Workshop on

**Locally adaptive
filters in signal and image processing**

**November 24-26, 2008**

**EURANDOM, Eindhoven, The Netherlands**

**ABSTRACTS**

**Markus van
Almsick (Technische
Universiteit Eindhoven)**

**An A-Priori Model of Line Propagation **

To regularize lines in 2-dimensional images, it is feasible to have a general, a priori model reflecting the properties of lines. Based on three simple and generally applicable assumptions we introduce a stochastic line propagation model with its resulting Fokker-Planck equation and Green's function. The line model implies a line diffusion scheme that is not simply another anisotropic diffusion method of scalar-valued luminosity functions, but a mechanism for the anisotropic diffusion of oriented line segments in a 3-dimensional space that encodes position and orientation. An application example will demonstrate the merits of this novel regularization approach.

**Maxime Boucher (Montreal Neurological
Institute)**

**Adaptative Depth Potentials to study the Geometry of Folding Patterns**

Folding patterns are formed as a successive alignment of folds on a surface (e.g.: human cortical surface). The perceptual organization of these folding patterns is an important research topic in medical shape analysis. In this presentation, we study how differences between similar folding patterns can be identified. More precisely, we want to identify which fold is shallower or deeper. We propose to use a shape operator based on Poisson's equation with curvature as the source term. We first show that the solution to this Poisson's equation integrates curvature at different scales to map differences in depth on a surface. Damping then allow us to choose which size of area to be used to map differences in depth. We thus can adapt our shape operator to the size of the features to be studied at a very low cost. Poisson's equation is numerically solved by solving a single linear system. Our shape operator does not require to deform the surface or to solve a time-iterated area-minimizing flow or a heat flow and is therefore orders of magnitude faster to compute than other measures of depth.

As an illustration, depth potentials are used to measure differences in depth among similar folds of the human cerebral cortex. We used surface extracted from 189 subjects, with one group affected with very mild to mild Alzheimer's Disease. Using Depth Potentials, tensor analysis, F-statistics and random field theory, we were able to determine on these surfaces which fold is shallower with the presence of Alzheimer's disease.

**Bernhard Burgeth (Saarland University)
**Joint work with Luis Pizarro, Stephan Didas, and Joachim Weickert

**Coherence-Enhancing Diffusion Filtering for Matrix Fields**

Coherence-enhancing diffusion (CED) filtering is an adaptive
technique that allows for the completion of interrupted lines and enhancement of
flow-like features in greyscale and color images.

It utilises a so-called structure tensor, a concept to obtain directional
information of image structures.

In this work we extend the CED-filtering concept to fields of symmetric matrices
as they appear, for example, in diffusion tensor magnetic resonance imaging (DT-MRI).

To this end we provide a generic structure tensor concept for such matrix fields
that relies on the operator-algebraic properties of symmetric matrices.

In numerical experiments we explore the gap-closing and enhancement properties
of the matrix-valued CED-filtering on artificial as well as on real 3D DT-MRI
data.

**Laurent Demaret (Helmholtz Zentrum München)**

**Local Adaptive Triangulations and Image Compression **

This talk is concerned with an approximation method of geometric images based on locally adaptive Delaunay triangulations. Classical spectral decompositions -like Fourier or wavelets- are not optimal for bivariate functions composed of smooth geometrical objects delimited by singularities supported in regular curves. Here we focus on a highly non-linear approximation scheme: local adaptive refinement of continuous, piecewise affine functions over Delaunay triangulations are performed and allow shape preserving and sparse representations of the target functions. In this talk the design and implementation of a new efficient compression scheme of the corresponding information, which makes use of local redundancies in the triangulation, are discussed. In particular, suitable contextual encoding of the positions of the vertices and of the greyscale values is proposed, which takes into account the specific local geometrical structure of the triangulation and combines it with appropriate combinatorial encoding. Finally some examples are shown where our method significantly outperforms JPEG2000.

**Xavier Descombes
(INRIA)**

**Counting a population by objects detection using marked point processes
**

In this talk, I will first introduce marked point processes for analyzing high resolution images. I will then focus on multiple objects detection for applications concerning the counting of individuals in a given population. A simple generic model will be derived. It consists of a non overlapping prior and an activity obtained from a local spatial filter based on a simple geometric description of the objects. A first application concerning tree counting will be described using a RJMCMC algorithm for optimizing the model. I will than address a multiple birth and death process, derived from a continuous dynamic, as an alternative to the RJMCMC scheme. Finally, an application concerning flamingo counting will be presented.

