Ridges in Image and Data Analysis

Transcription

1 Ridges in Image and Data Analysis

2 Computational Imaging and Vision Managing Editor MAX A. VIERGEVER Utrecht University, Utrecht, The Netherlands Editorial Board OLIVIER D. FAUGERAS, INRIA, Sophia-Antipolis, France JAN J. KOENDERINK, Utrecht University, Utrecht, The Netherlands STEPHEN M. PIZER, University of North Carolina, Chapel Hill, USA SABURO TSUJI, Osaka University, Osaka, Japan STEVEN W. ZUCKER, McGill University, Montreal, Canada Volume 7

3 Ridges in Image and Data Analysis by David Eberly SAS Institute, Inc., Cary, North Carolina, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN DOI / ISBN (ebook) Printed on acid-free paper All Rights Reserved 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 Contents Preface IX 1 Introduction 1.1 A History of Ridges. 1.2 Reading Strategies. 2 Mathematical Preliminaries 2.1 Linear Algebra Eigenvalues and Eigenvectors Gram-Schmidt Orthonormalization Symmetric, Unitary, and Definite Matrices Maxima of Quadratic Forms 2.2 Differential Calculus...; Derivative and Index Notation Summation Convention Directional Derivatives Local Extrema of Functions 2.3 Tensors Cartesian Coordinates General Coordinates Tensor Calculus Curves Surfaces Manifolds v

6 VI 3 Ridges in Euclidean Geometry Generalized Local Extrema Height Ridge Definition I-Dimensional Ridges in lr? Continuous Formulations Differential Geometric Relationships Ridge Tangents I-Dimensional Ridges in lr Continuous Formulations Differential Geometric Relationships Ridge Tangents I-Dimensional Ridges in lr n Continuous Formulations Differential Geometric Relationships Ridge Tangents Dimensional Ridges in lr Continuous Formulations Ridge Tangents Dimensional Ridges in lr Continuous Formulations Ridge Tangents d-dimensional Ridges in Jan Continuous Formulations Ridge Tangents Ridges in Riemannian Geometry Generalized Local Extrema Height Ridge Definition I-Dimensional Ridges in lr I-Dimensi.onal Ridges in lr I-Dimensional Ridges in lr n 69

7 Vll Dimensional Ridges in lr? Dimensional Ridges in lr d-dimensional Ridges in lr n 72 5 Ridges of Functions Defined on Manifolds Height Ridge Definition I-Dimensional Ridges in M2 C lr d-dimensional Ridges in Mn C lrp Maximal Curvature Ridge Definitions Curvature Extrema of Planar Curves Curvature Extrema of Surfaces Extensions and Generalizations 94 6 Applications to Image and Data Analysis Medical Image Analysis Linear Scale Space Boundary Measurements Medial Measurements Cores Nonlinear Scale Space Molecular Modeling DNA Structure Protein Structure X-Ray Crystallography Electron Density Maps Ridges of Electron Density Fluid Flow Vector Field Analysis Newtonian Viscous Fluids Pressure Ridges

8 Vlll 7 Implementation Issues Bridging the Gap Between Theory and Practice B-spline Interpolation Definitions Basis Matrices Direct Implementation Generalized and Optimized Spline Calculation Polynomial Construction A voiding Intermediate Calculations Computing Data On-Demand Putting It All Together Eigensystem Solvers Symbolic Tridiagonalization QR Iteration with Explicit Shifting Generalized Eigensystems Ridge Construction Multidimensional Bisection Minimization without Derivatives Manifold Extraction Arbitrarily Spaced Data.200 Bibliography 203 Index 211

