Restoration of images with rotated shapes

Transcription

1 Numer Algor (008) 48:49 66 DOI 0.007/s y ORIGINAL PAPER Restoration of images with rotated shapes S. Setzer G. Steidl T. Teuber Received: 6 September 007 / Accepted: 3 January 008 / Published online: 4 March 008 Springer Science + Business Media, LLC 008 Abstract Methods for image restoration which respect edges and other important features are of fundamental importance in digital image processing. In this paper, we present a novel technique for the restoration of images containing rotated (linearly transformed) rectangular shapes which avoids the round-off effects at vertices produced by known edge-preserving denoising techniques. Following an idea of Berkels et al. our approach is also based on two steps: the determination of the angles related to the rotated shapes and a subsequent restoration step which incorporates the knowledge of the angles. However, in contrast to Berkels et al., we find the smoothed rotation angles of the shapes by minimizing a simple quadratic functional without constraints which involves only first order derivatives so that we finally have to solve only a linear system of equations. Moreover, we propose to perform the restoration step either by quadratic programming or by solving an anisotropic diffusion equation. We focus on a discrete approach which approximates derivatives by finite differences. Particular attention is paid to the choice of the difference filters. We prove some relations concerning the preservation of rectangular shapes for our discrete setting. Finally, we present numerical examples for the denoising of artificial images with rotated rectangles and parallelograms and for the denoising of a real-world image. Keywords Anisotropic diffusion Diffusion tensor Anisotropic ROF model Wulff shapes Difference filters Image denoising Dedicated to Prof. M. Tasche on the occasion of his 65th Birthday. S. Setzer G. Steidl (B) T. Teuber Department of Mathematics and Computer Science, University of Mannheim, Mannheim, Germany steidl@math.uni-mannheim.de

2 50 Numer Algor (008) 48:49 66 Introduction In image denoising one is typically interested in removing noise while preserving important structures such as edges. Since this goal cannot be achieved with linear filters various nonlinear strategies have been proposed in recent years, e.g., wavelet-based methods [8], stochastic methods [9], variational methods like the Rudin-Osher-Fatemi model [], PDE-based methods like the isotropic nonlinear diffusion of Perona and Malik [9], or Weickert s anisotropic edge enhancing diffusion [6]. In this paper, we focus on variational and PDE-based approaches. Most of these methods cause a significant rounding artefact at corners formed by sharp edges. A typical example is shown in Fig.. One suitable approach to overcome this drawback was the anisotropic modification of the Rudin-Osher-Fatemi model pointed out by Esedoglu and Osher [6], see also [5, 7], and further developed by Berkels et al. [3]. Note that a related approach was given from the PDE point of view by Weickert [5]. In [3], the authors deal with images f containing noisy rotated rectangular shapes. They suggest to find the denoised version u of f by computing the minimizer of the functional F(u,α) := ( f u) dxdy + λ R(α) u dxdy + μ α + μ α dxdy with the rotation matrices cos (α(x, y)) sin (α(x, y)) R (α(x, y)) :=. () sin (α(x, y)) cos (α(x, y)) More precisely, they propose to find u by iterating the following alternating procedure: for fixed u(x, y) compute the minimizer of F u (α) := R(α) u dxdy + μ α + μ α dxdy, () and for fixed α(x, y) find the minimizer of E α (u) := ( f u) dxdy + λ R(α) u dxdy. (3) The main idea here is that when rotating the gradient the smallest L -norm is attained for the rotated gradient being parallel to one of the axes. Consequently, the solution of F u (α) assigns to each point a smoothed Sometimes these processes are already denoted as anisotropic. In our nomenclature, this notion is reserved for processes driven by matrix-valued diffusion tensors.

