Front-End Vision and Multi-Scale Image Analysis : Multi-Scale Computer Vision Theory and Applications, Written in Mathematica

by Haar Romeny, Bart M. Ter

• ISBN13: 9781402015038
• ISBN10: 1402015038
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-10-01
• Publisher: SPRINGER

Summary

A breakthrough in interactive teaching of multi-scalemethods for image analysis"Front-End Vision and Multi-Scale Image Analysis" is a tutorial inmulti-scale methods for computer vision and image processing. Itbuilds on the cross fertilization between human visual perception andmulti-scale computer vision (' scale-space') theory and applications.The multi-scale strategies recognized in the first stages of the humanvisual system are carefully examined, and taken as inspiration for themany geometric methods discussed. All chapters are written inMathematica, a spectacular high-level language for symbolic andnumerical manipulations.The book presents a new and effective approach to quickly masteringthe mathematics of computer vision and image analysis. The typicallyshort code is given for every topic discussed, and invites the readerto spend many fascinating hours 'playing' with computer vision."Front-End Vision and Multi-Scale Image Analysis" is intended forundergraduate and graduate students, and allwith an interest incomputer vision, medical imaging, and human visual perception.

Table of Contents

Front-End Vision and Multi-Scale Image Analysis	xiii
The purpose of this book	xiii
Scale-space theory is biologically motivated computer vision	xiv
This book has been written in Mathematica	xvi
Acknowledgements	xviii
Apertures and the notion of scale	1	(12)
Observations and the size of apertures	1	(1)
Mathematics, physics, and vision	2	(3)
We blur by looking	5	(4)
A critical view on observations	9	(3)
Summary of this chapter	12	(1)
Foundations of scale-space	13	(24)
Constraints for an uncommited front-end	13	(2)
Axioms of a visual front-end	15	(6)
Dimensional analysis	15	(1)
The cooking of a turkey	16	(2)
Reynold's number	18	(1)
Rowing: more oarsmen, higher speed?	19	(2)
Axiomatic derivation of the Gaussian kernel	21	(2)
Scale-space from causality	23	(2)
Scale-space from entropy maximization	25	(2)
Dervatives of sampled, observed data	27	(4)
Scale-space stack	31	(1)
Sampling the scale-axis	32	(3)
Summary of this chapter	35	(2)
The Gaussian kernel	37	(16)
The Gaussian kernel	37	(1)
Normalization	38	(1)
Cascade property, selfsimilarity	39	(1)
The scale parameter	40	(1)
Relation to generalized functions	40	(3)
Separability	43	(1)
Relation to binomial coefficients	43	(1)
The Fourier transform of the Gaussian kernel	44	(2)
Central limit theorem	46	(2)
Anisotropy	48	(1)
The diffusion equation	49	(1)
Summary of this chapter	50	(3)
Gaussian derivatives	53	(18)
Introduction	53	(1)
Shape and algebraic structure	53	(4)
Gaussian derivatives in the Fourier domain	57	(2)
Zero crossings of Gaussian derivative functions	59	(1)
The correlation between Gaussian derivatives	60	(4)
Discrete Gaussian kernels	64	(1)
Other families of kernels	65	(2)
Higher dimensions and separability	67	(2)
Summary of this chapter	69	(2)
Multi-scale derivatives: implementations	71	(20)
Implementation in the spatial domain	71	(2)
Separable implementation	73	(1)
Some examples	74	(4)
N-dim Gaussian derivative operator implementation	78	(1)
Implementation in the Fourier domain	79	(4)
Boundaries	83	(2)
Advanced topic: speed concerns in Mathematica	85	(4)
Summary of this chapter	89	(2)
Differential structure of images	91	(46)
The differential structure of images	91	(1)
Isophotes and flowlines	92	(4)
Coordinate systems and transformations	96	(6)
Directional derivatives	102	(1)
First order gauge coordinates	103	(5)
Gauge coordinate invariants: examples	108	(7)
Ridge detection	108	(2)
Isophote and flowline curvature in gauge coord	110	(3)
