Curvelet Regularized Nonlinear Anisotropic

Curvelet Regularized Nonlinear Anisotropic Diffusion

Jianwei Ma1 and Gerlind Plonka2 1

Laboratoire LMC-IMAG, University Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France [email protected] 2 Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany [email protected]

Abstract In this paper, two models for curvelet-regularized nonlinear diffusion are proposed for discontinuity-preserving denoising, using a new tight frame of curvelets and a nonlinear diffusion scheme. Numerical experiments show their good performance to recover the shape of edges and some important detailed components of the image while we use less iterations, in comparison to traditional methods. Mathematics Subject Classification 2000. 65T60, 65M06, 65M12, 94A12. Key words. Curvelets, nonlinear diffusion, regularization, discontinuity-preserving, denoising.

1

Introduction

Image denoising is one important operation in image processing. The tools for attaching this problem come from very different fields like Computational Harmonic Analysis (CHA) and partial differential equations (PDEs). They serve the same denoising purpose; removing the noise from the observed signal without sacrificing important discontinuity structures such as edges. Generally, tools from the CHA such as wavelets have promising properties for singularity analysis and efficient computational complexity, but suffer from the pseudo-Gibbs artifacts and shift/rotation variance. New approaches like complex wavelets and curvelets attempt to overcome these problems at least partially. Application of PDEs such as nonlinear diffusion, variational methods and level sets are almost free from the above lacks of CHA, but cost heavy computational burden. The two directions have been opposed for long time, but recent research is increasingly focusing on combinations of both. Total variation (TV) regularization has been combined with wavelets to reduce the pseudo-Gibbs artifacts resulted from wavelet shrinkage [4, 6, 8, 11]. The relations between wavelet shrinkage, nonlinear diffusion, and regularization have been explored [15, 17, 22]. This connection provides a fruitful exchange of ideas between the two directions. Recently, diffusion-inspired wavelet 1

shrinkage with improved rotation invariance has been considered [12]. On the other hand, a new wavelet-inspired explicit scheme that permits larger time steps while preserving convergence has been also proposed for nonlinear diffusion [14]. Within the last years, Cand`es and Donoho [1, 2, 3] developed a new geometric multiscale transform, the so-called curvelet transform, which allows an optimal sparse representation of objects with C 2 -singularities. The needle-shape elements of this transform own very high directional sensitivity and anisotropy. The transform represents edges and singularities along curves much more efficiently than traditional wavelet transforms. For a smooth object f with discontinuities along smooth curves, the best m-term approximation f˜m by curvelet thresholding obeys
f − f˜m
22 ≤ Cm−2 (log m)3 , while for wavelets the decay rate is only m−1 . It has been shown that the new transform is superior to tensor-product wavelet transforms in the field of image processing [10, 16]. The idea of image smoothing by nonlinear diffusion can be explained as follows. Let u0 be an observed noisy image which is known to be the sum of the original image f and some Gaussian noise n, u0 (x) = f (x) + n(x), x = (x1 , x2 ) ∈ Ω. Here Ω ⊂ R2 denotes a rectangle. We consider the diffusion process (see [13]) ∂u = ∇ · (g(|∇u|)∇u) ∂t

(1.1)

with the given noisy signal u0 as initial condition u(x, 0) = u0 (x),

x∈Ω

(1.2)

