LIC-Based Regularization of Multi-Valued Images David Tschumperlé CNRS UMR 6072 (GREYC/ENSICAEN) - Image Team ICIP’2005, Genova, 11-14 September 2005
Data Regularization • Aim of regularization consists in transforming a noisy signal into a more regular one, while preserving the important image informations (discontinuities). Regularization of a noisy 1D signal Regularization of a noisy 2D image • Several regularization methods exist in the litterature. ⇒ Non-linear diffusion PDE’s are particularly efficient for this task.
PDE Framework in Image Processing • PDE = Partial Differential Equation : ∂t = ∂ 2 I ∂x 2 + ∂ 2 I ∂I ∂y 2 • Using PDE’s in Image Processing : - I represents the data (1D signals or 2D/3D images) we want to process. - t is an extra time variable corresponding to the PDE iterations. ⇒ Iterative Algorithm - One starts from an image I ( t =0) which evolves until convergence, or after a finite number of iterations ( t = t end ). I ( t =0) = I 0 ∂I ( x,y,t ) = β ( x,y,t ) ∂t
PDE’s and Image Regularization • Convolution and linear PDE’s (Koenderink[84], Alvarez-Guichard-etal[92], ...) : 4 πt e − x 2+ y 2 1 ∂I I ( t ) = I ( t =0) ∗ G σ where G σ = ⇐ ⇒ ∂t = ∆ I 4 t • Nonlinear PDE’s (Perona-Malik[90], Alvarez [92], ...) : ∂I ∂t = div ( c ( �∇ I � ) ∇ I ) with c : R − → R Noisy scalar image linear PDE Perona-Malik PDE
PDE Regularization of Scalar Images • A wide range of PDE-based methods have been proposed since Perona-Malik[90], for scalar image regularization I : Ω → R . (Alvarez, Aubert, Barlaud, Blanc-Feraud, Charbonnier, Chan, Cohen, Deriche, Kornprobst, Malladi, Munford, Morel, Nordström, Osher, Perona-Malik, Rudin, Sapiro, Sochen, Weickert,...) Noisy scalar image I : Ω → R Regularized image, using PDE
PDE Regularization of Multivalued Images • Image I : Ω → N of multivalued points : vectors ( N = R n ), matrices ( N = M n ). ( Blomgren-Chan[98], Kimmel-Malladi-Sochen[98], Sapiro-Ringach[96], Tschumperle- Deriche[01,03], Weickert[97,03], ...) Color Image ( N = R 3 ) Scalar PDE, channel by channel Multivalued PDE Noisy 2D vector field ( N = R 2 ) Regularized field
Principle of PDE-based Regularization • PDE regularization is mainly based on local image smoothing. • Local image smoothing is done as follows : – On a edge, smoothing is done only along the edge, to preserve it. – On homogeneous regions, smoothing is done isotropically (in all directions).
How the smoothing is done ? • Let I : Ω → R n be a noisy multi-valued image. • Smoothing depends on the local geometry of I . Computation of the smoothed i ∇ I i ∇ I T structure tensor field G σ = G ∗ G σ : ∀ ( x, y ) ∈ Ω , G ( x , y ) = � i • Eigenvalues λ + , λ − and eigenvectors θ + , θ − of G describe the local configuration of I at point ( x, y ) . ⇒ Definition of a diffusion tensor field T from G that will tell how the smoothing is performed. For instance : 1 f 1 ( s ) = 1+ s p T = f 1 ( λ + + λ − ) θ − θ T − + f 2 ( λ + + λ − ) θ + θ T with + 1 f 2 ( s ) = √ 1+ s q
How the smoothing is done ? (2) • Then, the smoothing itself is performed by the application of one or several PDE iterations : ∂I i ∂I i ∂t = div ( T ∇ I i ) or ∂t = trace ( TH i ) ⇒ The smoothing behavior of the PDE process follows then the tensor field T : Application of a diffusion PDE on a color image, following a synthetic tensor field T . ⇒ Efficient regularization of images when T is correctly defined.