3 Numer Algor (008) 48: Fig. Left: noisy image corrupted by white Gaussian noise with standard deviation 50. Middle: denoised image using edge enhancing diffusion, see [6]. Right: denoised image using the method proposed in this paper angle α, representing the orientation of the gradients in the image. Using this information in the second term of (3), we see that minimizing E α (u) leads to good denoising results for objects which are aligned in the directions α and α + π. The functional F u(α) depends on α and also on the trigonometric functions cos α and sin α of α. Moreover, we see that R(α(x, y)) u(x, y) has the same value for α(x, y) + k π, k Z so that E α(u) depends only on α modulo π. Corresponding restrictions on α while computing a minimum of F u in () were not addressed in [3]. To cope with the significant loss of contrast in this process, the authors propose to apply in addition Bregman iterations, a kind of inverse scale space method. The aim of this paper is to present a simple model for the restoration of images containing rotated (linearly transformed) rectangles. The approach is based on the idea of Berkels et al. in so far as we also iterate two steps: an angle adaptation step and a restoration step which takes the former angle computation into account. In the angle adaptation step we find the smoothed rotation angles of the shapes contained in the image. To this end, we minimize a simple quadratic functional without constraints so that we finally have to solve only a linear system of equations. Attention is paid to the fact that the angles are only needed modulo π. For the restoration step we propose two methods, namely either to minimize a discrete version of (3) by quadratic programming or to solve an anisotropic diffusion equation related to the Euler- Lagrange equation of (3) by finite difference methods. In both steps, we focus on discrete techniques which approximate derivatives by forward differences. Therefore, we have to pay attention to the careful choice of the difference filters. Moreover, we consider some results of Esedoglu and Osher [6] from our discrete point of view. On the one hand, this makes the approaches simpler and also manageable for other operators than the gradient. On the other hand structures like Wulff shapes in R m do not carry over to Z m. This paper is organized as follows: Section describes our mathematical model. In particular, we explain the angle adaptation in Subsection. and the two restoration variants in the Subsections. and.3. InSection3, we prove some useful relations concerning the preservation of discrete rectangular shapes. Finally, in Section 4, we present various numerical examples for the

4 5 Numer Algor (008) 48:49 66 denoising of artificial images with rotated rectangles and parallelograms as well as for the denoising of a real-world image. The paper concludes with a summary and a sketch of ongoing and future work in Section 5. Mathematical model Our denoising algorithm is based on iterations of two steps, the angle adaptation step and the restoration step which we describe in the following subsections. For the restoration step, we propose either to apply quadratic programming methods or to solve an anisotropic diffusion equation. In our discrete setting, we assume for convenience that f :{0,...,n } {0,...,n } R is a quadratic image. Alternatively, we may reshape f = ( f (x, y)) n x,y=0 columnwise into a vector f :{0,...,N } R,whereN = n.. Angle adaptation Instead of the functional () one may try to compute the smoothed rotation angles α by minimizing the simpler quadratic functional (α u α) dxdy + μ α dxdy, (4) where α u (x, y) is the angle of the gradient of u at (x, y). Note that α(x, y) = cos(α(x, y)) + sin(α(x, y)). (5) Unfortunately, the functional (4) does not take α modulo π into account so that we have to look for corresponding modifications in the rest of this subsection. To find the desired rotation angles α we first have to compute the rotation angles α u of the given image u R n,n andthentosmooththem.wefocuson the discrete setting. Computation of α u We compute α u (x, y) as the angle of the discrete gradient of u in (x, y), (x, y) {0,...,n } {0,...,n }. More precisely, we compute this angle with respect to the smoothed image u σ := K σ u obtained by convolving u with the Gaussian K σ of standard deviation σ. As discrete partial derivatives of u σ we use with H 0 := u σ,x := H u σ H T 0, u σ,y := H 0 u σ H T (6)..., H :=... 0.

