Adaptive Forward-Backward Diffusion Flows in Image Processing


Abstract:


The nonlinear diffusion model introduced by Perona and Malik in 1990 is well suited to preserve salient edges while restoring noisy images. This model overcomes well-known edge smearing effects of the heat equation by using a gradient dependent diffusion function. Despite providing better denoising results, the analysis of the PM scheme is difficult due to the forward-backward nature of the diffusion flow. We study a related adaptive forward-backward diffusion equation which uses a mollified inverse gradient term en-grafted in the diffusion term of a general nonlinear parabolic equation. We prove a series of existence, uniqueness and regularity results for viscosity, weak and dissipative solutions for such forward-backward diffusion flows. In particular, we introduce a novel functional framework for wellposedness of flows of total variation (TV) type. A set of synthetic and real image processing examples are used to illustrate the properties and advantages of the proposed adaptive forward-backward diffusion flows.


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Spatial regularization in the diffusion coefficient alters discontinuities in a synthetic corner image. (a) Original synthetic image of size 31 x 31, a square (gray value = 160) at the bottom right corner with uniform background (gray value = 219). (b) Input image obtained by adding Gaussian std = 30 to the original image. This noisy image is used as the initial value u0 for the nonlinear PDEs with C1 diffusion coefficient and K = 20. Results of PMADE (1) with 20 iterations in (left) image (right) surface format (c), and GRADE (3) with 20 iterations in (left) image (right) surface format (d). The intersection of red dotted lines indicate the exact corner location of the square.

Diffusion process for a simple synthetic image. (a) Original synthetic image of size 31 × 31, a square (2 × 2, gray value = 1) at the center with uniform background(gray value = 0). (b) Input image obtained by adding Gaussian noise !n = 30 to the original image. This noisy image is used as the initial value u0. (c) Diffusion coefficient C1 in (2), with K = 20. This acts as a discontinuity detector and stops the diffusion spread across edges. (d) Flux function C1(|grad u| ) · |grad u| . (e) Result of heat equation with 20 iterations in (left) image (right) surface format. (f) Result of PMADE equation (1) with 20 iterations in (left) image (right) surface format. The white dotted lines indicate the influence region at the center.


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Inverse mollification function when combined with power growth numerator can stop diffusion across edges. Solution of the PDE (*) on noisy synthetic Brain image (noise standard deviation std = 30) with power growth phi(|grad u|)= |grad u|^p, p = 1, 2, 3, 4, 5 (left to right) without inverse mollification (a) g = 0, and with (b) g (x) = x^2 . In both cases we used K = 10^−4 and terminal time 100. It is clear visually that the inverse mollification has an effect in keeping homogenous regions separated by strong edges and avoids leakage.

Reference:

V. B. S. Prasath, J. M. Urbano, D. Vorotnikov. Analysis of adaptive forward-backward diffusion flows with applications in image processing. Inverse Problems, 31, 105008 (30pp), September 2015. doi:10.1088/0266-5611/31/10/105008

Preprint 15-07, Department of Mathematics, University of Coimbra, Portugal.

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