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A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement

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A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement Linkoping¨ Studies in Science and Technology Dissertation No. 1171 A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement Bjorn¨ Svensson Department of Biomedical Engineering Linkopings¨ universitet SE-581 85 Linkoping,¨ Sweden http://www.imt.liu.se Linkoping,¨ April 2008 A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement c 2008 Bjorn¨ Svensson Department of Biomedical Engineering Linkopings¨ universitet SE-581 85 Linkoping,¨ Sweden ISBN 978-91-7393-943-0 ISSN 0345-7524 Printed in Linkoping,¨ Sweden by LiU-Tryck 2008 Abstract Filtering is a fundamental operation in image science in general and in medical image science in particular. The most central applications are image enhancement, registration, segmentation and feature extraction. Even though these applications involve non-linear processing a majority of the methodologies available rely on initial estimates using linear filters. Linear filtering is a well established corner- stone of signal processing, which is reflected by the overwhelming amount of literature on finite impulse response filters and their design. Standard techniques for multidimensional filtering are computationally intense. This leads to either a long computation time or a performance loss caused by approximations made in order to increase the computational efficiency. This dis- sertation presents a framework for realization of efficient multidimensional filters. A weighted least squares design criterion ensures preservation of the performance and the two techniques called filter networks and sub-filter sequences significantly reduce the computational demand. A filter network is a realization of a set of filters, which are decomposed into a structure of sparse sub-filters each with a low number of coefficients. Sparsity is here a key property to reduce the number of floating point operations required for filtering. Also, the network structure is important for efficiency, since it deter- mines how the sub-filters contribute to several output nodes, allowing reduction or elimination of redundant computations. Filter networks, which is the main contribution of this dissertation, has many po- tential applications. The primary target of the research presented here has been local structure analysis and image enhancement. A filter network realization for local structure analysis in 3D shows a computational gain, in terms of multiplica- tions required, which can exceed a factor 70 compared to standard convolution. For comparison, this filter network requires approximately the same amount of multiplications per signal sample as a single 2D filter. These results are purely al- gorithmic and are not in conflict with the use of hardware acceleration techniques such as parallel processing or graphics processing units (GPU). To get a flavor of the computation time required, a prototype implementation which makes use of filter networks carries out image enhancement in 3D, involving the computation of 16 filter responses, at an approximate speed of 1MVoxel/s on a standard PC. Popul¨arvetenskaplig Sammanfattning Filtrering ar¨ en av de mest grundlaggande¨ operationerna inom bildanalys. Detta galler¨ sarkilt¨ for¨ analys av medicinska bilder, dar¨ bildforb¨ attring,¨ geometrisk bild- anpassning, segmentering och sardragsextraktion¨ ar¨ centrala tillampningar.¨ Dessa tillampningar¨ kraver¨ i allmanhet¨ icke-linjar¨ filterering, men de flesta metoder som finns bygger and¨ a˚ pa˚ berakningar¨ som har sin grund i linjar¨ filtrering. Sedan en lang˚ tid tillbaka ar¨ en av grundstenarna i signalbehandling linjara¨ filter och dess design. Standardmetoder for¨ multidimensionell filtrering ar¨ dock i allmanhet¨ berakningsintensiva,¨ vilket resulterar i antingen langa˚ exekveringstider eller samre¨ prestanda pa˚ grund av de forenklingar¨ som maste˚ goras¨ for¨ att minska berakningskomplexiteten.¨ Den har¨ avhandlingen presenterar ett ramverk for¨ re- alisering av multidimensionella filter, dar¨ ett minsta-kvadrat kriterium sakerst¨ aller¨ prestanda och de tva˚ teknikerna filternat¨ och filtersekvenser anvands¨ for¨ att avsevart¨ minska berakningskomplexiteten.¨ Ett filternat¨ ar¨ en realisering av ett flertal filter, som delas upp i en struktur av glesa delfilter med fa˚ koefficienter. Glesheten ar¨ en av de viktigaste egenskaperna for¨ att minska antalet flyttalsoperationer som kravs¨ for¨ filtrering. Natverksstrukturen¨ ar¨ ocksa˚ viktig for¨ hogre¨ effektivitet, eftersom varje delfilter da˚ samtidigt kan bidra till flera filter och darmed¨ reducera eller eliminera redundanta berakningar.¨ Det viktigaste bidraget i avhandlingen ar¨ filternaten,¨ en teknik som kan tillampas¨ i manga˚ sammanhang. Har¨ anvands¨ den framst¨ till analys av lokal struktur och bildforb¨ attring.