and periodic boundary conditions. Here the time t acts as a scale parameter for filtering. The choice of the diffusion function g(x) = const corresponds to linear diffusion which is known to lead to a strong smoothing of u with increasing t. Typically, g(x) is a non-negative decreasing function with limx→∞ g(x) = 0. The diffusivity g controls the smoothing process by admitting strong diffusion if the gradient ∇u is small (possibly caused by noise) and by slowing down (or even stop) the smoothing for large gradients. Frequently applied diffusivitiesin (1.1) are Perona-Malik diffusivity g(x) := 1/(1+x2 /γ 2 ), Charbonnier diffusivity g(x) := 1/ 1 + x2 /γ 2 , and TV diffusivity g(x) := 1/x with suitable chosen contrast parameter γ (see [20]). In spite of having many desirable theoretical and computational properties, one serious problem of the diffusion model is that it is very sensitive to noise (see [23]). The gradient-based model in (1.1) possibly misconstrues the true edges and heavy noise, which leads to undesirable diffusion in regions where there is no true edge. Another problem is that staircasing effects arise around smooth edges [20]. A remedy to the deficiencies is suggested by introducing a regularization of the model (see [5, 18]). A typical improved model is based on Gaussian regularization ∂u (1.3) = ∇ · (g(|∇(Gσ  u)|)∇u) ∂t where Gσ is a Gaussian kernel with variance σ. The Gaussian filtering acts as a pre-processing to reduce the influence of noise during the diffusion process. The regularization can also be done with other kernels, see e.g. a wavelet regularization in [19]. The goal of this paper is to propose two different models wich are hybrid methods using curvelet shrinkage and nonlinear diffusion for discontinuity-preserving denoising. In section 2, we introduce the curvelet transform and derive an algorithm for the efficient computation of the discrete curvelet coefficients in the periodic case. In section 3, we propose two new 2

hybrid models for image smoothing. In the first model we apply a curvelet shrinkage method and afterwards a diffusion process in order to reduce the pseudo-Gibbs phenomenon. This idea is closely related to approaches in [8, 11]. We also compare our approach with the model obtained by curvelet shrinkage followed by a projected regularization method. The second model relates to the regularized diffusion (1.3), where we propose to replace the Gaussian filtering by a curvelet transform. Section 4 is devoted to numerical results for image denoising obtained with the new models.

2

Curvelet transform

The first generation of curvelets is based on block ridgelets [1]. Apart from the blocking effects, however, the application is limited, because the geometry of ridgelets is itself unclear as they are not true ridge functions in digital images. In this paper, we apply the second-generation Discrete Curvelet Transform DCuT [13,14], which is considerably simpler to use. Let V (t) and W (r) be a pair of smooth, non-negative real-valued window functions, such that V is supported on [−1, 1] and W on [ 12 , 2]. The windows need to satisfy the admissibility conditions ∞ 

V 2 (t − l) = 1,

∞ 

t ∈ R,

W 2 (2−j r) = 1,

r > 0.

j=−∞

l=−∞

These conditions are satisfied taking e.g. the scaled Meyer windows (see [7],   1   1 |t| ≤ 1/3   π cos[ 2 ν(5 − 6r)] V (t) = cos[ π2 ν(3t − 1)] 1/3 ≤ |t| ≤ 2/3 , W (r) = π cos[    2 ν(3r − 4)] 0 else  0 where ν is a smooth function satisfying  0 x≤0 ν(x) = , 1 x≥1

ν(x) + ν(1 − x) = 1,

p. 137) 5/6 ≤ r ≤ 4/3 2/3 ≤ r ≤ 5/6 , 4/3 ≤ r ≤ 5/3 else

x ∈ R.

1 −ix,ξ dx. Now, for Let the Fourier transform of f ∈ L2 (R2 ) be defined by f (ξ) := 2π R2 f (x)e 2 j ≥ 0 let the window Uj (ξ), ξ = (ξ1 , ξ2 ) ∈ R in frequency domain be given by Uj (ξ) = 2−3j/4 W (2−j |ξ|) V (2j/2 θ),

ξ ∈ R2 ,

where (|ξ|, θ) denotes the polar coordinates corresponding to ξ. The support of Uj is a polar ’wedge’ determined by supp W (2−j ·) = [2j−1 , 2j+1 ] and supp V (2j/2 ·) = [−2−j/2 , 2−j/2 ]. The system of curvelets is now indexed by three parameters; a scale 2−j , j ∈ N0 ; an equispaced (j,l) sequence of rotation angles θj,l = 2πl · 2−j/2 , 0 ≤ l ≤ 2j/2 − 1, and a position xk = −j , k 2−j/2 )T , (k , k ) ∈ Z2 , where R Rθ−1 (k 2 denotes the rotation matrix with angle θ 1 2 1 2 θj,l j,l . j,l The curvelets are defined by (j,l)

