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Medical imaging often requires a preprocessing step where lters are applied that remove noise while preserving semantically important structures such as edges. This may help to simplify subsequent tasks such as segmentation. One class of recent adaptive denoising methods consists of methods based on nonlinear partial differential equations (PDEs). In the present paper we survey our recent results on PDE-based preprocessing methods that may be applied to medical imaging problems. We focus on nonlinear diffusion filters and variational restoration methods. We explain the basic ideas, sketch some algorithmic aspects, illustrate the concepts by applying them to medical images such as mammograms, computerized tomography (CT), and magnetic resonance (MR) images. In particular we show the use of these filters as preprocessing steps for segmentation algorithms.

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... Particularly the anisotropic diffusion model originally introduced by Perona and Malik [16] and further developed by [1], [5]. Other anisotropic diffusion techniques such as curve evolution methods based on geometric scale space [10]- [12], and the construction of diffusion tensors that contain the information of both the modulus and directions of the gradients [21]- [23] have also been developed. The basic idea of most anisotropic diffusion techniques is to employ the gradient (both modulus and direction) of the intensity to detect the edges between regions then smooth the image within the homogeneous region and along the edges but not across the boundaries of such regions. ...

... Therefore, diffusion filtering only happens within different homogeneous regions, but not across the boundaries of such regions. The smoothing approach is very similar to the idea of "anisotropic diffusion" [1], [16], [21]- [23]. Many advantages can be achieved for image segmentation and denoising in this piecewise smooth approach, such as simultaneous segmentation and smoothing of noisy images, detection of triple junctions by using multiple level-set functions [20] (or the approach in [18]), and smoothing the images with complex features [19], [20]. ...

... The key point of the anisotropic diffusion approach is how to construct the diffusion tensor . Weickert proposed two different ways to choose the diffusion tensor for different diffusion goals, namely the edge-enhancing and coherence-enhancing anisotropic diffusions [21]- [23]. Among the PDE diffusion approaches for image denoising, the anisotropic approach gives the highest performance. ...

Recently, Chan and Vese developed an active contour model for image segmentation and smoothing by using piecewise constant and smooth representation of an image. Tsai et al. also independently developed a segmentation and smoothing method similar to the Chan and Vese piecewise smooth approach. These models are active contours based on the Mumford-Shah variational approach and the level-set method. In this paper, we develop a new hierarchical method which has many advantages compared to the Chan and Vese multiphase active contour models. First, unlike previous works, the curve evolution partial differential equations (PDEs) for different level-set functions are decoupled. Each curve evolution PDE is the equation of motion of just one level-set function, and different level-set equations of motion are solved in a hierarchy. This decoupling of the motion equations of the level-set functions speeds up the segmentation process significantly. Second, because of the coupling of the curve evolution equations associated with different level-set functions, the initialization of the level sets in Chan and Vese's method is difficult to handle. In fact, different initial conditions may produce completely different results. The hierarchical method proposed in this paper can avoid the problem due to the choice of initial conditions. Third, in this paper, we use the diffusion equation for denoising. This method, therefore, can deal with very noisy images. In general, our method is fast, flexible, not sensitive to the choice of initial conditions, and produces very good results.

... The circuit is structured as a Cellular Nonlinear Network (CNN). Its elementary cell is derived from the functional introduced by Weickert in [1], by applying the synthesis method defined in [2]. In the processed image (obtained by numerical simulations of the model), the significant information present in the original image is preserved, whereas both noise and the less relevant details of the original image are smoothed. ...

... The function Γ is assumed to be convex, in order to guarantee the function û (x,y) to be unique [4]. The noise-removal task, in particular, can be accomplished by minimizing the functional introduced by Weickert in [1] where the (convex) function Γ is defined as follows: ...

In this paper the minimization of a functional defined in the context of biomedical image processing is obtained through a cellular nonlinear network (CNN). The CNN performs noise removal in grey-level biomedical images. The performances of the circuit at behavioural level are shown in a case study concerning retinal images.

... Color image is resized with [150*150] using imresize function, after that, median filter [3 3] is used for eliminating noise from images, edges of the image and is preserved using this filter, so this another advantages of median filter [9]. ...