¨ En realisering med filternat¨ for¨ estimering av lokal struktur i 3D kan utforas¨ med en faktor 70 ganger˚ farre¨ multiplikationer jamf¨ ort¨ med falt- ning. Detta motsvarar ungefar¨ samma mangd¨ multiplikationer per sampel som for¨ ett enda vanligt 2D filter. Eftersom filternat¨ enbart ar¨ en algoritmisk teknik, finns det ingen motsattning¨ mellan anvandning¨ av filternat¨ och acceleration med hjalp¨ av hardvara,˚ som till exempel parallella berakningar¨ eller grafikkortspro- cessorer. En prototyp har implementerats for¨ filternatsbaserad¨ bildforb¨ attring¨ av 3D-bilder, dar¨ 16 filtersvar beraknas.¨ Med denna prototyp utfors¨ bildforb¨ attring¨ i en hastighet av ungefar¨ 1MVoxel/s pa˚ en vanlig PC, vilket ger en uppfattning om de berakningstider¨ som kravs¨ for¨ den har¨ typen av operationer. Acknowledgements I would like to express my deepest gratitude to a large number of people, who in different ways have supported me along this journey. I would like to thank my main supervisor, Professor Hans Knutsson, for all the advice during the past years, but also for giving me the opportunity to work in his research group, a creative environment dedicated to research. You have been a constant source of inspiration and new ideas. My co-supervisor, Dr. Mats Andersson, recently called me his most hopeless PhD student ever, referring to my lack of interest in motor cycles. Nevertheless, we have had a countless number of informal conversations, which have clarified my thinking. Your help has been invaluable and very much appreciated. My other co-supervisor, Associate Professor Oleg Burdakov, has taught me basi- cally everything I know about non-linear optimization. Several late evenings of joint efforts were spent on improving optimization algorithms. I would like to thank ContextVision AB for taking an active part in my research and in particular for the support during a few months to secure the financial sit- uation. The input from past and present members of the project reference group, providing both an industrial and a clinical perspective, has been very important. All friends and colleagues at the department of Biomedical Engineering, espe- cially the Medical Informatics group and in particular my fellow PhD students in the image processing group for a great time both on and off work. All friends from my time in Gothenburg and all friends in Linkoping,¨ including the UCPA ski party of 2007 and my floorball team mates. You have kept my mind busy with more important things than filters and tensors. My family, thank you for all encouraging words and for believing in me. Kristin, you will always be my greatest discovery. The financial support from the Swedish Governmental Agency for Innovation Sys- tems (VINNOVA), the Swedish Foundation for Strategic Research (SSF), Con- textVision AB, the Center for Medical Image Science and Visualization (CMIV) and the Similar Network of Excellence is gratefully acknowledged. Table of Contents 1 Introduction 1 1.1 Motivations . 1 1.2 Dissertation Outline . 2 1.3 Contributions . 3 1.4 List of Publications . 4 1.5 Abbreviations . 5 1.6 Mathematical Notation . 6 2 Local Phase, Orientation and Structure 7 2.1 Introduction . 7 2.2 Local Phase in 1D . 8 2.3 Orientation . 10 2.4 Phase in Higher Dimension . 12 2.5 Combining Phase and Orientation . 14 2.6 Hilbert and Riesz Transforms . 16 2.7 Filters Sets for Local Structure Analysis . 18 2.8 Discussion . 19 3 Non-Cartesian Local Structure 21 3.1 Introduction . 21 3.2 Tensor Transformations . 22 3.3 Local Structure and Orientation . 24 3.4 Tensor Estimation . 27 3.5 Discussion . 29 4 Tensor-Driven Noise Reduction and Enhancement 31 4.1 Introduction . 31 4.2 Noise Reduction . 33 4.2.1 Transform-Based Methods . 34 4.2.2 PDE-Based Methods . 35 4.3 Adaptive Anisotropic Filtering . 38 4.3.1 Local Structure Analysis . 39 4.3.2 Tensor Processing . 40 4.3.3 Reconstruction and Filter Synthesis . 41 4.4 Discussion . 42 5 FIR Filter Design 45 viii Table of Contents 5.1 Introduction . 45 5.2 FIR Filters . 47 5.3 Weighted Least Squares Design . 48 5.4 Implementation . 53 5.5 Experiments . 54 5.6 Discussion . 56 6 Sub-Filter Sequences 57 6.1 Introduction . 57 6.2 Sequential Convolution . 58 6.3 Multi-linear Least Squares Design . 59 6.3.1 Alternating Least Squares . 62 6.4 Generating Robust Initial Points . 62 6.4.1 Implementation . 65 6.5 Experiments . 67 6.6 Discussion . 67 7 Filter Networks 69 7.1 Introduction . 69 7.2 Graph Representation . 71 7.3 Network Structure and Sparsity . 73 7.4 Least Squares Filter Network Design . 76 7.5 Implementation . 77 7.6 Experiments . 79 7.7 Discussion . 81 8 Review of Papers 83 8.1 Paper I: On Geometric Transformations of Local Structure Tensors 83 8.2 Paper II: Estimation of Non-Cartesian Local Structure Tensor
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