The Euclidean k-Distance Transformartion in Arbitrary Dimensions - a Separable Implementation

The signed k-distance transformation (k-DT) computes the k nearest prototypes from each location on a discrete regular grid within a given D dimensional volume. We propose a new k-DT algorithm that divides the problem into D 1-dimensional problems and compare its accuracy and computational complexity to the existing raster-scanning and propagation approaches

Distance transformations: fast algorithms and applications to medical image processing

Medical image processing is a demanding domain, both in terms of CPU and memory requirements. The volume of data to be processed is often large (a typical MRI dataset requires 10 MBytes) and many processing tools are only useful to the physician if they are available as real-time applications, i.e. if they run in a few seconds at most. Of course, a large part of these demands are - and will be - handled by the development of more powerful hardware. On the other hand, when faced with non-linear computational complexity, the development of improved algorithms is obviously the best solution. Distance transformations, a powerful image analysis tool used in a number of problems such as image registration, requires such improvements. A distance map is an image where the value of each pixel is the distance from this pixel to the nearest pixel belonging to a given set or object. A distance transformation (DT) is an algorithm that computes a distance map from a binary image representing this set of pixels. This definition is global in the sense that it requires finding the minimum on a set of distances computed between all image pixels and all object pixels. Therefore, a direct application of the definition usually leads to an unacceptable computational complexity. Numerous algorithms have been proposed to localize this definition of distance to the nearest pixel and allow a faster DT computation, but up to now, none of them combines both exactness and linear complexity. Numerous applications of distance transformations to image analysis and pattern recognition have been reported and those related to medical image processing are explored in what follows. Chapter 1 introduces a few basic concepts, a typical application of distance transformations in pattern recognition and the key challenges in producing a DT algorithm. Chapter 2 contains an exhaustive critical review of published algorithms. The strong and weak points of the most popular ones are discussed and the core principles for our original algorithms are derived. Chapters 3, 5, 6, 8 and 10 present original distance transformation algorithms. Each of those chapters is organized in a somewhat similar fashion. First we describe the algorithm. Then we evaluate its computational complexity and compare it to the state of the art. Chapter 4, 7, 9 and 11 each present an application to a particular problem in medical image processing, using the algorithm developed in the previous chapter. Ideally, the description of any medical image processing problem should include a medical justification of the need for an automated processing, a complete review of the state of the art in the field, a detailed description of the proposed processing method, and an evaluation of the accuracy of the results and their medical significance. Because of both time and space constraints in this thesis, such an exhaustive work will only be presented for the application in chapter 4, while the other applications will be described more briefly. Chapter 3 describes a new exact Euclidean distance transformation using ordered propagation. It is based on a variation of Ragnelmam's approximate Euclidean DT. We analyze the error patterns for approximate Euclidean DT using finite masks, and we derive a rule defining, for any pixel location, the size of the neighborhood that guarantees the exactness of the DT. This algorithm is particularly well-suited to implement mathematical morphology operations, which are examined in details. In Chapter 4, we apply the algorithm of chapter 3 to the segmentation of neuronal fibers from microscopic images of the sciatic nerve. In particular, it is used to determine the thickness of the myelin sheath surrounding the center of the fiber. This study was carried out in collaboration with the Neural Rehabilitation Engineering Laboratory, UCL. Chapter 5 proposes another exact Euclidean distance transformation, based on the explicit computation of the Voronoi division of the image. Possible error locations are detected at the corners of the Voronoi polygons and corrected if needed. This algorithm is shown to be the fastest exact EDT to date. It approaches the theoretical optimal complexity, a CPU time proportional to the number of pixels on which the distance is computed. Chapter 6 investigates how the algorithms of chapters 3 and 5 can be extended to 3 dimensional images. It shows the limitations of both approaches and proposes an hybrid algorithm mixing the method of chapter 5 and Saito's. In Chapter 7, the 3D Euclidean DT is applied to the registration of MR images of the brain where the matching criterion is the distance between the surfaces of similar objects (skin, cortex, ventricular system, ...) in both images. Examples are shown, from projects with the Neuro-physiology Laboratory, UCL, and with the Positron Tomography Laboratory, UCL. Chapter 8 discusses an extension of the distance transformation concept: geodesic distances on non-convex domains. Because geodesic distances are based on the notion of paths, a trade-off has to be introduced between the accuracy with which straight lines are represented and the way curves of the domain are followed. It is shown that, whatever the trade-off chosen, there is an efficient implementation of the geodesic DT by propagation. By back-tracking the geodesic distance propagation, one can find the shortest path between a target and a starting point. In chapter 9, this is used to plan the optimal path for the camera movements in virtual endoscopy, a work done in collaboration with the Surgical Planning Laboratory, Harvard Medical School, Boston. Chapter 10 extends the Euclidean distance transformation from finding the nearest object pixel to finding the k nearest object pixels. It is shown that this can be done with a complexity increasing linearly with k. In Chapter 11, the k-DT is used as a fast implementation of the k Nearest Neighbors (k-NN) classification between different tissue types in multi-modal MR imaging. This is illustrated through the classification of multiple sclerosis lesions from T1-T2 images, provided by the Radiology unit, St-Luc Hospital, UCL, via the Positron Tomography Laboratory, UCL. Finally, a general conclusion is drawn. It reviews the main contributions of the thesis, its applications and explores some new domains in which their applications could also be useful. Ultimately, the publications related to this thesis are briefly reviewed