... In recent years, nonlinear diffusion methods (Perona and Malik, 1990) have proved to be very effective in image processing, especially in biomedical applications (Weickert and Schnorr, 2000;Catte et al., 1992;Weickert, 2001;Niessen et al., 1994;Gilboa et al., 2001;Krim et al., 1999;Lysaker et al., 2003;Dong et al., 2003), since they usually require a preprocessing step to remove noise while preserving fine structures such as edges and boundaries in the images. It has been shown that the partial differential equation (PDE)-based method preserves image transitions better than spatially averaging techniques such as moving average and Gaussian filtering (Weickert, 1999;Moisan et al., 2002). ...

... Here, n denotes the outward normal to the domain's boundary ∂Ω, and 〈g·∇u, n〉 indicates the inner product ∫ ∂Ω (g·∇u) · n ds. In this model the diffusivity has to be such that g (|∇u| 2 ) → 0 when |∇u| → ∞ and g (|∇u| 2 ) → 1 when |∇u| → 0. Notwithstanding the practical success of the Perona-Malik model, it presents some serious theoretical problems such as (i) ill-posedness (Nitzberg and Shiota, 1992; Weickert and Schnörr, 2000); (ii) non-uniqueness and instability (Catté et al., 1992; Kichenassamy, 1997); (iii) excessive dependence on numerical regularization (Benhamouda, 1994; Fröhlich and Weickert, 1994). The last observation motivated an enormous amount of research towards the incorporation of the regularization directly into the partial differential equation (PDE), to avoid too much implicit reliance on the numerical schemes. ...

The interpretation and measurement of the architectural organization of mitochondria depend heavily upon the availability of good software tools for filtering, segmenting, extracting, measuring, and classifying the features of interest. Images of mitochondria contain many flow-like patterns and they are usually corrupted by large amounts of noise. Thus, it is necessary to enhance them by denoising and closing interrupted structures. We introduce a new approach based on anisotropic nonlinear diffusion and bilateral filtering for electron tomography of mitochondria. It allows noise removal and structure closure at certain scales, while preserving both the orientation and magnitude of discontinuities without the need for threshold switches. This technique facilitates image enhancement for subsequent segmentation, contour extraction, and improved visualization of the complex and intricate mitochondrial morphology. We perform the extraction of the structure-defining contours by employing a variational level set formulation. The propagating front for this approach is an approximate signed distance function which does not require expensive re-initialization. The behavior of the combined approach is tested for visualizing the structure of a HeLa cell mitochondrion and the results we obtain are very promising.

... and Malik [5] introduced a nonlinear diffusion model for image denoising and seg- mentation in 1987, many other nonlinear diffusion models have been proposed and have proved to be very powerful in the processing of 2-D and 3-D images678910111213 ). Also there exist many nonlinear reaction-diffusion systems applied to image restoration, texture generation, and halftoning. ...

The optimal error estimate O(hk+1) for a popular nonlinear diffusion model widely used in image processing is proved for the standard kth-order (k ≥ 1) conforming tensor-product finite elements in the L2-norm. The optimal L2-estimate is obtained by the integral identity technique [1–3] without using the classic Nitsche duality argument [4].

... This is because the role of shape recovery has always been a critical component in 2-D and 3-D medical and nonmedical imagery. This assists largely in medical therapy, and object detection/tracking in industrial applications, respectively (see the recent book by Suri et al. [4] and the references therein and also see Weickert et al. [23], [24] and the references therein). Shape recovery of medical organs in medical images is more difficult compared to other imaging fields. ...

Partial differential equations (PDEs) have dominated image processing research. The three main reasons for their success are: (1) their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; (2) their ability to solve PDEs in the level set framework using finite difference methods; and (3) their easy extension to a higher dimensional space. The paper is an attempt to summarize PDEs and their solutions applied to image diffusion. The paper first presents the fundamental diffusion equation. Next, the multi-channel anisotropic diffusion imaging is presented, followed by tensor non-linear anisotropic diffusion. We also present the anisotropic diffusion based on PDE and the Tukey/Huber weight function for image noise removal. The paper also covers the recent growth of image denoising using the curve evolution approach and image denoising using histogram modification based on PDE. Finally, the paper presents non-linear image denoising. Examples covering both synthetic and real world images are presented.