**Michael Felsberg (Linköping University)**

**Adaptive Filtering using Channel Representations **

This talk continues in the spirit of the previous one by Hanno Scharr. First, I will give a detailed introduction to channel representations and their relation to density estimation in order to explain the methodology of channel smoothing. Next, I will shed light onto different variants of channel smoothing: -orientation adaptive channel smoothing with applications to coherence- enhancing filtering -channel smoothing with alpha-synthesis for improving stability of edge-enhancing filtering -channel smoothing using graph-cuts for improving edge-enhancing filtering at corners I will then introduce a rather novell variant of channel-based processing, channel-coded feature maps (CCFMs). CCFMs are related to scale-space theory and reconstructing images from CCFMs results in a very efficient non-linear image enhancement scheme. Several results are shown and are compared to a novel variant of anisotropic diffusion, based on iterated adaptive filtering.

**Luc Florack (Eindhoven University of Technology)**

**Scale
Space Representations Locally Adapted to the Geometry of Base and Target
Manifold**

We generalize the Gaussian multi-resolution image paradigm for a Euclidean domain to general Riemannian base manifolds and also account for the codomain by considering the extension into a fibre bundle structure. We elaborate on aspects of parametrization and gauge, as these are important in practical applications. We subsequently scrutinize two examples that are of interest in bio-mathematical modeling, viz. scale space on the unit sphere, used among others for codomain regularization in the context of high angular resolution diffusion imaging (HARDI), and retino-cortical scale space, proposed as a biologically plausible model of the human visual pathway from retina to striate cortex.

**Erik Franken (Technische Universiteit Eindhoven)**

**Diffusion on the 2D and 3D Euclidean Motion Group for Enhancement of
Crossing Elongated Structures**

Abstract: The enhancement of elongated structures in noisy image data is relevant for many (bio)medical imaging applications. Existing generic image processing methods cannot appropriately handle elongated structures that cross each other. In this work we propose to solve this problem by using so-called orientation scores, which can be considered as functions on the Euclidean motion group SE(2) or SE(3) voor 2D resp. 3D images. This observation makes it possible to map many well-known image processing algorithms to the orientation score domain SE(n). To ensure that the effective operation on the image is rotationally invariant, we consider linear and nonlinear diffusion equations on SE(n) that are left-invariant. We have developed a coherence-enhancing diffusion method for curve enhancement in images via orientation scores, where the diffusion process is controlled by a number of estimated local properties in the orientation score. This method is shown to be beneficial for enhancing e.g. 2D microscopy images with crossing fibres. We also discuss how these methods can be applied to orientation scores of 3D images, which is shown to be useful for processing high angular resolution diffusion imaging (HARDI) data.

**Hartmut Führ (RWTH Aachen)**

**The relation between scale space representation and continuous wavelet
transforms on Lie groups **

It is well-known, and easily verified, that differentiating a Gaussian scale-space representation of a function along the time variable results in the continuous wavelet transform of the same function with respect to the Mexican Hat wavelet. We investigate this phenomenon in a more general setting, showing how recently defined scale spaces on Lie groups such as the Heisenberg group relate to continuous wavelet transforms on these groups. For example, the same argument as for the reals exhibits a relationship between scale space on the Heisenberg groups and the Mexican Hat wavelet defined by A. Mayeli. We demonstrate how the connection allows to derive energy preservation properties for the wavelet transform. If time permits, we will also comment on the euclidean motion group.

**Yaniv Gur
(Tel Aviv University)**

**Fast Affine-Invariant DT-MRI Denoising **

Abstract I present a novel framework for regularization of symmetric positive-definite (SPD) tensors (e.g., diffusion tensors). This framework is based on a local differential geometric approach and a unique decomposition of SPD matrices, the Iwasawa decomposition. The manifold of SPD matrices, $P_n$, is parameterized in this framework via the Iwasawa coordinate system. Then, it is turned into a Riemannian manifold by introducing a natural affine-invariant metric. Via the Beltrami framework a functional over the tensor field, accompanied by a suitable data fitting term, is defined. The variation of this functional with respect to the Iwasawa coordinates leads to a set of coupled equations of motion for these coordinates. Then, by means of the gradient descent method, these equations of motion define a PDE system of Laplace-Beltrami equations, that define the regularization flow over $P_n$. For diffusion tensors, in particular, the regularization flow is defined by a set of six coupled Laplace-Beltrami equations. It turns out to be that the local coordinate approach via the Iwasawa coordinate system results in very simple numerics that leads to fast convergence of the algorithm. I will present regularization results of real in vivo DT-MRI datasets and the resulting fibers tractography.