5 Numer Algor (008) 48: In other words, u σ,x is computed by taking forward differences of u σ in vertical direction, the usual smoothing in horizontal direction, and assuming Neumann boundary conditions. If we suppose pixel distances h x, h y not equal to, then H must be multiplied by /h x and H T by /h y to obtain consistent discretizations of the derivatives in x- and y-directions in (6). Note that H 0, H are also the filter matrices related to the undecimated Haar wavelet transform, cf. [8]. Alternatively, one can also apply symmetric differences which leads to similar numerical results. In the vector reshaped version, (6) reads Dx H0 H u σ = u D σ = u y H H σ, (7) 0 where denotes the tensor product of matrices. For (x, y) {0,...,n } {0,...,n }, let u σ (x, y) :=(u σ,x (x, y) + u σ,y (x, y))/. If u σ (x, y) =0 we set α u (x, y) := 0, and otherwise arccos u σ,x(x,y) u σ if u (x,y) σ,y (x, y) 0, α u (x, y) = π arccos u σ,x(x,y) u σ if u (x,y) σ,y (x, y) <0. Next, we want to smooth α u in an appropriate way. Smoothing of α u Since we want to take only α modulo π into account, it makes sense to use the π -periodic functions c u (x, y) := cos(4α u (x, y)), s u (x, y) = sin(4α u (x, y)) in the smoothing process. Now we are looking for vectors (c v, s v ) T such that cu cv u u. s u s v We do not require that c v + s v =. Having (5) in mind, we solve arg min u cu cv + μ( cv + s c v,s v s u s v ). v (x,y) Z n Let I N denote the N N identity matrix. Reshaping the matrices columnwise and using G := diag( u ) with the discrete partial derivative operators Dx IN H :=, (8) D y H I N this is equivalent to finding the minimizers of G(c u c v ) + μ Dx c D v and G(s u s v ) + μ Dx s y D v. y These minimizers are given by the solutions of ( G T G + μ ( D T x D x + D T y D y) ) c v = G T Gc u, ( G T G + μ ( D T x D x + D T y D y) ) s v = G T Gs u. (9)

6 54 Numer Algor (008) 48:49 66 The matrix D T x D x + D T y D y is the discretization of the negative Laplacian with centered differences and Neumann boundary conditions. It is a diagonal dominant matrix in R N,N of the rank N. The matrix G T G = 0 is a diagonal matrix with the nonnegative diagonal entries u(x, y).then G T G + μ(d T x D x + D T y D y) is irreducible diagonal dominant and consequently invertible. Moreover, it is easy to check that the maximum principle c v (x, y) max (x,y) Z n c u (x, y) is fulfilled (similarly for s v ) by considering for example the Jacobi iteration process to solve (9). Note that we have in general c v (x, y) + s v (x, y) =. Once we have found c v and s v, we compute the smoothed version α of α u as α(x, y) := 0 if (c v (x, y), s v (x, y)) =0and α(x, y) = arccos 4 π arccos Now we turn to the restoration step.. Restoration by quadratic programming c v (x,y) (c v (x,y),s v (x,y)) if s v (x, y) 0, c v (x,y) (c v (x,y),s v (x,y)) if s v (x, y) <0. We consider a discrete version of (3) which reads for columnwise reshaped images as follows: { } arg min u f u + λ R(α) Dx u D, (0) y where R(α) := diag(cos α) diag(sin α) diag(sin α) diag(cosα) and D x and D y are chosen as in (7). It is well-known, cf. [4, 4] and Section 3, that the minimizer û of (0) is given by û = f λ(d T x DT y ) ˆV,where ˆV is R(α)T the solution of the dual problem f λ ( D T x y) DT R(α)T V min subject to V. () This is a quadratic problem with linear constraints which can be solved by corresponding optimization methods. For heavy noise, our choice of D x and D y due to their kernels may lead to checkerboard effects. These effects can be avoided by solving, for a small constant ν, the slightly modified problem arg min u { f u + λ R(α) ( Dx D y ) u + ν (H H )u }. By [3, 6] and the following Section 3, the restoration via minimization of (0) is useful for images containing rotated rectangles. Of course, the approach can

