G_{i+1,j}^k+G_{i-1,j}^k+G_{i,j+1}^k+G_{i,j-1}^k\end{array}$$

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$$\begin{array}{c}{\left({\mathcal{N}}_{im}\right)}_{i,j}={\overline{G}}_{i+1,j}^k{\left(U_{im}^k\right)}_{i+1,j}+{\overline{G}}_{i-1,j}^k{\left(U_{im}^{k+1}\right)}_{i-1,j}+{\overline{G}}_{i,j+1}^k{\left(U_{im}^k\right)}_{i,j+1}\end{array}$$

$$\begin{array}{c}+{\overline{G}}_{i,j-1}^k{\left(U_{im}^{k+1}\right)}_{i,j-1}+{\lambda }_2{\left({\widehat{U}}_{im}\right)}_{i,j}+2{\lambda }_3{\left(U_{im}^k\right)}_{i,j}\end{array}$$

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$$\begin{array}{c}{\left({\mathcal{D}}_{im}\right)}_{i,j}={\lambda }_2+2{\lambda }_3\left({\left(U_{real}^k\right)}_{i,j}^2+{\left(U_{im}^k\right)}_{i,j}^2\right)+{\overline{G}}_{i+1,j}^k+{\overline{G}}_{i-1,j}^k+{\overline{G}}_{i,j+1}^k+{\overline{G}}_{i,j-1}^k\end{array}$$

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5. Experimental Results

In this section, results are shown using both synthetic and real-world examples. All algorithms in this section were executed on a computer with an Intel Core i7 2.5 GHz processor, 16 GB of RAM, and a 64-bit installation of Debian GNU/Linux 9 (stretch). The proposed model was implemented using the C/C++ programming language and the high-performance vector mathematics library Blitz++ [39]. The code and experiments can be found at https://github.com/clirlab/wrapped_phase_denoising (accessed on 27 April 2023).

5.1. Quality of Results

Let us begin by analyzing the outcomes of synthetic images polluted with varying levels of additive Gaussian noise. A wrapped phase image with a vertical discontinuity in the middle is depicted in Figure 1a,c, with signal-to-noise ratio (SNR) values of $74.41$ dB and $43.34$ dB, respectively. The proposed model was applied to both images, resulting in the restored images shown in Figure 1b,d. The parameters were fixed at $\beta =0.001$, ${\lambda }_1={\lambda }_2=2.5$, and ${\lambda }_3=5$ for both problems, and the noise was successfully eliminated while the discontinuity was preserved.

In addition, Figure 2 depicts a noisy and restored image of a real problem. Again, the restoration is excellent, as are the edges. This problem comes from a brain MRI scan.

5.2. Algorithm Performance

In this section, the performance of the gradient descent and fixed-point numerical algorithms in solving the proposed variational model is analyzed.

Table 1. Comparison between the gradient descent method and the fixed-point method.

Numerical Method	Problem	# of Iterations	CPU Time (s)
Gradient Descent	Figure 1a	1649	130
	Figure 1c	5159	431
Fixed Point	Figure 1a	104	18
	Figure 1c	307	55

displays the CPU time and number of iterations required by each algorithm to solve the problems presented in Figure 1a,c. Iterations were terminated for both algorithms when the relative residual was less than $ϵ={10}^{-7}$ with $ϵ\equiv \left|\right|u^{k+1}-u^k{\left|\right|}_2/\left|\right|u^k{\left|\right|}_2$.

As anticipated, Table 1 demonstrates that the fixed-point method is roughly one order of magnitude faster than the gradient descent method.

5.3. Comparison against Other Denoising Algorithms

A natural question is whether state-of-the-art image denoising algorithms provide an acceptable restoration of $U_{real}$ and $U_{im}$ as well as the wrapped phase map.