ϕj,l,k (x) := ϕj (Rθj,l (x − xk

)),

x = (x1 , x2 ) ∈ R2 ,

where ϕ j (ξ) := Uj (ξ), i.e., Uj is the Fourier transform of ϕj . Observe that in spatial domain, (j,l) ϕj,l,k is of rapid decay away from a 2−j by 2−j/2 rectangle with center xk and orientation θj,l with respect to the vertical axis in x. Further, introducing the real-valued, non-negative low-pass window W0 by  W0 (r)2 + W (2−j r)2 = 1, j≥0

3

let the coarse scale curvelet be given by ϕ−1,0,k (x) := ϕ−1 (x − k),

ϕ −1 (ξ) := W0 (|ξ|),

being nondirectional. For simplification, let µ = (j, l, k) be the collection of the triple index. The system of curvelets (ϕµ ) forms a tight frame in L2 (R2 ), i.e., each function f ∈ L2 (R2 ) can be represented by  f= cµ (f ) ϕµ . µ

Using Parseval‘s identity, the curvelet coefficients are given by   (j,l) ˆ f (ξ) ϕˆµ (ξ)dξ = fˆ(ξ) Uj (Rθj,l ξ) eixk ,ξ dξ. cµ (f ) := f, ϕµ  = R2

R2

(2.1)

In practical implementations one would like to have Cartesian arrays instead of the polar tiling of frequency plane (see Figure 1). Cartesian coronae are based on concentric squares (instead of circles) and shears. Cand`es at al. [3] applied a pseudo-polar grid by replacing the window Wj (ξ) := W (2−j ξ) by a window of the form

j ≥ 0, Wj (ξ) = χ[0,∞) (ξ1 ) φ2 (2−j−1 ξ1 ) − φ2 (2−j ξ1 ), where the one-dimensional window φ satisfies 0 ≤ φ ≤ 1, supp φ ⊂ [−2, 2] and φ(r) = 1 for r ∈ [−1/2, 1/2]. (Here, χ[0,∞) (ξ1 ) denotes the characteristic function of [0, ∞).) As before, φ can be taken to be a scaled Meyer window. With Vj (ξ) := V (2j/2 ξ2 /ξ1 ) the Cartesian window, j (ξ) := 2−3j/4 W j (ξ) Vj (ξ) U can be determined, being analogous to Uj and determining the frequencies near the wedge {(ξ1 , ξ2 ) : 2j ≤ ξ1 ≤ 2j+1 , −2j/2 ≤ ξ2 /ξ1 ≤ 2j/2 }. Let tan θj,l := l 2−j/2 , l = −2j/2 , . . . , 2j/2 − 1 be the set of equispaced slopes and put (j,l)

ϕ µ (x) = ϕ j,l,k (x) := ϕ j (SθTj,l (x − x k

x = (x1 , x2 ) ∈ R2 ,

)),

 (ξ) := U j (ξ) ϕ j

:= Sθ−T (k1 2−j , k2 2−j/2 ) =: Sθ−T kj , and with j,l j,l  1 0 . − tan θ 1 (j,l)

as the cartesian counterpart of ϕj,l,k , where x k the shear matrix  Sθ =

Observe that the angles θj,l range between −π/4 and π/4 and are not equispaced here, but the slopes. The set of curvelets ϕ µ needs to be completed by symmetry and by rotation by ±π/2 radians in order to obtain the whole family. We find the cartesian counterpart of the coefficients in (2.1) by   (j,l) −1 i˜ xk ,ξ ˆ ˜ ˜j (ξ) eikj ,ξ dξ. f (ξ) Uj (Sθj,l ξ) e fˆ(Sθj,l ξ) U µ  = dξ =  cµ (f ) = f, ϕ R2