7 Numer Algor (008) 48: be simply modified for rectangles which are linearly transformed by a constant transform matrix A R, : { } arg min u f Dx u + λ R(α) (A I N ) u D. () y In particular, we will consider images with rotated parallelograms which can s be obtained by applying a shear matrix A(s) =. 0.3 Restoration by anisotropic diffusion Another way to perform the restoration step is based on the Euler Lagrange equation of (3). For this, we replace ϕ(x, y) := (x, y) = x + y in (3) by the differentiable function ϕ ε (x, y) := x + ε + y + ε with ε and obtain Ẽ α (u) = ( f u) dxdy + λ ϕ ε (R(α) u) dxdy. (3) The minimizer of Ẽ α (u) has to fulfill the Euler Lagrange equation given by with the diffusion tensor 0 = u f λ div (D α ( u) u) (4) ψ( eα, u ) 0 D α ( u) := R(α) T 0 ψ( e α, u ) R(α), (5) where ψ(x) := / x + ε and e α := (cos α, sin α) T. Then one may compute the steady state u(x, t) for t of the reaction diffusion equation t u = f u + λ div(d α ( u) u) with initial condition u(, 0) = f and homogeneous Neumann boundary conditions. On the other hand, the Euler Lagrange equation can be rewritten as u f = div(d α ( u) u). λ This can be interpreted as a fully implicit time discretization of the diffusion equation t u = div(d α ( u) u) (6) with initial image u(, 0) = f and homogeneous Neumann boundary conditions. In our numerical examples, we will restrict our attention to a time explicit scheme to solve (6). Of course, such a scheme will not lead to a minimizer of (3). For details see [3]. However, we propose this scheme as an alternative denoising technique which leads to very good results which we were not able to improve by adding e.g. the reaction term.

8 56 Numer Algor (008) 48:49 66 More precisely, in our experiments we will solve (6) by finite difference methods with an explicit Euler discretization of the time derivative. Here we have to be very careful with the discretization of the spatial derivatives to obtain comparable results with those from quadratic programming. We apply the first order difference filters proposed in [, 7]: H 0 := , H := These filters which result in 5 5 spatial masks in (6) are optimized (among all 5 5 masks) with respect to rotation invariance and lead to sharp edges. Again, if we suppose pixel distances h x, h y =, we have to multiply H by /h x to obtain a consistent discretization of the derivative in x-direction. Moreover, to avoid possible checkerboard effects we use the following method suggested in [, Section 9.5]: the fully discrete scheme of (6) reads u n+ = (I N + τ M) u n with discretization Mu of div(d α ( u) u) andtimestepsizeτ. This scheme was modified in [] to ( ) u n+ = I N + τ M + ν(ĩ I N ) u n with a small weight parameter ν and the usual Neumann boundary modifications. The low pass filter Ĩ is represented by the 5-band Toeplitz matrix with band 6 (, 4, 0, 4, ) and is a discretization of the identity filter I N of consistency order 4. The idea behind the additional term Ĩ I N is to add a filter, which on the one hand does no harm to the consistency of the scheme, but on the other hand eliminates the checkerboard effects. For further explanation and numerical examples demonstrating the influence of the additional term, see [, Section 9.5]. As in the previous subsection, we can also handle linearly transformed rectangles by solving (6) with the diffusion tensor ψ( eα, A u ) 0 D α ( u) := A T R(α) T 0 ψ( e α, A u ) R(α) A. 3 Shape preservation The functional (3) with constant rotation α = 0 can be considered as a special case of the more general functional ( f u) dx + λ ϕ( u) dx (7)