Affine invariant corner detection	113	(2)
A curvature illusion	115	(2)
Second order structure	117	(10)
The Hessian matrix and principal curvatures	119	(1)
The shape index	120	(2)
Principal directions	122	(1)
Gaussian and mean curvature	123	(3)
Minimal and zero Gaussian curvature surfaces	126	(1)
Third order image structure. T-junction detection	127	(4)
Fourth order image structure: junction detection	131	(1)
Scale invariance and natural coordinates	132	(2)
Irreducible invariants	134	(2)
Intermezzo: Tensor notation	135	(1)
Summary of this chapter	136	(1)
Natural limits on observations	137	(6)
Limits on differentiation: scale, accuracy and order	137	(4)
Summary of this chapter	141	(2)
Differentiation and regularization	143	(10)
Regularization	143	(1)
Regular tempered distributions and test functions	144	(3)
An example of regularization	147	(1)
Relation regularization Gaussian scale-space	148	(4)
Summary of this chapter	152	(1)
The front-end visual system-the retina	153	(14)
Introduction	153	(1)
Studies of vision	154	(2)
The eye	156	(1)
The retina	157	(3)
Retinal receptive fields	160	(2)
Sensitivity profile measurement of a receptive field	162	(3)
Summary of this chapter	165	(2)
A scale-space model for the retinal sampling	167	(12)
The size and spatial distribution of receptive fields	167	(5)
A scale-space model for the retinal receptive fields	172	(5)
Summary of this chapter	177	(2)
The front-end visual system - LGN and cortex	179	(18)
The thalamus	179	(2)
The lateral geniculate nucleus (LGN)	181	(2)
Corticofugal connections to the LGN	183	(2)
The primary visual cortex	185	(6)
Simple cells	187	(1)
Complex cells	188	(1)
Directional selectivity	189	(2)
Intermezzo: Measurement of neural activity in the brain	191	(4)
Electro-Encephalography (EEG)	191	(1)
Magneto-Encephalography (MEG)	192	(1)
Functional MRI (fMRI)	193	(1)
Optical imaging with voltage sensitive dyes	194	(1)
Positron Emission Tomography (PET)	194	(1)
Summary of this chapter	195	(2)
The front-end visual system - cortical columns	197	(18)
Hypercolumns and orientation structure	197	(3)
Stabilized retinal images	200	(2)
The concept of local sign	202	(2)
Gaussian derivatives and Eigen-images	204	(4)
Plasticity and self-organization	208	(2)
Higher cortical visual areas	210	(1)
Summary of this chapter	211	(1)
Vision dictionary	211	(4)
Further reading on the web	212	(3)
Deep structure I. watershed segmentation	215	(26)
Multi-scale measurements	215	(1)
Scale selection	216	(2)
Normalized feature detection	218	(1)
Automatic scale selection	219	(2)
λ-Normalized scale selection	220	(1)
Is this really deep structure?	220	(1)
Edge focusing	221	(4)
Simplification followed by focusing	221	(1)
Linking in 1D	222	(3)
Follicle detection in 3D ultrasound	225	(6)
Fitting spherical harmonics to 3D points	229	(2)
Multi-scale segmentation	231	(8)
Dissimilarity measure in scale-space	231	(1)
Watershed segmentation	232	(2)
Linking of regions	234	(3)
The multi-scale watershed segmentation	237	(2)
Deep structure and nonlinear diffusion	239	(2)
Non-linear diffusion watershed segmentation	239	(2)
Deep structure II. catastrophe theory	241	(16)
Catastrophes and singularities	241	(1)
Evolution of image singularities in scale-space	242	(1)
Catastrophe theory basics	243	(7)
Functions	243	(1)
Characterization of points	243	(1)
Structural equivalence	244	(1)
Local characterization of functions	244	(1)
Thom's theorem	245	(1)
Generic property	246	(1)
Dimensionality	246	(1)
Illustration of the concepts	247	(3)
Catastrophe theory in scale-space	250	(6)
Generic events for differential operators	251	(3)
Generic events for other differential operators	254	(1)
Annihilations and creations	255	(1)
Summary of this chapter	256	(1)
Deep structure III. topological numbers	257	(20)
Topological numbers	257	(6)
Topological numbers in scale-space	258	(1)
