ADI splitting methods for image osmosis models

Transcription

1 ADI splitting methods for image osmosis models L. Calatroni, C. Estatico, N. Garibaldi, S. Parisotto 3 CMAP, Ecole Polytechnique 98 Palaiseau Cedex, France Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 646, Genova, Italy 3 Cambridge Centre for Analysis, Wilberforce Road, CB3 0WA, University of Cambridge, UK luca.calatroni@polytechnique.edu, estatico@dima.unige.it, garibaldi9nicola@gmail.com, sp75@cam.ac.uk Abstract. We consider Alternating Directional Implicit ADI) splitting schemes to compute efficiently the numerical solution of the PDE osmosis model considered by Weickert et al. in [0] for several imaging applications. The discretised scheme is shown to preserve analogous properties to the continuous model. The dimensional splitting strategy traduces numerically into the solution of simple tridiagonal systems for which matrix factorisation techniques are used to improve upon the performance of classical implicit methods even for large time steps. Applications to shadow removal and inpainting problems are presented.. Introduction Several imaging tasks can be formulated as the problem of finding a reconstructed version of a degraded and possibly under-sampled image. Removing signal oscillations image denoising) and optical aberrations image deblurring) due to acquisition and transmission faults are common problems in applications such as medical imaging, astronomy, microscopy and many more. Further, the image interpolation problem of filling in missing or occluded parts of the image using the information from the surrounding areas is called image inpainting [3, 8]. More sophisticated imaging tasks aim to decompose the acquired image into its component structures, such as cartoon and texture, or into the product of an intrinsic component not subject to lightchanges e.g. illumination) and its counterpart describing illumination changes only, see [4]. The problem of shadow removal belongs to this latter class [5]. For a given image f, it can be formulated as find u s.t. fx, y) = lx, y)ux, y), x, y) Ω, ) where Ω R denotes the image domain, lx, y) the illumination image and u the reflectance image, i.e. the intrinsic image not subject to illumination changes []. Due to the ill-posedness of the problem ), in [] a maximum-likelihood approach is used. In [9, 0], a PDE-based approach computing the solution of ) as the stationary state of a linear parabolic PDE is considered. Due to its similarities with the analogous physical process, such model is called image osmosis. Image osmosis. Given a rectangular image domain Ω R with Lipschitz boundary Ω and a positive initial gray-scale image f : Ω R +, for a given vector field d : Ω R the linear

2 image osmosis model proposed in [9, 0] is a drift-diffusion PDE which computes for every t 0, T ], T > 0 a regularised family of images {ux, t)} t>0 of f by solving: t u = u divdu) on Ω 0, T ] ux, 0) = fx) on Ω u du, n = 0 on Ω 0, T ], where, denotes the Euclidean scalar product and n the outer normal vector on Ω. The extension to RGB images is straightforward by letting the process evolve for each colour channel. In [0, Proposition ] the authors prove that any solution of ) preserves the average gray value and the non-negativity at any time t > 0. Furthermore, by setting d := ln v) for a reference image v > 0, the steady state w of ) turns out to be a minimiser of the energy functional: ˆ u ) Eu) = v dx, 3) v Ω and can be expressed as a rescaled version of v via the formula wx) = µ f µ v vx), where µ f and µ v are the average of f and v over Ω, respectively. In [0] the osmosis model ) is used as a non-symmetric variant of diffusion for several visual computing applications data compression, face cloning and shadow removal). A general fully discrete framework for this problem is studied in [9] where, under some standard conditions on the semi-discretised finite differences operators used, the conservation properties described above are shown to be still valid upon discretisation. We synthesise these results as follows: Theorem. [9] For a given f R N +, consider the semi-discretised linear osmosis problem: u0) = f, u t) = Aut), t > 0, 4) where A R N N is an irreducible non-symmetric) matrix with zero-column sum and nonnegative off-diagonal entries. For k 0, consider the time-discretisation schemes u k+ = P u k : P := I + τa), for τ < max a i,i ) ; Forward Euler) P := I τa), for any τ > 0. Backward Euler) Then, in both cases, P is an irreducible, non-negative matrix with strictly positive diagonal entries and unitary column sum and for ut) the following properties hold true: i) For every t > 0, the evolution preserves positivity and the average gray value of f; ii) The unique steady state is the eigenvector of P associated to eigenvalue. For large τ > 0, the numerical realisation via implicit Euler method requires the inversion of a non-symmetric penta-diagonal matrix, which may be highly costly for large images. Alternatively, problem 4) can be solved using dimensional splitting methods, [6]. ADI splitting methods. Given a domain Ω R s, a dimensional splitting method decomposes the semi-discretised operator A of initial boundary value problem 4) into the sum: A = A 0 + A A s 5) such that the components A j, j =,..., s, encode the linear action of A along the space direction j =,..., s, respectively. The term A 0 may contain additional contributes coming from mixed directions and non-stiff nonlinear terms. Alternating directional implicit ADI) schemes are time-stepping methods that treat the unidirectional components A j, j implicitly )