Locally Adaptable Mathematical Morphology Using Distance Transformations

We investigate how common binary mathematical morphology operators can be adapted so that the size of the structuring element can vary across the image pixels. We show that when the structuring elements are balls of a metric, locally adaptable erosion and dilation can be e±ciently implemented as a variant of distance trans- formation algorithms. Opening and closing are obtained by a local threshold of a distance transformation, followed by the adaptable dilation

Automatic segmentation and measurement of axones in microscopic images

We propose a method for the automatic segmentation, recognition and measurement of neuronal fibers in microscopic images of nerves. This permits a quantitative analysis of the distribution of the areas of the fibers, while nowadays such morphometrical methods are limited by the practical impossibility to process large amounts of fibers in histological routine. First, the image is thresholded to provide a coarse classification between myelin (black) and non-myelin (white) pixels. The resulting binary image is simplified using connected morphological operators. These operators simplify the zonal graph, whose vertices are the connected areas of the binary image. An appropriate set of semantic rules allow us to identify a number of white areas as axon candidates, some of which are isolated, some of which are connected. To separate connected fibers candidates sharing the same neighboring black area - we evaluate the thickness of the myelin ring around each candidate area through Euclidean distance transformation by propagation with a stopping criterion on the pixels in the propagation front. Finally, properties of each detected fibers are computed and false alarms are suppressed. The computational cost of the method is evaluated and the robustness of the method is assessed by comparison to the manual procedure. We conclude that the method is fast and accurate for our purpose

Fast and exact signed Euclidean distance transformation with linear complexity

We propose a new signed or unsigned Euclidean distance transformation algorithm, based on the local corrections of the well-known 4SED algorithm of Danielsson. Those corrections are only applied to a small neighborhood of a small subset of pixels from the image, which keeps the cost of the operation low. In contrast with all fast algorithms previously published, our algorithm produces perfect Euclidean distance maps in a time linearly proportional to the number of pixels in the image. The computational cost is close to the cost of the 4SSED approximation

Fast Euclidean morphological operators using local distance transformation by propagation

We propose a new method to compute the morphological dilation of a binary image with a circular structuring element of any given size, on a discrete lattice. The algorithm is equivalent to applying a threshold on an exact Euclidean distance map, but computations are restricted to a minimum number of pixels. The complexity of this dilation algorithm is compared to the complexity of the commonly used approximation of circular structuring elements and found to have a similar cost, while providing better results

Unsupervised Selective Anisotropic Feature Diffusion for Cortical Surface Registration

In this paper we use the theory of geometric curve and surface evolution to diffuse an object preserving their important features. We do that using a curvature flow method with a novel influence function. The proposed influence function takes profit from the characteristics of the level set representation to perform a directional diffusion improving other influence functions results that evolve in such a way that though they diffuse noise, important features are also lost after some iterations. We show that if the geometry of the object is important for our purposes our technique outperforms the classical ones