9 Numer Algor (008) 48: with R m and with a finite gauge function ϕ : R m R, i.e., a positively homogeneous, convex function which satisfies ϕ(0) = 0 and ϕ(x) >0 for x = 0. Since ϕ is positively homogeneous and convex, it follows that ϕ is sublinear which means that it fulfills in addition ϕ(x + y) ϕ(x) + ϕ(y). Moreover, since ϕ is finite and positively homogeneous, its conjugate function ϕ is the indicator function of a closed bounded convex set C ϕ containing the origin, called Wulff shape of ϕ, which is given as follows: C ϕ := { x R m : x, y ϕ(y) y R m}. Since ϕ is lower semicontinuous (lsc), we have that ϕ = ϕ so that ϕ is the support function of C ϕ, i.e., ϕ(x) = sup y C ϕ x, y. (8) The polar function ϕ o of a gauge function ϕ is defined by ϕ o x, y (x) := sup y =0 ϕ(y) and is a gauge function, too. Then C ϕ canbealsorewrittenas (9) C ϕ := { x R m : ϕ o (x) }. For example, we have for ϕ(x) := x that C ϕ ={x R m : x }. For m =, thesetc ϕ represents the square centered at the origin with sides of length parallel to the x-andy-axis. In [6], it was proven for the continuous setting that if f is the characteristic function of the Wulff shape of ϕ, i.e., f = Cϕ and λ is small enough, then the minimizer û of (7) preserves this form which means that û = c Cϕ with an appropriate constant c > 0. Moreover, it was shown that the minimizer of (7) with ϕ(x) = x also preserves rectangles with sides parallel to the axes. For a vector U := N,m U i+kn R mn,wesetu i=0,k=0 i := m U i+kn k=0 Rm. We are interested in the discrete counterpart to (7) given by f u + λ ( ϕ ((Lu) i ) ) N i=0 (0) with a finite lsc gauge function ϕ : R m R and a matrix L R mn,n.the typical example for L R N,N in the previous section was L = (D T x DT y.the )T penalizing term J(u) := ( ϕ ((Lu) i ) ) N is also a finite lsc gauge function and is therefore the support function of i=0 C J := { v R N : v, w J(w) w R N}. () It is well-known, cf. [4, 0], that the unique minimizer of (0) is given by û = f λ ˆv, ()

10 58 Numer Algor (008) 48:49 66 where λ ˆv is the orthogonal projection of f onto C λj.usingthatλ ˆv C λj if and only if ˆv C J we have that ˆv := arg min v CJ f λv. Concerning orthogonal projection the following lemma will be useful, see [6] for the continuous setting with L =. Lemma 3. Let J be the support function of the closed convex set C J defined by (). Then we have that λ ˆv is the orthogonal projection of f onto C λj if and only if ˆv C J and f λ ˆv, ˆv = J( f λ ˆv). (3) We add the brief proof for our discrete approach. Proof Let λ ˆv be the orthogonal projection of f onto C λj. Then we obtain by the projection theorem that f λ ˆv, y λ ˆv 0 y CλJ. Together with (8)thisimpliesthat λj( f λ ˆv) = sup y, f λ ˆv = f λ ˆv, λˆv y C λj and hence (3). Assume conversely that ˆv C J fulfills (3). Then λ ˆv C λj and we obtain by the definition of C λj that for all y C λj f λ ˆv, y λj( f λ ˆv) 0 f λ ˆv, y +λj( f λ ˆv) = f λ ˆv, λˆv y which yields by the projection theorem that λ ˆv is the orthogonal projection of f onto C λj. To determine ˆv we need a more manageable form of C J which is given in the following lemma. Lemma 3. The set C J defined by () coincides with { C J := v = L T V : ( ϕ o (V i ) ) } N. (4) Note that the condition on V is equivalent to V i C ϕ for all i=0,...,n. Proof. First we see that v C J must fulfill v R(L T ) since otherwise we can use the orthogonal decomposition v = v 0 + v with v 0 N (L), v 0 = 0 and v R(L T ) to obtain the contradiction i=0 v, v 0 = v 0 + v,v 0 = v 0 J(v 0) = 0.