3 and the A 0 component, if present, explicitly in time. Having in mind a standard finite difference space discretisation, this splitting idea translates into reducing the s-dimensional original problem to s one-dimensional problems. For D applications, we set s =. Let u i,j denote an approximation of u in the grid of size h at point i )h, j )h); the discretisation of ) considered in [9, 0] reads: u i,j = u i+,j u i,j + u i,j h + u i,j+ u i,j + u i,j h h h u i+,j + u i,j d,i+,j d,i,j+ u i,j+ + u i,j ) u i,j + u i,j d,i,j 6) ) u i,j + u i,j d,i,j =: A + A )ut) which can be splitted into the sum of A and A where no mixed and/or nonlinear terms appear A 0 = 0). Following [6], we recall now the two ADI schemes considered in this paper. The Peaceman-Rachford scheme. The first method considered is the second-order accurate Peaceman-Rachford ADI scheme. For every k 0 and time-step τ > 0, compute an approximation u k+ via the updating rule: u k+/ = u k + τ A u k + τ A u k+/, u k+ = u k+/ + τ A u k+ + τ 7) A u k+/. Heuristically, forward and backward Euler are applied alternatively in a symmetrical way, thus resulting in second-order accuracy [6]. The Douglas scheme. A more general ADI decomposition accommodating also the general case A 0 0 in 5) is the Douglas method. For k 0, τ > 0 and θ [0, ], the updating rule reads y 0 = u k + τau k y j = y j + θτa j y j A j u k ), j =, 8) u k+ = y. In words, the numerical approximation in each time step is computed by applying at first a forward Euler predictor and then it is stabilised by intermediate steps where just the unidirectional components A j of the splitting 5) appear, weighted by θ whose size balances the implicit/explicit behaviour of these steps. The time-consistency order of the scheme is equal to two for A 0 = 0 and θ = /, and it is of order one otherwise. For s = both schemes 7) and 8) are unconditionally stable, see [6]. However, for large time steps, the time-accuracy may suffer due to the presence of the explicit steps. Scope of the paper. In this work, we apply schemes 7) and 8) to solve 4) and show that similar conservation properties as in Theorem hold. Compared to standard implicit methods used in [9, 0], such schemes renders a much more efficient and accurate numerical solution.. ADI methods for image osmosis We show that ADI schemes 7) and 8) show similar conservation properties as in Theorem. { }). Proposition. Let f R N + and τ < max max a i,i, max a i,i Then, the Peaceman- Rachford scheme 7) on the splitting 6) preserves the average gray value, the positivity and converges to a unique steady state.