R2

For our application, we need the curvelet transform for functions (images) on the square. For that purpose we consider the N -periodization of ϕ µ ,  ϕ pµ (x) := ϕ µ (x − N n), x ∈ R2 , n = (n1 , n2 ), n∈Z2

4

Figure 1: Discrete curvelet tiling in frequency plane. The support is a ‘parabolic’ pseudo-polar wedge. where N ∈ N is fixed. The N -periodic function f ∈ L2 (Ω), Ω = [0, N ]2 , can now be written in the form  f=  cD pµ , µ (f ) ϕ µ∈M



with  cD µ (f )

 f (x) ϕ pµ (x) dx

:= Ω

=

R2

f (x) ϕ µ (x) dx.

(2.2)

Observe that in the periodic case we have ϕ pj,l,(k1 ,k2 ) (x) = ϕ pj,l,(k1 +2j N,k2 ) (x). Further, assuming that l is of the form l = ±2t (2r + 1) with r, t ∈ N0 , it follows ϕ pj,l,(k1 ,k2 ) (x) = ϕ pj,l,(k

1 ,k2 +2

2
j/2−t N )

(x).

Hence, in the N -periodic case the index set M is of the form M

= {(−1, 0, k) : k = 0, . . . , N − 1} ∪ {(j, l, (k1 , k2 )) : j ∈ N0 , l = −2j/2 , . . . , 2j/2 − 1, k1 = 0, . . . , 2j N − 1, k2 = 0, . . . , 22j/2−t N − 1}.

Using the Fourier series of f , f (x) =



2πim,x/N

dm (f ) e

1 dm (f ) := 2 N

,

m∈Z2



f (x) e−2πim,x/N dx,

Ω

it follows with kj := (k1 2−j , k2 2−j/2 )  cD µ (f )

=



 dm (f )

m∈Z2

= N



R

2

e2πim,x/N ϕ µ (x) dx

 ( dm (f ) ϕ µ

m∈Z2

= N



2πm ) N

j ( 2π S T m) e− dm (f ) U N θj,l 2

m∈Z

5

2πi SθT m,kj  N j,l

.

The numerical computation has been extensively described in [3]. However, the periodization of the curvelet system explained above has not been considered there. One can apply the following procedure for numerical evaluation of  cD cD µ (f ). Fix a j0 ∈ N and compute  µ (f ) for µ ∈ Mj0 , D where Mj0 = {µ ∈ M : j ≤ j0 }. Analogously, compute  cµ (f ) in the other three quadrants, if ϕ µ (x1 , x2 ) is replaced by ϕ µ (−x1 , −x2 ), ϕ µ (x2 , x1 ), and ϕ µ (−x2 , −x1 ), respectively. Algorithm. 1. Apply 2D FFT to compute the Fourier coefficients dm (f ) of f . j compute the product 2. For all m with SθTj,l m ∈ supp U j ( 2π S T m). dm (f ) U N θj,l 3. Apply the inverse 2D FFT to obtain the discrete coefficients  cD µ (f ). −1 In the application, f is usually given as a discrete set,i.e. f = (fi,j )N i,j=0 . For computing dm (f ), one can interpolate the f by piecewise linear spline functions whose Fourier transform is known. In the last step, an unequispaced FFT can be used. The forward and inverse DCuT have the same computational cost of O(N 2 log N ) for an (N × N ) image (see e.g. [3]).

3

Curvelet regularized nonlinear diffusion

We want to propose two different models. −1 Model 1. Let u0 = (u0,i,j )N i,j=0 be an observed discrete noisy signal on the square Ω = [0, N ]. We first apply curvelet shrinkage to the noisy signal u0 , and then apply a ‘projected’ nonlinear diffusion to the curvelet-processed signal to reduce the pseudo-Gibbs and curvelet-like artifacts. Let τ be a hard thresholding function defined by a fixed threshold λ  x, x ≥ λ, τ (x) = 0, x < λ. After applying the periodic curvelet transform and shrinkage to u0 , we obtain  τ ( cD pµ , uc = µ (u0 )) ϕ

(3.1)

µ∈M

where  cD µ (u0 ) are the curvelet coefficients of u0 as in (2.2). The indices of the curvelet coefficients retained after hard thresholding are recorded in Λ, i.e., Λ := {µ ∈ M : | cD µ (u0 )| ≥ λ}. Let the linear subspace V ⊂ L2 (Ω) of functions be given by V := {v :  cD µ (v) = 0, ∀µ ∈ Λ}.