11 Numer Algor (008) 48: Thus, { C J = v = L T V : V, Lw ( ϕ ((Lw) i ) ) N w R N}. (5). Let v C J. By definition of ϕ o in (9), we see that V i, W i ϕ o (V i )ϕ(w i ) W i R m so that we get for W i = (Lw) i by assumption N N N V, Lw = V i,(lw) i ϕ o (V i )ϕ((lw) i ) ϕ((lw) i ). i=0 i=0 By (5) this yields v C J. 3. Conversely, let v = L T V C J. We have to show that there exists a Ṽ such that v = L T Ṽ and ( ϕ o (Ṽ i ) ) N i=0. The functional l V (Lw) := V, Lw is linear on R(L) R mn and satisfies i=0 l V (Lw) p(lw) w R N, where p : R mn R is the sublinear function p(w) := ( ϕ (W i ) ) N i=0. By the Hahn-Banach theorem l V can be extended to a linear functional LṼ(W) := Ṽ, W on R mn which fulfills Ṽ, W p(w) W R mn. (6) i=0 Now LṼ(Lw) = l V (Lw) for all w R N, i.e., L T V,w = L T Ṽ,w for all w R N which implies that v = L T V = L T Ṽ. Let i 0 {0,...,N } so that ϕ o (Ṽ i0 ) = max{ϕ o (Ṽ i ) : i = 0,...,N }. Then we have by (6)forall W with W i := 0 for i = i 0 that Ṽ, W = Ṽ i0, W i0 ϕ W i0, W i0 R m, ϕ o Ṽi0, W i0 Ṽ i0 = sup W i0 =0 ϕ W i0 and we are done by assumption on Ṽ i0. It is not clear to us how to define some kind of Wulff shapes on a discrete grid for general ϕ. However, we may consider rectangles on Z and ask if they are preserved by the minimizer of (0) forϕ(x) = x and L = (D T x DT y )T.Of course this depends on the concrete choice of the difference matrices D x and D y. In the following, we provide an example. Similarly as in [6], we use that () and the Lemmas 3. and 3. imply that û is the minimizer of (0) ifand only if the following relations are fulfilled: i) û = f λ ˆv, ii) ˆv = L T ˆV, where ˆV i C ϕ for all i = 0,...,N, iii) J(û) = û, ˆv.

12 60 Numer Algor (008) 48:49 66 Example 3.3 Let ϕ(x, y) = x + y so that ii) is equivalent to ˆV. Let R := {x 0 +,, x 0 + a} {y 0 +,, y 0 + b} with x 0, y 0 0 and x 0 + a, y 0 + b n and let f = R be the image f :{0,...,n } {0,..., n } R with f (x, y) = if (x, y) R and f (x, y) = 0 otherwise. Furthermore, we use L := H0 H, where we slightlymodify the matrices in (6) H H 0 by h 0 (0, 0) = h (0, 0) = 0 and h 0 (n, n ) =, h (n, n ) =, i.e., H is the forward difference matrix with zero boundary assumptions and H 0 ˆV is the corresponding smoothing matrix. Consider ˆV = ˆV defined by ( ) y 0 y x I, y J, x I, y J, ( ) y 0+b y x I, y J 3, ( ) y 0 y (x x 0) x I a, y J, ˆV (x, y) = (x x 0) a x I, y J, ( ) y 0+b y (x x 0) x I a, y J 3, ( ) y 0 y+ x I 3, y J, x I 3, y J, ( ) y 0+b y+ x I 3, y J 3, and ( ) x 0 x x I, y J, x I, y J, ( ) x 0+a x x I 3, y J, ( ) x 0 x (y y 0) x I b, y J, ˆV (x, y) = (y y 0) b x I, y J, ( ) x 0+a x (y y 0) x I b 3, y J, ( ) x 0 x+ x I, y J 3, x I, y J 3, ( ) x 0+a x+ x I 3, y J 3, where I :={0,, x 0 }, I :={x 0 +,, x 0 + a}, I 3 :={x 0 + a +,, n }, and J :={0,, y 0 }, J :={y 0 +,, y 0 + b}, J 3 :={y 0 + b +,, n }. Of course ˆV fulfills ii). Further we see that ˆv = L T ˆV = ( a + ) R. Then we b obtain for û = f λ ˆv = ( ) (a + b) λ R = c R (7) ab