4 Proof. We write the Peaceman-Rachford 7) iteration as u k+ = I τ ) A I + τ ) A I τ ) A I + τ ) A u k, and observe that for i =, the matrices P i := I τ A ) i and P + i := I + τ A i), are non-negative and irreducible with unitary column sum and positive diagonal entries being each A i a one-dimensional implicit/explicit discretised osmosis operator satisfying the assumptions of Theorem. Therefore, at every implicit/explicit half-step and using the restriction on τ, the average gray-value and positivity are conserved. Furthermore, the unique steady state is the eigenvector of the operator P := P P + P P + associated to the eigenvalue to one. The following lemma is useful to prove a similar result for the Douglas scheme 8). Lemma. If C = c i,j ) R N N and D = d i,j ) R N N s.t. N i= c i,j =: c and N i= d i,j =: d for every j =,..., N, then, the matrix B := CD has column sum equal to cd. Proof. Writing each element b i,j in terms of the elements of C and D, we have that for every j: b i,j = i= c i,k d k,j = d N ) k,j c i,k = ) d k,j c = c d k,j = cd. i= k= k= i= k= k= Proposition. Let f R N +, τ > 0 and θ [0, ]. Then, the Douglas scheme 8) applied to the splitted semi-discretised scheme 6) preserves the average gray value. Proof. For every k 0, we write the Douglas iteration 8) as [ u k+ = I θτa ) I θτa ) ) ] I θτa ) + τa θτa ) ) = I + τi θτa ) I θτa ) A u k =: I + τp P A where both P and P are non-negative, irreducible, with unitary column sum and strictly positive diagonal entries by Theorem, while A = A + A is zero-column sum being the standard discretised osmosis operator 6). By Lemma, the operator B := P P A has zero column sum and the operator P := I + τb has unitary column sum. Thus, for every k 0: u k u k N i= u k+ i = N i= j= p i,j u k j = N N ) p i,j u k j = N j= i= u k j. j= Remark. There is no guarantee that the non-diagonal entries of the operator B are nonnegative. { Therefore, it is not possible to apply directly Theorem to conclude that the iterates u k } remain positive and that they converge to a unique steady state. However, our k 0 numerical tests suggest that both properties remain valid. A rigorous proof of these properties is left for future research. 3. Numerical results In the following Algorithm and we present the pseudocode for the Peaceman-Rachford 7) and the Douglas 8) schemes, respectively, when applied to images of size m n pixels, with N := mn. We recall that in [9, 0], the non-splitted Euler Implicit method is solved iteratively by BiCGStab, which is influenced by the tolerance and maximum iterations required.

5 Due to the structure of the ADI matrices A and A, a tridiagonal LU factorisation can be used for a better efficiency after a permutation on matrix A based on [7] to reduce the bandwidth to one: this is convenient for images with large m, where non-splitted methods become prohibitively expensive. In the non-splitted case, one LU factorisation plus T τ system resolutions for the penta-diagonal matrix A, with lower and upper bandwidth equal to m, costs Om 3 n + 4T τ m n) flops overall. In the splitted case, the computational cost is reduced to O6mn ) + T τ 0mn 8)) flops because of the use of tridiagonal -bandwidth matrices. For the same reason, the inverse splitted operators A and A require less storage space than the non-splitted operator A while the inverse of A has significantly more non-zeros entries. Since τ is constant, all the LU factorisations can be computed only once before the main loop. Algorithm : LU Peaceman-Rachford Input : u 0, A = A + A ; Parameters : τ, T ; foreach c in color channel do p c = symrcma c ); E c = I τ A c ) ; E c = I τ A c p c ; p c ) ) ; [L c, U c ] = lu I 0.5 τ A c ); [L c, U c ] = lu I 0.5 τ A c p c ; p c ) ). Algorithm : LU-Douglas Input : u 0, A = A + A ; Parameters : τ, T, θ [0, ]; foreach c in color channel do p c = symrcma c ); E c = τ A c ; E c = τ A c p c ; p c ) T c = τ θ A c ; T c = τ θ A c p c ; p c ) ; [L c, U c ] = lu I τ θ A c ); [L c, U c ] = lu I τ θ A c p c ; p c ) ). for k = 0 to T τ do z = E c u k c ; y p c ) = U c \ L c \ z p c )); z p c ) = E c y p c ); u k+τ c = U c \ L c \ z ); for k = 0 to T τ do y 0 = E c u k c ; y 0 p c ) = E c u k c p c ) ; z = u k c + y 0 + y 0 T c u k c ; y = U c \ L c \ z ); z p c ) = y p c ) T c u k c ; u k+τ c p c ) = U c \ L c \ z p c )); return u T. return u T. As a toy example, we evolved the osmosis model ) starting from an initial constant image, with a known steady state. In Table we compare the non-splitting and splitting solutions at T = 000 for different fixed time-steps τ and θ = {0.5, } where exact-in-time reference solution is computed by Exponential Integrators []. For a comparison with [9], the non-splitted approach is solved by BiCGStab in our case tol=e-5 and maxiter=e6) and also test the direct LU factorisation. The order of the methods for the Energy 3) error E.E.) are plotted in Figure a. For larger time-steps, ADI methods are 0 faster than non-splitting methods. Finally, in Figure b the ADI numerical solution of the shadow-removal problem as in [0] is shown, starting from a shadowed image u and setting d = ln u) = 0 on the shadow boundary. There, in order to correct the diffusion artefacts, an inpainting correction e.g. []) is applied. 4. Conclusions In this paper, we consider the image osmosis model proposed in [9, 0] for several imaging applications. We proposed an ADI splitting approach to decompose the -D finite difference operator into two one-dimensional tridiagonal operators. We proved that the splitting numerical schemes preserve properties holding for the continuous model. Numerically, these methods are efficient and accurate. We tested the proposed methods for a shadow removal problem, with an inpainting post-processing step. Future research directions include a rigorous stability and convergence analysis of the Douglas method 8) and the formulation of a nonlinear) model driving the diffusion on the shadow mask to avoid the inpainting correction.