(3.2)

Then, the model 1 can be formulated as follows ∂u = ∇ · (g(|∇PV (u)|) ∇PV (u)) ∂t with the curvelet shrinkage signal uc in (3.1) as initial condition u(x, 0) = uc (x), 6

x ∈ Ω,

(3.3)

and with periodic boundary conditions. Here PV (u) denotes the projection of u onto V . That means, only the coefficients with absolute value smaller than λ will be changed by the diffusion process. Let T be the periodic curvelet transform and T −1 be the inverse transform, then we can write PV (u) = T −1 τ −1 T (u), where the τ −1 denotes a so-called inverse thresholding function, which only takes the coefficients outside Λ, i.e.,  0, x ≥ λ −1 τ (x) := x, x < λ. We apply the Euler-forward scheme for discretization of (3.3) and forward/backward differences for approximation of ∇PV u. Using this model, one only needs a few iterations of diffusion after the curvelet shrinkage to get a satisfying result. The projection PV ensures that significant components of the signal remain unchanged during the diffusion while only the highfrequency components are revised to eliminate the artifacts. This strategy avoids that the narrow peaks/textures are smoothed too much as in conventional diffusion. It can better remain the signal amplitude of detailed components while reducing the pseudo-Gibbs oscillations at the same time, in comparison to those directly using curvelet shrinkage or diffusion. It should be noted that a similar strategy has been used in TV-minimization based wavelet/ridgelet transform [8, 11]. In that work, the TV minimization does not set the insignificant wavelet/ridgelet coefficients to zero, but typically inputs optimal small values to cancel the pseudo-Gibbs oscillations. Let us briefly compare TV-minimization with our model 1 proposed above. We can formulate the TV-minimization for curvelet transform as follows. For a function u with |∇u| ∈ L1 (Ω), the total variation functional is defined by  |∇u(x)| dx, T V (u) = Ω

where Ω is a rectangle (or square) in R2 in our application. The idea of TV-minimization (see [8]) is to remove the pseudo Gibbs oscillations by minimizing the functional  F (u) = |u − u0 |2 dx + λ T V (u) Ω

for u ∈ {uc + v, v ∈ V }, where u0 is the initial noisy signal and uc is the reconstructed function after curvelet thresholding in (3.1). Using uc as initial guess, i.e., u0 := uc , the constrained TV-minimization can be computed by a projected gradient descent scheme (see [8]) ul+1 = ul − tl PV (gT V (ul )),

(3.4)

where gT V (u) denotes the subgradient of T V at u and PV again denotes the projection of √ l gT V (u ) on V given in (3.2). The step size tl > 0 can be taken in the form tl = t0 / l in order to ensure convergence. It has been demonstrated that in the 1D case that space discrete TV diffusion and TV regularization/ minimization are identical if we identify the diffusion time with the regularization parameter, see [17]. However, is the constrained minimization scheme (3.4) equivalent to model 1 in (3.3)? We can deduce the gradient of total variation functional as in [23], 1 gT V (ul ) = −∇ · (g(|∇ul |) ∇ul ), g(x) = . x So (3.4) can be rewritten as ul+1 = ul + tl PV ∇ · (g(|∇ul |) ∇ul ) 7

which can be seen as discretization of ∂u = PV ∇ · (g(|∇u|) ∇u). ∂t This equation clearly shows the difference to our model 1 in (3.3), where regularization is used followed by evolution of diffusion. Model 2. A second idea to apply curvelet transform in the diffusion process, is to take the formula (1.3), but using curvelet regularization instead of classical Gaussian regularization, ∂u = ∇ · (g(|∇(Pλ u)|) ∇u) ∂t

