An extension of min/max flow framework

In this paper, the min/max flow scheme for image restoration is revised. The novelty consists of the fol- 24 lowing three parts. The first is to analyze the reason of the speckle generation and then to modify the 25 original scheme. The second is to point out that the continued application of this scheme cannot result 26 in an adaptive stopping of the curvature flow. This is followed by modifications of the original scheme 27 through the introduction of the Gradient Vector Flow (GVF) field and the zero-crossing detector, so as 28 to control the smoothing effect. Our experimental results with image restoration show that the proposed 29 schemes can reach a steady state solution while preserving the essential structures of objects. The third is 30 to extend the min/max flow scheme to deal with the boundary leaking problem, which is indeed an 31 intrinsic shortcoming of the familiar geodesic active contour model. The min/max flow framework pro- 32 vides us with an effective way to approximate the optimal solution. From an implementation point of 33 view, this extended scheme makes the speed function simpler and more flexible. The experimental 34 results of segmentation and region tracking show that the boundary leaking problem can be effectively 35 suppressed

PDE-based morphology for matrix fields : numerical solution schemes

Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets

PDE Based Enhancement of Color Images in RGB Space

International audienceA novel method for color image enhancement is proposed as an extension of scalar diffusion-shock filter coupling model, where noisy and blurred images are denoised and sharpened. The proposed model is based on using single vectors of the gradient magnitude and the second derivatives as a technique to relate different color components of the image. This model can be viewed as a generalization of Bettahar-Stambouli filter to multi-valued images. The proposed algorithm is more efficient than the mentioned filter and some previous works on color image denoising and deblurring without creating false colors

Theoretical foundations for 1-D shock filtering

While shock filters are popular morphological image enhancement methods, no well-posedness theory is available for their corresponding partial differential equations (PDEs). By analysing the dynamical system of ordinary differential equations that results from a space discretisation of a PDE for 1-D shock filtering, we derive an analytical solution and prove well-posedness. We show that the results carry over to the fully discrete case when an explicit time discretisation is applied. Finally we establish an equivalence result between discrete shock filtering and local mode filtering

Regularised Diffusion-Shock Inpainting

We introduce regularised diffusion--shock (RDS) inpainting as a modification of diffusion--shock inpainting from our SSVM 2023 conference paper. RDS inpainting combines two carefully chosen components: homogeneous diffusion and coherence-enhancing shock filtering. It benefits from the complementary synergy of its building blocks: The shock term propagates edge data with perfect sharpness and directional accuracy over large distances due to its high degree of anisotropy. Homogeneous diffusion fills large areas efficiently. The second order equation underlying RDS inpainting inherits a maximum--minimum principle from its components, which is also fulfilled in the discrete case, in contrast to competing anisotropic methods. The regularisation addresses the largest drawback of the original model: It allows a drastic reduction in model parameters without any loss in quality. Furthermore, we extend RDS inpainting to vector-valued data. Our experiments show a performance that is comparable to or better than many inpainting models, including anisotropic processes of second or fourth order

VARIATIONAL METHODS FOR IMAGE DEBLURRING AND DISCRETIZED PICARD\u27S METHOD

In this digital age, it is more important than ever to have good methods for processing images. We focus on the removal of blur from a captured image, which is called the image deblurring problem. In particular, we make no assumptions about the blur itself, which is called a blind deconvolution. We approach the problem by miniming an energy functional that utilizes total variation norm and a fidelity constraint. In particular, we extend the work of Chan and Wong to use a reference image in the computation. Using the shock filter as a reference image, we produce a superior result compared to existing methods. We are able to produce good results on non-black background images and images where the blurring function is not centro-symmetric. We consider using a general Lp norm for the fidelity term and compare different values for p. Using an analysis similar to Strong and Chan, we derive an adaptive scale method for the recovery of the blurring function. We also consider two numerical methods in this disseration. The first method is an extension of Picards method for PDEs in the discrete case. We compare the results to the analytical Picard method, showing the only difference is the use of the approximation versus exact derivatives. We relate the method to existing finite difference schemes, including the Lax-Wendroff method. We derive the stability constraints for several linear problems and illustrate the stability region is increasing. We conclude by showing several examples of the method and how the computational savings is substantial. The second method we consider is a black-box implementation of a method for solving the generalized eigenvalue problem. By utilizing the work of Golub and Ye, we implement a routine which is robust against existing methods. We compare this routine against JDQZ and LOBPCG and show this method performs well in numerical testing

Adaptive continuous-scale morphology for matrix fields

In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method

A new anisotropic diffusion method, application to partial volume effect reduction

The partial volume effect is a significant limitation in medical imaging that results in blurring when the boundary between two structures of interest falls in the middle of a voxel. A new anisotropic diffusion method allows one to create interpolated 3D images corrected for partial volume, without enhancement of noise. After a zero-order interpolation, we apply a modified version of the anisotropic diffusion approach, wherein the diffusion coefficient becomes negative for high gradient values. As a result, the new scheme restores edges between regions that have been blurred by partial voluming, but it acts as normal anisotropic diffusion in flat regions, where it reduces noise. We add constraints to stabilize the method and model partial volume; i.e., the sum of neighboring voxels must equal the signal in the original low resolution voxel and the signal in a voxel is kept within its neighbor's limits. The method performed well on a variety of synthetic images and MRI scans. No noticeable artifact was induced by interpolation with partial volume correction, and noise was much reduced in homogeneous regions. We validated the method using the BrainWeb project database. Partial volume effect was simulated and restored brain volumes compared to the original ones. Errors due to partial volume effect were reduced by 28% and 35% for the 5% and 0% noise cases, respectively. The method was applied to in vivo "thick" MRI carotid artery images for atherosclerosis detection. There was a remarkable increase in the delineation of the lumen of the carotid artery

Shock filters based on implicit cluster separation

One of the classic problems in low level vision is image restoration. An important contribution toward this effort has been the development of shock filters by Osher and Rudin (1990). It performs image deblurring using hyperbolic partial differential equations. In this paper we relate the notion of cluster separation from the field of pattern recognition to the shock filter formulation. A kind of shock filter is proposed based on the idea of gradient based separation of clusters. The proposed formulation is general enough as it can allow various models of density functions in the cluster separation process. The efficacy of the method is demonstrated through various examples