... The behaviour of solutions of the Perona-Malik problem have been investigated in several literature [9,27,34,44]. Unfortunately, it has been shown that the classical Perona-Malik equation behaves locally like backward diffusion process which is an ill-posed evolution problem [9,34,63]. To overcome this drawback, Catté et al. introduced a development approach to regularize the nonlinearity term in the equation (1.1) [9]. ...

... Notwithstanding the practical success of the Perona-Malik model, it presents some serious theoretical problems: (i) none of the classical well-posedness frameworks is applicable to the Perona-Malik model, i.e. we can not ensure well-posedness results [34,42]; (ii) uniqueness and stability with respect to the initial image should not be expected, i.e. solvability is a difficult problem, in general [15,21,22,25,36]; (iii) the regularizing effect of the discretization plays too much of an important role in the solution [6,17]. The latter is perhaps the key element in the success or failure of the model. ...

We introduce the use of mimetic methods to the imaging community, for the solution of the initial-value problems ubiquitous in the machine vision and image processing and analysis fields. PDE-based image processing and analysis techniques comprise a host of ap-plications such as noise removal and restoration, deblurring and enhancement, segmentation, edge detection, inpainting, registration, motion analysis, etc. Because of their favorable stability and efficiency properties, semi-implicit finite difference and finite element schemes have been the methods of choice (in that order of preference). We propose a new approach for the numerical so-lution of these problems based on mimetic methods. The mimetic discretization scheme preserves the continuum properties of the mathematical operators often encountered in the image processing and analysis equations. This is the main contributing factor to the improved performance of the mimetic method approach, as compared to both of the aforementioned popular numerical solution techniques. To assess the performance of the proposed approach, we employ the Catté-Lions-Morel-Coll model to restore noisy images, by solving the PDE with the three numerical solution schemes. For all of the benchmark images employed in our experiments, and for every level of noise applied, we observe that the best image restored by using the mimetic method is closer to the noise-free image than the best images restored by the other two methods tested. These results motivate further studies of the application of the mimetic methods to other imaging problems.

... This makes the interpretation of these images very difficult, especially if the image is used as input of a pattern classification algorithm. In medical imaging such an algorithm will normally try to find the type of tissue at each pixel or voxel.There are used approaches for Medical Image Preprocessing that include filters which are applied to remove noise while preserving semantically important structures such as edges-methods based on nonlinear Partial Differential Equations (PDEs)[13]. It is used for medical images such as mammograms, CT and MR images. ...

The content extraction from static images is an actual investigation task, because in the modern information
networks the amount of information, which is stored, shared and processed under this form, is vastly increasing.

... Image smoothing based on PDEs has become an active field. Particularly the anisotropic diffusion model originally was introduced by Perona and Malik [37] and further developed such as [38,39]. Besides the total variation model (the ROF model) proposed by Rudin, Osher and Fatemi in [40], has been extensively studied and proven to be efficient for preserving edges [41,42]. ...

... where α, β and are positive parameters. This diffusivity function is well-suited for denoising purposes [8] (see also [14,15,7,13] for similar filters) and the solution of the resulting diffusion equation is stable under perturbations of the initial data and the parameters. The corresponding diffusive filter is obtained by discretizing both, spatially and temporally, continuous equation (6). ...

... The PM model was designed with the explicit goal of achieving a good trade-off between noise removal and edge preservation. Since the pioneering study by Perona and Malik, many variance-based models have been proposed, including the anisotropic diffusion model proposed by Weickert [4] [5], structure tensor diffusion [6], the manifold diffusion method [7], adaptive anisotropic dif-fusion based on a structure tensor [8], anisotropic diffusion based on band pass signals [9], a modified PM model based on directional Laplacian [10], and automatic parameter selection anisotropic diffusion [11]. The TV model has been studied extensively, thereby demonstrating that it is efficient for removing noise and preserving edges [12] [13]. ...

Recently, variational and partial differential equation (PDE)-based algorithms have become very important for image restoration. In this study, we propose a new second order hyperbolic PDE model based on directional diffusion for image restoration. This hyperbolic PDE restoration model can simply diffuse along the edge’s tangential direction in the observed image, thereby removing noise while preserving the image edges and fine details, which avoids the staircase effect in the restored image. An effective numerical scheme is proposed for handling the computation of our approach using the finite difference method. Successful image restoration experiments demonstrated that the proposed second order hyperbolic PDE-based model obtains superior performance compared with other models at preserving edges and it avoids the staircase effect.