13 Numer Algor (008) 48: that Thus, for λ J(û) = (a )c + 4 c + (b )c + 4 c = (a + b) c = û, ˆv. ab, the function û in (7)istheminimizerof(0). (a+b) 4 Numerical examples Finally, we present numerical examples. All programs were written in MAT- LAB. Further, we have used the a primal-dual predictor-corrector interior point method implemented in the software package MOSEK [] to solve the quadratic problem with linear constraints (). To visualize the images we have Fig. Top left: noisy image with standard deviation 50. Top right: smoothed rotation angles used to denoise the image with parameters μ = and σ =.5. Bottom left: denoised image by quadratic programming with λ = 800 and ν = 0.. Bottom right: denoised image by anisotropic diffusion with ε = 0.00, time step size τ = 0., number of iterations itmax = 8000 and ν = 0.00

14 6 Numer Algor (008) 48: Fig. 3 Top left: noisy image containing sheared rectangles with fixed shear parameter s = 0.9 and white Gaussian noise of standard deviation 50. Top right: smoothed rotation angles used to denoise the image with parameters μ = and σ =.5. Bottom left: denoised image by quadratic programming with λ = 50 and ν = Bottom right: denoised image by anisotropic diffusion with ε = 0.00, time step size τ = 0., number of iterations itmax = 800 and ν = 0.0 used the MATLAB imagesc routine which incorporates an affine gray value scaling to use the full gray value map. The parameters are chosen with respect to the visually best results. The first two examples in Figs. and 3 demonstrate the denoising of rotated rectangles and parallelograms obtained by applying a shear matrix with s = 0.9. In both examples it suffices to perform only one angle adaptation and restoration step. We show the results for the restoration by quadratic programming and anisotropic diffusion. We observe a slight smoothing of the edges in Fig. 3 in the second approach. This can be reduced by choosing a smaller parameter ν with the disadvantage that checkerboard effects become visible. Since the images are depicted with the affine gray value scaling of the MATLAB imagesc routine we remark that the computed minimal/maximal gray values are given as follows: Fig. bottom left: , 0.3, Fig. bottom right: 7, 3.,

15 Numer Algor (008) 48: Fig. 4 Top left: original image of a city area as presented in [3]. Top right: smoothed rotation angles used to obtain the final image at bottom left with parameters μ = 0000 and σ = 0.8. The smoothed rotation angles belonging to the anisotropic diffusion at the bottom right look very similar. Bottom left: cartoon generated by quadratic programming with λ = 50 and ν = 0. Bottom right: cartoon generated by anisotropic diffusion with ε = 0., time step size τ = 0.05, number of iterations itmax = 700 and ν = 00 Fig. 3 bottom left: 0.5, 49.5, Fig. 3 bottom right: 7.8, 5.4. The original gray values ranged from 0 to 55. The third example in Fig. 4 depicts the cartoon extraction from a real-world image which was also presented in [3]. For comparison we found it useful to take the same image as in [3]. We have iterated our two steps three times. Finally, we consider an image from [] (Fig. 5) whichisoftenusedasa challenging example for the comparison of denoising algorithms, see, e.g., [6, 8]. We present the results after 4 iterations with quadratic programming and anisotropic diffusion. Moreover we show versions without and with rescaling to get an impression of the loss of contrast. Here the minimal/maximal