(3.5)

with the original noisy signal u0 as initial condition, i.e., u(x, 0) = u0 (x), x ∈ Ω. Here Pλ u = T −1 τ T (u) denotes a curvelet shrinkage of the signal u with threshold λ. Unlike classical Gaussian filtering, the curvelet pre-processing can effectively remove the noise while the edges are preserved very well, which leads to better discontinuity-preserving diffusion for denoising. An optimally selected parameter λ is important for the success of model 2, because the diffusion process is still ill-posed for too small λ , while image features are smeared for too large λ. Using the Eulerforward scheme for discretization of (3.5), a possible approach is to apply a large λ initially to suppress noise and then to reduce the threshold step by step in the further iterations. Other shrinkage rules (instead of hard thresholding) can be applied as well. The computation load is a problem for the regularized nonlinear diffusion because the Pλ is required at each iteration. An alternative idea is to replace ∇(Pλ u) by differences of curvelet coefficients  cD µ . This approach avoids the computation of inverse DCuT and of the gradient, which helps to speedup.

4

Numerical experiments

In the first test, we show the performance of the proposed methods for denoising an image with line singularities. Fig. 2(a) shows the image contaminated with heavy noise. Fig. 2(b) is obtained using the DCuT shrinkage. Although the DCuT is effective in recovering edges, it suffers from the pseudo-Gibbs and curvelet-like artifacts yet. Fig. 2(c) is obtained using the direct nonlinear diffusion with 200 iterations similarly as in [22]. The noise is clearly removed without the pseudo-Gibbs artifacts, but obvious staircasing effect arises along the edges. Another problem is its heavy computational burden, which is not suitable for time-critical application. Fig. 2(d) is the result obtained using the proposed model 1 with 50 iterations of diffusion. It can be seen that the artifacts in (b) and staircasing oscillations in (c) are much reduced. It should be noted that TV diffusivity is used in above tests (c)-(d). This unbounded diffusivity tends to produce blocky effects in smooth areas because taking a large diffusivity coefficient leads to oversmoothing. In the following tests for model 2, we adopt the Perona-Malik (PM) diffusivity. Fig. 2(e) is obtained by Gaussian-regularized PM diffusion [5] with 50 iterations. As we know, the result obtained directly using PM diffusion without regularization is far away from satisfaction due to the well-known ill-posedness of PM diffusion. (f) is obtained using the model 2 with 50 iterations showing a fantastic performance for edges-preserving denoising, in comparison to (e). Our next test shows denoising results for a magnetic resonance image of a human head. Fig. 3(a) is a noisy image. Fig. 3 (b) is obtained using DCuT. Fig. 3(c) is obtained using TV diffusion with 20 iterations. Fig. 3(d) is obtained using model 1 (with TV diffusivity) with 20 iterations. Fig. 3(e) and Fig. 3(f) are obtained using Gaussian-regularized diffusion and model 2 (with PM diffusivity) with 50 iterations, respectively. Fig. 4 shows the large close-up of hindbrain taken from Fig. 3 (b-f). It is clearly to see that the shape of edges is recovered well using our proposed methods. 8

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Denoising using proposed methods, in comparison to existing methods It should be noted that the revised version of model 2 mentioned in section 3 manipulates the diffusion in coefficient domain. Due to the existence of aliasing effect arising from the shift sensitivity of DCuT, the denoising achieved in current implementation is not satisfactory enough for immediate application. There is room for substantial progress in future research.

Acknowledgment The first author likes to thank for the support of the IDOPT project (CNRS-INRIA-UJF-INPG), as well as for the Guest Professorship at the Department of Mathematics of the University Duisburg-Essen.

9

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Denoising of a magnetic resonance image using proposed methods, in comparison to existing methods

(b)

(c)

(d)

(e)

(f)

Figure 4: From left to right: Close-up of hindbrain taken from Fig. 3 (b)–(f)

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