LIC : Line Integral Convolution • LIC has been proposed by Cabral & Leedom in 93 as a method to visualize vector flows F : R 2 → R 2 . ⇒ Starting from a pure noisy image I noise , compute for each pixel X = ( x, y ) an averaging of the image intensities along integral curves C X of F : ∀ ( x, y ) ∈ Ω , C X = X � + ∞ (0) ( x , y ) = 1 I LIC f ( p ) I noise ( C X ( p ) ) dp where N ∂ C X −∞ ( a ) F ( C X = ( a ) ) ∂a • From smoothing purposes, on may choose f ( p ) to be gaussian.
LIC : Line Integral Convolution (2) • Smooth locally the image in different directions, following a vector field. ⇒ This suggests this can be used for PDE-based smoothing following a tensor field.
Contribution : Mixing LIC’s and PDE’s • We propose a LIC-based process that smoothes an image along a tensor field T , where T is defined as in the PDE-based regularization processes. • We decompose a smoothing along a tensor T into several smoothing processes along vectors w θ = TU θ , where U θ = (cos θ, sin θ ) : – If T ( X ) is isotropic then w θ ( X ) = α U θ . – If T ( X ) is anisotropic and directed along U θ , then w θ ( X ) ≃ α U θ . ( X ) ≃ ˜ – If T ( X ) is anisotropic and orthogonal to U θ , then w θ 0 . ⇒ The more U θ represents a part of T , the higher will be the norm � w θ � .
LIC-based smoothing along diffusion tensors • One replace one PDE iteration by a multiple LIC computation : � π � dt � w θ ( X ) � = 1 I regul f ( a ) I noisy ( C θ ( X , a ) ) da d θ ( X ) N − dt � w θ ( X ) � 0 � � where f () is a gaussian, N = f ( a ) dadθ , and dt is the overall smoothing strength. � C θ = X ( X , 0) ∂ C θ w θ ( C θ ( X , a ) ) = T ( C θ ∂a ( X , a ) = ( X , a ) ) U ( θ )
Properties ⇒ Maximum principle is verified (only averaging of pixel values). ⇒ Very stable and fast algorithm compared to classical PDE implementations : Time step ( dt ) can be large, process remains stable. ⇒ Corners and curved structures are particularly well preserved. (b) PDE-based (a) Original image (c) LIC-based (explicit Euler scheme)
Preservation of curved structures • Smoothing processes are done around the corners , taking into account the curvature of the image structures. • LIC’s naturally provide sub-pixel accuracy for the smoothing.
Applications : Image Denoising “Baboon” (detail) 512x512 (1 iter., 19s) “Tunisia” (detail) 555x367 (1 iter., 11s)
Applications : Image Denoising (2) “Lena” (detail) 256x256 (1 iter., 6.4s) “Chris” (detail) 293x306 (1 iter., 5.6s)
Applications : Image Denoising (3) “Penguin” (detail) 355x287 (1 iter., 12.8s) “Farm” (detail) 460x365 (1 iter., 26s)
Applications : Image Inpainting and Reconstruction “Parrot” 500x500 (200 iter., 4m11s) “Owl” 320x246 (10 iter., 1m01s)
Applications : Image Interpolation (a) Original color image (b) Bloc interpolation (b) Linear interpolation (b) Bicubic interpolation (b) PDE-based interpolation
Applications : Image Interpolation (2) “Nude” (1 iter., 20s) “Forest” (1 iter., 5s)
Conclusions & Perspectives • Very simple and efficient regularization process for multi-valued images. • Mix between PDE and LIC based techniques. ⇒ Perspectives : ’Curvature-preserving PDE’ corresponding to our tensor-directed LIC formulation : Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE’s. D. Tschumperlé Research Report : “Les cahiers du GREYC”, No 05-01, January 2005.
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