16 64 Numer Algor (008) 48: Fig. 5 Top left: noisy image. Top right: smoothed rotation angles used to generate the final image on the left with parameters μ = 0000 and σ =.5. The smoothed rotation angles belonging to the anisotropic diffusion look very similar. Middle left: denoised image by quadratic programming with λ = 50 and ν = 0. without rescaling. Middle right: denoised image by anisotropic diffusion with ε = 0.5, time step size τ = 0., number of iterations itmax = 700 and ν = 300 without rescaling. Bottom: same as in the middle with identical rescaling

17 Numer Algor (008) 48: gray values are 06.4, 63.6 for quadratic programming and 95., 73.3 for anisotropic diffusion. 5 Summary and conclusions We have proposed a method for the restoration of rectangular shapes contaminated with heavy noise which avoids the round-off effects at vertices produced by known edge-preserving denoising techniques. As in a paper of Berkels et al. our procedure approximates the rotation angle of the shapes in a first step and uses this information in a second step to denoise the image without destroying vertices. Our angle adaptation uses only first order derivatives of the linearly smoothed image and requires to solve of a linear system of equations. For the second step, we have proposed two different methods, namely quadratic programming and an anisotropic diffusion process with the diffusion tensor adapted to the rotation angle. So far, the algorithm works for rectangles and linearly transformed rectangles, where the linear transform has to be known in advance. We have demonstrated this also numerically for sheared rectangles (parallelograms) with fixed shear parameter. Our ongoing work aims at adapting the shear parameter, too. Moreover, we want to generalize the approach to the restoration of arbitrary (polygonal) shapes. This will incorporate the application of more sophisticated corner detectors and their inclusion into the diffusion tensor of a diffusion equation or an appropriate functional. References. MegaWave: MegaWave homepage. MOSEK: The MOSEK optimization toolbox Berkels, B., Burger, M., Droske, M., Nemitz, O., Rumpf, M.: Cartoon extraction based on anisotropic image classification, vision, modeling and visualization proceedings. In: Vision, Modeling and Visualization, pp Springer (006) 4. Chambolle, A: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 0, (004) 5. Chambolle, A.: Total variation minimization and a class of binary MRF models. In: Rangarajan, A., Vemuri, B.C., Yuille, A.L. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition, EMMCVPR. Lecture Notes in Computer Science, vol Springer, New York (005) 6. Esedoglu, S., Osher, S.: Decomposition of images by anisotropic Rudin-Osher-Fatemi model. Commun. Pure Appl. Math. 57(), (004) 7. Hintermüller, W., Kunisch, W.: Total bounded variation regularization as a bilaterally constrained optimization problem. SIAM J. Appl. Math. 4(64), (004) 8. Mallat, S.: A Wavelet Tour of Signal Processing, nd edn. Academic, San Diego (999) 9. Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell., (990) 0. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (998). Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. A 60, (99)

18 66 Numer Algor (008) 48: Scharr, H.: Optimal Operators in Digital Image Processing. Ph.D. thesis, Interdisciplinary Center for Scientific Computing, Ruprecht-Karls-Universität Heidelberg (000) 3. Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vis., (000) 4. Steidl, G.: A note on the dual treatment of higher order regularization functionals. Computing 76, (006) 5. Weickert, J.: Anisotropic diffusion filters for image processing based quality control. In: Fasano, A., Primicerio, M. (eds.) Proceedings of the Seventh European Conference on Mathematics in Industry, pp Teubner, Stuttgart (994) 6. Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (998) 7. Weickert, J., Scharr, H.: A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J. Vis. Commun. Image Represent. 3( ), 03 8 (00) 8. Welk, M., Steidl, G., Weickert, J.: Locally analytic schemes: a link between diffusion filtering and wavelet shrinkage. Appl. Comput. Harmon. Anal. (008, in press) 9. Winkler, G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Applications of Mathematics, vol. 7. Springer, Berlin (995)