... Notwithstanding the practical success of the Perona-Malik model, it presents some serious theoretical problems: (i) none of the classical well-posedness frameworks is applicable to the Perona-Malik model, i.e. we can not ensure well-posedness results [34,42]; (ii) uniqueness and stability with respect to the initial image should not be expected, i.e. solvability is a difficult problem, in general [15,21,22,25,36]; (iii) the regularizing effect of the discretization plays too much of an important role in the solution [6,17]. The latter is perhaps the key element in the success or failure of the model. ...

In this paper, we propose a modification and a generalization of the theory developed by P. Perona and J. Malik for edge detection and image restoration, from the case of a single equation to the case of nonlinear reaction-diffusion system. We are interested in the existence of weak solutions for this system for which two main properties hold: the positivity of the solutions and the total mass of the components are preserved with time.

Partial Differential Equations (PDEs) have dominated image processing research recently. The three main reasons for their success are: first, their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; second, their ability to solve PDEs in the level set framework using finite difference methods; and third, their easy extension to a higher dimensional space.
This paper is an attempt to survey and understand the power of PDEs to incorporate into geometric deformable models for segmentation of objects in 2D and 3D in still and motion imagery. The paper first presents PDEs and their solutions applied to image diffusion. The main concentration of this paper is to demonstrate the usage of regularizers in PDEs and level set framework to achieve the image segmentation in still and motion imagery. Lastly, we cover miscellaneous applications such as: mathematical morphology, computation of missing boundaries for shape recovery and low pass filtering, all under the PDE framework. The paper concludes with the merits and the demerits of PDEs and level set-based framework for segmentation modeling. The paper presents a variety of examples covering both synthetic and real world images.

In this paper, we develop a new segmentation and smoothing model which has many advantages compared to Chan and Vese's active contours model. In our method, the curve evolution partial differential equations (PDEs) for different level set functions are decoupled and solved separately. This decoupling of the motion equations of the level set functions not only speeds up the segmentation process significantly, it also removes the difficulties associated with the initialization of the level sets in Chan and Vese's method. The proposed method can avoid the initial condition problem. Finally, we use in this paper the diffusion equation for denoising and therefore it can deal with very noisy images.

Segmentation of MR brain images using intensity values is severely limited owing to field inhomogeneities, susceptibility artifacts and partial volume effects. Edge based segmentation methods suffer from spurious edges and gaps in boundaries. A multiscale method to MRI brain segmentation is presented which uses both edge and intensity information. First a multiscale representation of an image is created, which can be made edge dependent to favor intra-tissue diffusion over inter-tissue diffusion. Subsequently a multiscale linking model (the hyperstack) is used to group voxels into a number of objects based on intensity. It is shown that both an improvement in accuracy and a reduction in image post-processing can be achieved if edge dependent diffusion is used instead of linear diffusion. The combination of edge dependent diffusion and intensity based linking facilitates segmentation of grey matter, white matter and cerebrospinal fluid with minimal user interaction. To segment the total brain (white matter plus grey matter) morphological operations are applied to remove small bridges between the brain and cranium. If the total brain is segmented, grey matter, white matter and cerebrospinal fluid can be segmented by joining a small number of segments. Using a supervised segmentation technique and MRI simulations of a brain phantom for validation it is shown that the errors are in the order of or smaller than reported in literature.

In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: architectural requirements like locality, recursivity and causality in the scale space, stability requirements like the comparison principle and morphological requirements, which correspond to shape-preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lanrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t → ∞ the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution a...

©1998 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. DOI: 10.1109/TIP.1998.661176

In contrast to acquisition-based noise reduction methods a postprocess based on anisotropic diffusion is proposed. Extensions of this technique support 3-D and multiecho magnetic resonance imaging (MRI), incorporating higher spatial and spectral dimensions. The procedure overcomes the major drawbacks of conventional filter methods, namely the blurring of object boundaries and the suppression of fine structural details. The simplicity of the filter algorithm permits an efficient implementation, even on small workstations. The efficient noise reduction and sharpening of object boundaries are demonstrated by applying this image processing technique to 2-D and 3-D spin echo and gradient echo MR data. The potential advantages for MRI, diagnosis, and computerized analysis are discussed in detail.