6 Table : Results from non-splitting and ADI splitting p-order methods. Input pixels. τ = 0. τ = τ = 0 Method θ p Time E.E. Time E.E. Time E.E. BiCGStab 5 d.) LU 5 d.) Douglas ADI s s s.84e-0.84e-0.84e s s s..00e-09.84e-09.84e s s s..9e-08.85e-08.85e-08 BiCGStab 5 d.) LU 5 d.) Douglas ADI P.-R. ADI s s s s. 5.56e e-5.00e-5.5e-4 3. s. 5.9 s s. 5.8 s. 5.58e e-3.00e-3.5e- 0.9 s. 5.7 s s s. 8.00e e-09.96e-.53e energy error Order Order BiCGStab 5d. = 0.5 BiCGStab 5d. = LU 5d. = 0.5 LU 5d. = Douglas ADI = 0.5 Douglas ADI = P-R ADI dt a) ADI orders for Energy Error. b) Shadow removal + inpainting. Figure : a: Energy error at T = 000 for several τ. b: Input with shadow boundary top); Douglas ADI solution and result after non-local inpainting [] bottom). Parameters: τ = 0; θ = 0.5. Image from: Acknowledgments LC acknowledges the joint ANR/FWF Project Efficient Algorithms for Nonsmooth Optimization in Imaging EANOI) FWF n. I48 / ANR--IS CE acknowledges the support of GNCS-INDAM. SP was supported by the UK Engineering and Physical Sciences Research Council EPSRC) grant EP/L0656/. References [] Arias P, Facciolo G, Caselles V and Sapiro G 0 ICJV [] Al-Mohy and A.H. Higham N J 0 SIAM J. Sci. Comput [3] Aubert G and Kornprobst P 006 Mathematical problems in image processing: Partial Differential Equations and the Calculus of Variations nd ed Applied Mathematical Sciences Springer) ISBN [4] Barrow H and Tanenbaum J M 978 Computer Vision Systems in A. Hanson and E. Riseman eds Academic Press) 3-6 [5] Finlayson G, Hordley S, Lu C and Drew M 00 ECCV in LNCS Springer) [6] Hundsdorfer W and Verwer J 003 Numerical solution of time-dependent advection-diffusion-reaction equations Computational Mathematics Springer) ISBN [7] Saad Y.003 Iterative Methods for Sparse Linear Systems nd ed SIAM) ISBN [8] Scho nlieb C-B 05 Partial Differential Equation Methods for Image Inpainting Cambridge Monographs on Applied and Computational Mathematics CUP) ISBN [9] Vogel O, Hagenburg K, Weickert and J Setzer S 03 SSVM in LNCS Springer) [0] Weickert J, Hagenburg and K BreußVogel O 03 EMMCVPR in LNCS Springer) [] Weiss Y 00 ICCV in IEEE Proceedings on ICCV 68-75