**Michal Haindl
(Institute of Information Theory and Automation)**

**Visual Data Recognition and Modelling
Based on Local Markovian Models**

An exceptional 3D wide-sense Markov model which can be solved analytically will be presented. Illumination invariants can be derived from its recursive statistics and exploited in content based image retrieval, supervised or unsupervised image recognition. Its modelling efficiency will be demonstrated on adaptive multispectral and multichannel image and video restoration, bidirectional texture function synthesis and compression, and enhancement applications.

**Thorsten Hohage (Georg-August-Universität
Göttingen)**

**Deconvolution-type problems in optical nanoscopy **

Over the last years new techniques (4Pi, STED and PALMIRA) have been discovered, which break Abb\'e's famous diffraction limit and allow imaging of living cells with an unprecedented resolution of up to 20-50 nm using visible light. In this talk we discuss some inverse problems of deconvolution type related to these techniques and regularization methods for their solution. Difficulties arise from non-negativity constraints, nonlinear phenomena, spatially varying point spread functions, adequate treatment of the Poissonian distribution of the data, and computational efficiency, particularly for three-dimensional objects. An emphasis will be on attempts to respect a spatially varying structure of the unknown object by choosing regularization parameters locally adaptive using multi-resolution test statistics.

**Michel Loubes (Université de Toulouse)**

**Regularization with maximum entropy with approximated moments **

We focus on maximum entropy regularization where the moments are unknown but can be estimated by a preliminiar estimator. This problem is of interest when dealing with inverse problems with unknown operator which are currently encountered in econometry , tomography or image processings. We propose a regularization method based on Approximated Maximum Entropy Estimator which enables to recover a tractable estimator.

**Joerg Polzehl (WIAS Berlin)**

**Structural adaptive smoothing in Diffusion Tensor Imaging
**Diffusion Tensor Imaging (DTI) data is characterized by a high noise level.
Thus, estimation errors of quantities like anisotropy indices or the main
diffusion direction used for fiber tracking are relatively large and may
significantly confound the accuracy of DTI in clinical or neuroscience
applications.

Besides pulse sequence optimization, noise reduction by smoothing can be pursued as a complementary approach to increase the accuracy of DTI. We suggest an anisotropic structural adaptive smoothing procedure, which is based on the Propagation-Separation method and preserves the structures seen in DTI and their different sizes and shapes.

**Yves Rozenholc
(Université Paris
Descartes)**

**Denoising of dynamic images : Application to angiogenesis evaluation using
DCE-CT **

Fighting the angiogenesis in a cancer tumor appears today as a promising way to cure cancer. Evaluating this new type of treatment requires measuring "in vivo" the parameters of the microcirculation in the tissue. To this aim, one may use a sequence of images (MRI, CT-scan, ...) to follow the injection of a contrast agent. Unfortunately, because the use of X-ray such sequences are of low signal to noise level

We propose a new method to denoise this kind of dynamical images avoiding filtering techniques. This method is based on the local comparison of the (full) temporal kinetics using adaptive nonparametric statistical tests. This method allows a significant increase of the noise to signal ratio, without lost of local or temporal information. In particular, this method shows excellent properties to protect homogeneity in the tissue.

It is possible to extend this technic to clusterize the dynamics existing in the image.

**Hanno Scharr (Forschungszentrum Jülich)**

**A Short Introduction to Diffusion-like Methods**

This contribution aims to give a basic introduction to diffusion-like methods. There are many different methods commonly used for regularization tasks. Some of them will be briefly introduced and their connection to diffusion and robust statistics shown. In addition to this we will go into some detail for diffusion-like methods in a narrower sense, i.e. methods based on PDEs similar to the diffusion PDE known from physics. Main issues here are which PDE to use, how diffusivities in such a PDE are constructed, and which discretization is most suitable for a given task.

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