Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

/pdf/anisotropic-diffusion-in-image-processing-z13u09qvsm.pdf

Anisotropic diffusion in image processing

A survey on evaluation methods for image segmentation

http://www.bioimageanalysis.org/wp/wp-content/uploads/formidable/Survey_unsupervisedEvaluation.pdf

Image segmentation evaluation: A survey of unsupervised methods

A general framework for multiresolution image fusion: from pixels to regions

The completion of interrupted lines or the enhancement of flow-like structures is a challenging task in computer vision, human vision, and image processing. We address this problem by presenting a multiscale method in which a nonlinear diffusion filter is steered by the so-called interest operator (second-moment matrix, structure tensor). An m-dimensional formulation of this method is analysed with respect to its well-posedness and scale-space properties. An efficient scheme is presented which uses a stabilization by a semi-implicit additive operator splitting (AOS), and the scale-space behaviour of this method is illustrated by applying it to both 2-D and 3-D images.

Coherence-Enhancing Diffusion Filtering

Heuristic linking models in multiscale image segmentation

Probabilistic segmentation of partial volume voxels

A validation methodology for image segmentation methods is explored that is based on quality constrained cost analysis. In this methodology a segmentation method is evaluated by the cost reduction it provides relative to the cost of a full-interactive (manual) segmentation. This cost is constrained by a quality threshold, so that less-than-perfect segmentations are allowed. In this way, segmentation methods which are entirely different in nature can be compared objectively. The validation methodology is presented in its most general form, along with an example of its application to the comparison of segmentation methods. >

A methodology for the validation of image segmentation methods

Validation of the interleaved pyramid for the segmentation of 3D vector images

An improvement in the design of the hyperstack, a three-dimensional image segmentation tool, is described. It is based on a multiresolution approach which is mathematically supported by the diffusion equation. The blurring strategy, used to build the scale space, is outlined, including some difficulties that occur in view of the transition from 2-D to 3-D. The existing prototype, a hyperstack grounded on isointensity following, is extended by two novel ideas: weighted linking and stand-alone parents. The result of a segmented 3D SPECT image (of a liver) is shown. Theoretical considerations concerning the addition of feature information to guide the segmentation process are briefly mentioned. A flexible way to obtain several output images from one single hyperstack is outlined and the reduction of the sampling rate by means of interpolation, which will decrease the total amount of processing time, is investigated. >

Multiresolution segmentation of 3D images by the hyperstack

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