In order to compare the results, the mean squared error (MSE) and the image fidelity index (IFI) [8] were evaluated for any variable $y\in {\mathbb{R}}^{m\times n}$.

$$\begin{array}{c}MSE=\frac{1}{m\ast n}\sum _{i=1}^m\sum _{j=1}^n{(y(i,j)-\overline{y}(i,j))}^2,\end{array}$$

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and

$$\begin{array}{c}IFI=1-\frac{{\sum }_{i=1}^m{\sum }_{j=1}^n{\left(y(i,j)-\overline{y}(i,j)\right)}^2}{{\sum }_{i=1}^m{\sum }_{j=1}^{n}y{(i,j)}^2}.\end{array}$$

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The closer the IFI is to 1, the more accurate the resulting estimate will be.

There are two methods for the denoising of interferograms, the most common of which is directly processing the wrapped phase image. The second method, which was utilized in this study, is to denoise independently each component of the complex image and then use both to reconstruct the wrapped phase image using the arctan-2 function. First, we demonstrate the results of the first approach using BM3D [40], Non-Local Means (NLM) [41], and the de facto method to denoise wrapped phase maps, Ströbel’s method [25], on the synthetic problems of Figure 1a,c and the real problem of Figure 2a. Then, we apply cutting-edge algorithms to the wrapped phase image.

The results of using the first method are shown in

Table 2. Comparison among different denoising algorithms.

	MSE		IFI
Model	${\mathit{U}}_{\mathit{real}}$	${\mathit{U}}_{\mathit{im}}$	${\mathit{U}}_{\mathit{real}}$	${\mathit{U}}_{\mathit{im}}$
Proposed model	0.0555	0.0542	0.8515	0.8551
Ströbel	0.1537	0.1489	0.5891	0.6021
BM3D	0.3696	0.3687	0.0125	0.0149
NLM	0.3478	0.3511	0.0706	0.0621

. The proposed model has by far the best restoration. Even Strobel’s method outperforms BM3D and NLM, demonstrating that the relationship between $U_{real}$ and $U_{im}$ must be preserved when denoising a wrapped phase map image. Figure 3 and Figure 4 illustrate that enforcing the Pythagorean trigonometric identity between the real and imaginary parts of the phase map improves the wrapped phase map restoration, thereby reducing inaccuracies in potential applications.

In

Table 3. CPU times of each algorithm when running the experiments of Figure 1.

Algorithm	$\mathit{SNR}=43.34\mathbf{dB}$	$\mathit{SNR}=74.41\mathbf{dB}$
Proposed model	6.9 s	63 s
Ströbel	5.5 s	6.2 s
BM3D	36.6 s	45.6 s
NLM	14.4 s	106 s

, we present the CPU times required by each model to denoise the problems depicted in Figure 1. As can be seen, the processing time of the proposed method is very competitive.

The results of using the second method are shown in Figure 5, where we compare it against two convolutional neural network (CNN) models for SAR image despeckling. In the work of Wang et al. [42], a deep-learning-based approach to remove speckle from SAR images using convolutional layers, batch normalization, a rectified linear unit activation function, and a componentwise division residual layer is presented. In Reference [43], however, a different strategy is taken by applying a CNN model to a recently proposed SAR speckle removal framework, constructing a reliable dataset of speckle-free SAR images and training a hybrid CNN on speckle-free SAR images.

We evaluate the proposed model against the two CNN-based models described previously. Figure 5 demonstrates that both CNN-based models obtain significantly worse MSE and IFI values than our model. Moreover, the reconstruction of our proposed model appears to be of superior quality to those of the other two.

6. Conclusions

This article presents a novel variational model for the denoising of wrapped phase maps. Regularization is achieved through total variation, so the model preserves discontinuities, and the fundamental Pythagorean trigonometric identity between the real and imaginary portions of the phase map is enforced, significantly enhancing the phase map restoration. In addition, the existence and uniqueness of our model’s solution are analyzed, and a rapid fixed-point algorithm for the numerical implementation is presented. Also provided is a theoretical analysis of this fixed-point method to demonstrate its convergence. In conclusion, restoration results on synthetic and real-world examples are presented as evidence of the restoration quality, and comparisons are made with well-known denoising techniques such as BM3D, NLM, and Strobel. In addition, comparisons are made with cutting-edge CNN-based models. In both instances, the proposed variational model produces the best results.