Topological number for a signal	259	(1)
Topological number for an image	259	(1)
The winding number on 2D images	260	(3)
Topological numbers and catastrophes	263	(2)
The deep structure toolbox	265	(6)
Detection of singularities	265	(1)
Linking of singularities	265	(3)
Linking of contours	268	(1)
Detection of catastrophes	268	(1)
General discrete geometry approach	269	(2)
From deep structure to global structure	271	(4)
Image representations	271	(1)
Hierarchical pre-segmentation	272	(1)
Perceptual grouping	273	(1)
Matching and registration	274	(1)
Image databases	274	(1)
Image understanding	275	(1)
Summary of this chapter	275	(2)
Deblurring Gaussian blur	277	(8)
Deblurring	277	(1)
Deblurring with a scale-space approach	277	(4)
Less accurate representation, noise and holes	281	(3)
Summary of this chapter	284	(1)
Multi-scale optic flow	285	(26)
Introduction	285	(1)
Motion detection with pairs of receptive fields	286	(3)
Image deformation by a discrete vectorfield	289	(1)
The optic flow constraint equation	290	(2)
Scalar and density images	292	(1)
Derivation of multi-scale optic flow constraint equation	292	(11)
Scalar images, normal flow	296	(5)
Density images, normal flow	301	(2)
Testing the optic flow constraint equations	303	(2)
Cleaning up the vector field	305	(2)
Scale selection	307	(2)
Discussion	309	(1)
Summary of this chapter	310	(1)
Color differential structure	311	(18)
Introduction	311	(1)
Color image formation and color invariants	311	(3)
Koenderink's Gaussian derivative color model	314	(6)
Implementation	320	(5)
Combination with spatial constraints	325	(2)
Summary of this chapter	327	(2)
Steerable kernels	329	(16)
Introduction	329	(1)
Multi-scale orientation	330	(1)
Orientation analysis with Gaussian derivatives	331	(1)
Steering with self-similar functions	332	(4)
Steering with Cartesian partial derivatives	336	(2)
Detection of stellate tumors	338	(4)
Classical papers and student tasks	342	(1)
Summary of this chapter	343	(2)
Scale-time	345	(16)
Introduction	345	(1)
Analysis of prerecorded time-sequences	346	(3)
Causal time-scale is logarithmic	349	(2)
Other derivations of logarithmic scale-time	351	(2)
Real-time receptive fields	353	(1)
A scale-space model for time-causal receptive fields	354	(5)
Conclusion	359	(1)
Summary of this chapter	360	(1)
Geometry-driven diffusion	361	(32)
Adaptive Smoothing and Image Evolution	361	(1)
Nonlinear Diffusion Equations	362	(2)
The Perona & Malik Equation	364	(2)
Scale-space implementation of the P&M equation	366	(4)
The P&M equation is ill-posed	370	(2)
Von Neumann stability of numerical PDE's	372	(1)
Stability of Gaussian linear diffusion	373	(3)
A practical example of numerical stability	376	(2)
Euclidean shortening flow	378	(1)
Grayscale invariance	379	(1)
Numerical examples shortening flow	379	(3)
Curve Evolution	382	(1)
Duality between PDE-and curve evolution	383	(3)
Mathematical Morphology	386	(3)
Mathematical morphology on grayvalued images	389	(1)
Mathematical morphology versus scale-space	390	(1)
Summary of this chapter	390	(3)
Epilog	393	(2)
A. Introduction to Mathematica	395	(18)
Quick overview of using Mathematica	395	(2)
Quick overview of the most useful commands	397	(4)
Pure functions	401	(1)
Pattern matching	401	(3)
Some special plot forms	404	(1)
A faster way to read binary 3D data	405	(2)
What often goes wrong	407	(3)
Suggested reading	410	(2)
Web resources	412	(1)
B. The concept of convolution	413	(6)
Convolution	413	(3)
Convolution is a product in the Fourier domain	416	(3)
C. Installing the book and packages	419	(4)
Content	419	(1)
Installation for all systems	420	(1)
Viewing the book in the Help Browser	420	(1)
Sources of additional applications	421	(2)
D. First Start with Mathematica: Tips & Tricks	423	(2)
Evaluation	423	(1)
Images	423	(1)
Programming	424	(1)
3D	424	(1)
References	425	(30)
Index	455