9 Preface The concept of ridges has appeared numerous times in the image processing literature. Sometimes the term is used in an intuitive sense. Other times a concrete definition is provided. In almost all cases the concept is used for very specific applications. When analyzing images or data sets, it is very natural for a scientist to measure critical behavior by considering maxima or minima of the data. These critical points are relatively easy to compute. Numerical packages always provide support for root finding or optimization, whether it be through bisection, Newton's method, conjugate gradient method, or other standard methods. It has not been natural for scientists to consider critical behavior in a higher-order sense. The concept of ridge as a manifold of critical points is a natural extension of the concept of local maximum as an isolated critical point. However, almost no attention has been given to formalizing the concept. There is a need for a formal development. There is a need for understanding the computation issues that arise in the implementations. The purpose of this book is to address both needs by providing a formal mathematical foundation and a computational framework for ridges. The intended audience for this book includes anyone interested in exploring the usefulness of ridges in data analysis. While the concepts appear most often in image analysis, there is nothing that prevents them from being used in any other area that requires analysis of data such as statistics, engineering, the physical sciences, or for that matter, the social sciences. The book is structured to be read in a variety of ways; the introduction chapter describes these strategies. However, for those interested in a full understanding of the concept, I recommend reading the entire book from front to back. The second chapter on mathematical preliminaries is a bit heavy, but necessary. The ridge chapters themselves are fairly mathematical. The area of computational vision is seeing a rebirth due to the emphasis on formal mathematical foundations for the problems at hand. Many of the outstanding problems of vision just cannot be solved with a simple, heuristic approach. I have consiqered alternate, non-mathematical ways to describe the material in this book, but I do not think it is possible. The problems whose solutions require ridges necessarily require the mathematics background. As with most large works, many people, directly or indirectly, contribute to them. A large portion of the chapter on applications was possible only through the hard work and contributions of the Medical Image Display and Analysis Group at the University of North Carolina. The group, headed by Professor Stephen M. Pizer, has produced over the last five years a significant body of knowledge on the theory of cores and their application to medical image analysis. He and Christina Burbeck have also contributed significantly to the understanding of human vision with their theories. The other group which has contributed greatly to these ideas is the 3D Computer Vision Group at Utrecht University, headed by Professor Max Viergever.

10 x David Eberly Many interesting discussions with the group members, Max Viergever, Bart ter Haar Romeny, Luc Florack, Alfons Salden, Twan Maintz, Petra van den Elsen, and others, have led to clarification of many of the ideas contained here. Also, Tony Lindeberg of the Royal Institute of Technology in Stockholm has contributed many fine ideas to the topic. Figures were produced by Jacob Furst as part of his dissertation work at UNC on computing cores of 3D medical images. He continues to make good progress at developing algorithms for efficient and rapid calculation of cores as 2D manifolds living in a 4D Riemannian space. Not many people are brave enough to tackle the intricate geometric details of such a setting, but Jacob has been very successful at it. Figures and were created by Daniel Fritsch during the course of his dissertation work at UNC. Now as a researcher and faculty member in the Department of Radiation Oncology, Dan's ideas on optimal scale cores are being successfully used towards accurate and rapid image registration techniques, especially in portal imaging. Figures through were contributed by Stephen Aylward and are part of his dissertation work at UNC on segmentation of the arterial tree in the brain. He works jointly with neurosurgeon Elizabeth Bullitt on developing computer algorithms for image-guided minimally invasive surgery, in particular on percutaneous rhizotomy for correcting trigeminal neuralgia. Stephen's ideas on pseudo cores were an attempt to improve on my true core algorithm which was too slow and too sensitive to starting conditions to be useful in the applications. Figures and are contributions from Matthew McAuliffe and are part of his biomedical engineering dissertation work at UNC designed to produce accurate and rapid segmentation programs for medical images. His work is extremely important in handling the problems with ridge and core construction due to interference of neighboring objects. The data sets for figures 6.2.9a and 6.2.9b were provided by Juraj Horacek (and originally obtained from Professor Jane Richardson at Duke University). Figures through were obtained from his interactive molecular modeling program (affectionately known to the X-window title manager as "Ridges-R-Us", but soon to be renamed I suppose). The modeler is part of his dissertation work at UNC on providing computer tools to aid molecular biologists in determining protein structure. Juraj's work is very good and has helped to improve the ridge algorithms by properly handling branching structures. My original ridge algorithms typically terminated when any type of branching appeared likely. The remainder of the figures were built using my C++ image analysis libraries, MAGIC. Any undue simplicity and lack of sophistication in these are my fault. Finally, it is always good to acknowledge sources of funding that helped contribute to a work. After all, without those sources most of us would have no time to spend

11 Ridges in Image and Data Analysis Xl on creativity. The funding is very much appreciated. During my three years as a graduate student at UNC ( ), Professor Stephen Pizer provided my stipend through National Institutes of Health grant NIH P01-CA The National Science Foundation was kind enough to let me carryover a grant from my previous university; grant DMS provided summer salary for my first year (1991) at UNC. After I graduated, I remained at UNC for one year as a Research Associate Professor with funding provided by Dr. Elizabeth Bullitt. Since leaving UNC, in my spare time I have completed this book and developed the robust algorithms herein, most notably the continuous formulation for ridges and the adaptation of minimization and extraction algorithms to it. While my current employer, SAS Institute, is not directly involved with this book, their wonderful environment has made it possible for me to continue exploring my ideas. I hope to eventually make ridge algorithms a standard component in their statistical and numerical software. Dave Eberly SAS Institute, Inc. June 1996