Variational segmentation and nonlinear diffusion approaches have been very active research areas in the fields of image processing and computer vision during the last years. In the present paper, we review recent advances in the development of efficient numerical algorithms for these approaches. The performance of parallel implementations of these algorithms on general-purpose hardware is assessed. A mathematically clear connection between variational models and nonlinear diffusion filters is presented that allows to interpret one approach as an approximation of the other, and vice versa. Numerical results confirm that, depending on the parametrization, this approximation can be made quite accurate. Our results provide a perspective for uniform implementations of both nonlinear variational models and diffusion filters on parallel architectures.

A global edge detection algorithm based on variational regularization is presented and analysed. The algorithm can also be viewed as an anisotropic diffusion method. These two quite different methods are thereby unified from the original outlook. This puts anisotropic diffusion, as a method in early vision, on more solid grounds; it is just as well founded as the well-accepted standard regularization techniques. The unification also brings the anisotropic diffusion method an appealing sense of optimality, thereby intuitively explaining its extraordinary performance. The algorithm to be presented, moreover, has the following attractive properties:1. It only requires the solution of a single boundary value problem over the entire image domain — almost always a very simple (rectangular) region.2. It converges to the solution of interest.The first of these properties implies very significant advantages over other existing regularization methods; the computation cost is typically cut by an order of magnitude or more. The second property represents considerable advantages over the existing diffusion methods; it removes the problem of deciding when to stop, as well as that of actually stopping the diffusion process.

I propose a diffusion process that operates on the jet-space of an image. This process uses variable conductance diffusion as an alternative to Gaussian scale in order to smooth differential measurements in a manner that preserves structures of interest. The process is presented within a general framework that suggests a wide range of possibilities for segmenting images on the basis of homogeneity of local shape. Previous work has shown how first-order geometry is used to locate ridges and valleys in greyscale objects. In this paper I use apply these principles to first and second-order geometry in order to find boundaries and skeletons of objects. Examples of first and second-order segmentations of medical images are given. This method appears to offer a reliable and accurate means of segmenting images and is shown to preserve the orthogonal group properties of properly constructed geometric invariants.

Many image processing problems are ill-posed and must be
regularized. Usually, a roughness penalty is imposed on the solution.
The difficulty is to avoid the smoothing of edges, which are very
important attributes of the image. The authors first give sufficient
conditions for the design of such an edge-preserving regularization.
Under these conditions, it is possible to introduce an auxiliary
variable whose role is twofold. Firstly, it marks the discontinuities
and ensures their preservation from smoothing. Secondly, it makes the
criterion half-quadratic. The optimization is then easier. The authors
propose a deterministic strategy, based on alternate minimizations on
the image and the auxiliary variable. This yields two algorithms, ARTUR
and LEGEND. The authors apply these algorithms to the problem of SPECT
reconstruction

Segmentation algorithms are presented which combine regularization by nonlinear partial differential equations (PDEs) with a watershed transformation with region merging. We develop efficient algorithms for two well-founded PDE methods. They use an additive operator splitting (AOS) leading to recursive and separable filters. Further speed-up can be obtained by embedding AOS schemes into a pyramid framework. Examples demonstrate that the preprocessing by these PDE techniques eases and stabilizes the segmentation. The typical CPU time for segmenting a 2562 image on a workstation is less than 2 seconds.

. The paper deals with a nonuniform diffusion filtering of magnetic resonance (MR) tomograms. Alternative digital schemes for
discrete implementation of the nonuniform diffusion equations are analyzed and tested. A novel locally adaptive conductance
for the geometry-driven diffusion (GDD) filtering is proposed. It is based on a measure of the neighborhood unhomogeneity
adopted from the optimal orientation detection of linear symmetry. The algorithm performance is evaluated on the basis of
pseudoartificial 2D MR brain phantom and using the signal-to-noise ratio, as well as HC measure, developed for image discrimination
characterization. Three filtering methods are applied to MR images acquired by the fast 3D FLASH sequence. The results obtained
are quantitatively and visually compared and discussed.

This article surveys deformable models, a promising and vigorously researched computer-assisted medical image analysis technique. Among model-based techniques, deformable models offer a unique and powerful approach to image analysis that combines geometry, physics and approximation theory. They have proven to be effective in segmenting, matching and tracking anatomic structures by exploiting (bottom-up) constraints derived from the image data together with (top-down) a priori knowledge about the location, size and shape of these structures. Deformable models are capable of accommodating the significant variability of biological structures over time and across different individuals. Furthermore, they support highly intuitive interaction mechanisms that, when necessary, allow medical scientists and practitioners to bring their expertise to bear on the model-based image interpretation task. This article reviews the rapidly expanding body of work on the development and application of deformable models to problems of fundamental importance in medical image analysis, including segmentation, shape representation, matching and motion tracking.

. This paper proposes a diffusion scheme for multi-spectral images which incorporates both spatial derivatives and feature-space classification. A variety of conductance terms are suggested that use the posterior probability maps and their spatial derivatives to create resistive boundaries that reflect objectness rather than intensity differences alone. A theoretical test case is discussed as well as simulated and real magnetic resonance dual echo images. We compare the method for both supervised and unsupervised classification. Keywords: Scale Space, Anisotropic Diffusion, Feature-Space classification, Magnetic Resonance Imaging. 1 Introduction Multi-spectral images arise in a number of contexts, either directly, such as RGB components of a colour image, indirectly, by registration of multiple modalities such as intensity and range data, or by postprocessing to generate a feature vector at every pixel for input for example into a Neural Network classifier. In Medical Image P...

Regularization may be regarded as diffusion filtering with an implicit time discretization where one single step is used. Thus, iterated regularization with small regularization parameters approximates a diffusion process. The goal of this paper is to analyse relations between noniterated and iterated regularization and diffusion filtering in image processing. In the linear setting, we show that with iterated Tikhonov regularization noise can be better handled than with noniterated. In the nonlinear framework, two filtering strategies are considered: total variation regularization and the diffusion filter of Perona and Malik. It is established that the Perona-Malik equation decreases the total variation during its evolution. While noniterated and iterated total variation regularization is well-posed, one cannot expect to find a minimizing sequence which converges to a minimizer of the corresponding energy functional for the Perona-Malik filter. To address this shortcoming, a novel regu...

. This paper gives an overview of scale-space and image enhancement techniques which are based on parabolic partial differential equations in divergence form. In the nonlinear setting this filter class allows to integrate a-priori knowledge into the evolution. We sketch basic ideas behind the different filter models, discuss their theoretical foundations and scale-space properties, discrete aspects, suitable algorithms, generalizations, and applications. 1 Introduction During the last decade nonlinear diffusion filters have become a powerful and well-founded tool in multiscale image analysis. These models allow to include a-priori knowledge into the scale-space evolution, and they lead to an image simplification which simultaneously preserves or even enhances semantically important information such as edges, lines, or flow-like structures. Many papers have appeared proposing different models, investigating their theoretical foundations, and describing interesting applications. For a...

Seale-Spaee Theories in Computer Vision

- M Nielsen
- P Johansen
- O F Olsen
- J Weiekert

M. Nielsen, P. Johansen, O.F. Olsen, J. Weiekert (Eds.), Seale-Spaee Theories in Computer Vision, Leeture Notes in ComputerSeience, Vol. 1682, Springer, Berlin, 1999.

Classifieation and uniqueness ofinvariantgeometrie fiows

- P J Olver
- G Sapiro
- A Tannenbaum
- C R Aead
- Sei

P.J. Olver, G. Sapiro, A. Tannenbaum: C. R. Aead. Sei. Paris 319, Serie I, 339-344, 1994. Classifieation and uniqueness ofinvariantgeometrie fiows,

Malik:Seale-spaee and edge-deteetion

- P Perona

P. Perona, 12(7), 629-639, 1990. J. Malik:Seale-spaee and edge-deteetion,IEEE Trans. PatternAnal. Mach. Intell.

Gliek: Loeal geometry variableconduetanee diffusionfor

- D.-S Luo
- M A King
- Filtering
- Trans

D.-S. Luo, reeonstruetion M.A. King, filtering, IEEE Trans. Nuclear Sei. 41, 2800-2806, 1994. S. Gliek: Loeal geometry variableconduetanee diffusionfor post-

book on Computer Academic Nonlineardiffusionfiltering

- J Weickert

J. Weickert: book on Computer Academic Nonlineardiffusionfiltering, In B. Jähne, Vol. 2: Signal Processing 1999. H. Haußecker,P. Geißler and Pattern (Eds.), Recognition, Hand-Vision and Applications, Press, San Diego, 423-450,

from phase transitions A. Kumar, to aetive vision

- S Kiehenassamy

S. Kiehenassamy, from phase transitions A. Kumar, to aetive vision, Areh. Rat. Meeh. Anal. 134, 275-301, 1996. P. Olver, A. Tannenbaum,A. Yezzi: Conformaleurvature fiows:

Springer, first and second-order Information Berlin, patches using geometry-limited in Medical Imaging, diffusion

- Rt Whitaker
- A F Characterizing
- Gmitro
- Science
- H H Notesin
- Barrett

RT. Whitaker:Characterizing A.F. Gmitro Science, Vol. 687, Springer, first and second-order (Eds.), Information Berlin, patches using geometry-limited in Medical Imaging, diffusion, NotesIn H.H. Barrett, in Computer Processing 149~167, 1993. Lecture 11

Nonlinear multiseale analysis of 3D eehoeardiographie IEEE Trans

- A Sarti
- K Mikula
- F Sgallari

A. Sarti, K. Mikula, F. Sgallari: Nonlinear multiseale analysis of 3D eehoeardiographie IEEE Trans. Medieal Imaging 18, 453-466, 1999. sequenees,

Viergever: segment at ion of three-dimensionalMR brain images

- W J Niessen
- K L Vineken
- J Weickert
- B M Romeny

W.J. Niessen, K.L. Vineken, J. Weickert, B.M. ter Haar Romeny, M.A. Viergever: segment at ion of three-dimensionalMR brain images, Int. J. Comput. Vision 31, 185-202, 1999. Multiseale

A study of a eonvex variational diffusion approach for image segment at ion and feature extraetion

- C Sehnörr

C. Sehnörr: A study of a eonvex variational diffusion approach for image segment at ion and feature extraetion,J. Math. Imag. Vision 8(3), 271-292, 1998.

Simmons:Multi-spectral L. Florack, Notes in Computer probabilisticdiffusionusing Bayesian

- S R Arridge
- B In
- Romeny
- Computer

S.R. Arridge, In B. ter Haar Romeny, Computer A. Simmons:Multi-spectral L. Florack, Notes in Computer probabilisticdiffusionusing Bayesian (Eds.), c1assification, Theory J. Koenderink,M. ViergeverScale-Space Berlin, 224-235, in Vision, LectureScience, Vol. 1252, Springer,1997.

diffusion - a unified regularization and diffusion approachto

- N Nordström

N. Nordström: edge detection, Biased anisotropie Image and Vision Computing 8(4), 318-327, 1990. diffusion - a unified regularization and diffusion approachto

Tannenbaum: Classifieation and uniqueness ofinvariant geometrie fiows

- P J Olver
- G Sapiro

P.J. Olver, G. Sapiro, A. Tannenbaum: Classifieation and uniqueness ofinvariant geometrie fiows,
C. R. Aead. Sei. Paris 319, Serie I, 339-344, 1994.

Loeal geometry variable conduetanee diffusion for postreeonstruetion filtering

- D.-S Luo
- M A King
- S Gliek

D.-S. Luo, M.A. King, S. Gliek: Loeal geometry variable conduetanee diffusion for postreeonstruetion filtering, IEEE Trans. Nuclear Sei. 41, 2800-2806, 1994.

- M Nielsen
- P Johansen
- O F Olsen

M. Nielsen, P. Johansen, O.F. Olsen, J. Weiekert (Eds.), Seale-Spaee Theories in Computer Vision,
Leeture Notes in Computer Seience, Vol. 1682, Springer, Berlin, 1999.

Seale-spaee and edge-deteetion

- P Perona
- J Malik

P. Perona, J. Malik: Seale-spaee and edge-deteetion, IEEE Trans. Pattern Anal. Mach. Intell.
12(7), 629-639, 1990.

Anisotropie Diffusion in Image Proeessing

- J Weiekert

J. Weiekert: Anisotropie Diffusion in Image Proeessing, Teubner, Stuttgart, 1998.