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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Presents recent significant and rapid development in the field of 2D and 3D image analysis
2D and 3D Image Analysis by Moments, is a unique compendium of moment-based image analysis which includes traditional methods and also reflects the latest development of the field.
The book presents a survey of 2D and 3D moment invariants with respect to similarity and affine spatial transformations and to image blurring and smoothing by various filters. The book comprehensively describes the mathematical background and theorems about the invariants but a large part is also devoted to practical usage of moments. Applications from various fields of computer vision, remote sensing, medical imaging, image retrieval, watermarking, and forensic analysis are demonstrated. Attention is also paid to efficient algorithms of moment computation.
Key features:
- Presents a systematic overview of moment-based features used in 2D and 3D image analysis.
- Demonstrates invariant properties of moments with respect to various spatial and intensity transformations.
- Reviews and compares several orthogonal polynomials and respective moments.
- Describes efficient numerical algorithms for moment computation.
- It is a "classroom ready" textbook with a self-contained introduction to classifier design.
- The accompanying website contains around 300 lecture slides, Matlab codes, complete lists of the invariants, test images, and other supplementary material.
2D and 3D Image Analysis by Moments, is ideal for mathematicians, computer scientists, engineers, software developers, and Ph.D students involved in image analysis and recognition. Due to the addition of two introductory chapters on classifier design, the book may also serve as a self-contained textbook for graduate university courses on object recognition.
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Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Authors' biographies
Acknowledgements
About the companion website
Chapter 1: Motivation
1.1 Image analysis by computers
1.2 Humans, computers, and object recognition
1.3 Outline of the book
References
Chapter 2: Introduction to Object Recognition
2.1 Feature space
2.2 Categories of the invariants
2.3 Classifiers
2.4 Performance of the classifiers
2.5 Conclusion
References
Chapter 3: 2D Moment Invariants to Translation, Rotation, and Scaling
3.1 Introduction
3.2 TRS invariants from geometric moments
3.3 Rotation invariants using circular moments
3.4 Rotation invariants from complex moments
3.5 Pseudoinvariants
3.6 Combined invariants to TRS and contrast stretching
3.7 Rotation invariants for recognition of symmetric objects
3.8 Rotation invariants via image normalization
3.9 Moment invariants of vector fields
3.10 Conclusion
References
Chapter 4: 3D Moment Invariants to Translation, Rotation, and Scaling
4.1 Introduction
4.2 Mathematical description of the 3D rotation
4.3 Translation and scaling invariance of 3D geometric moments
4.4 3D rotation invariants by means of tensors
4.5 Rotation invariants from 3D complex moments
4.6 3D translation, rotation, and scale invariants via normalization
4.7 Invariants of symmetric objects
4.8 Invariants of 3D vector fields
4.9 Numerical experiments
4.10 Conclusion
Appendix 4.A
Appendix 4.B
Appendix 4.C
References
Chapter 5: Affine Moment Invariants in 2D and 3D
5.1 Introduction
5.2 AMIs derived from the Fundamental theorem
5.3 AMIs generated by graphs
5.4 AMIs via image normalization
5.5 The method of the transvectants
5.6 Derivation of the AMIs from the Cayley-Aronhold equation
5.7 Numerical experiments
5.8 Affine invariants of color images
5.9 Affine invariants of 2D vector fields
5.10 3D affine moment invariants
5.11 Beyond invariants
5.12 Conclusion
Appendix 5.A
Appendix 5.B
References
Chapter 6: Invariants to Image Blurring
6.1 Introduction
6.2 An intuitive approach to blur invariants
6.3 Projection operators and blur invariants in Fourier domain
6.4 Blur invariants from image moments
6.5 Invariants to centrosymmetric blur
6.6 Invariants to circular blur
6.7 Invariants to
N
-FRS blur
6.8 Invariants to dihedral blur
6.9 Invariants to directional blur
6.10 Invariants to Gaussian blur
6.11 Invariants to other blurs
6.12 Combined invariants to blur and spatial transformations
6.13 Computational issues
6.14 Experiments with blur invariants
6.15 Conclusion
Appendix 6.A
Appendix 6.B
Appendix 6.C
Appendix 6.D
Appendix 6.E
Appendix 6.F
Appendix 6.G
References
Chapter 7: 2D and 3D Orthogonal Moments
7.1 Introduction
7.2 2D moments orthogonal on a square
7.3 2D moments orthogonal on a disk
7.4 Object recognition by Zernike moments
7.5 Image reconstruction from moments
7.6 3D orthogonal moments
7.7 Conclusion
References
Chapter 8: Algorithms for Moment Computation
8.1 Introduction
8.2 Digital image and its moments
8.3 Moments of binary images
8.4 Boundary-based methods for binary images
8.5 Decomposition methods for binary images
8.6 Geometric moments of graylevel images
8.7 Orthogonal moments of graylevel images
8.8 Conclusion
Appendix 8.A
References
Chapter 9: Applications
9.1 Introduction
9.2 Image understanding
9.3 Image registration
9.4 Robot and autonomous vehicle navigation and visual servoing
9.5 Focus and image quality measure
9.6 Image retrieval
9.7 Watermarking
9.8 Medical imaging
9.9 Forensic applications
9.10 Miscellaneous applications
9.11 Conclusion
References
Chapter 10: Conclusion
10.1 Summary of the book
10.2 Pros and cons of moment invariants
10.3 Outlook to the future
Index
End User License Agreement
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Guide
Cover
Table of Contents
Begin Reading
List of Illustrations
Preface
Figure 1 The number of moment-related publications as found in SCOPUS.
Chapter 1: Motivation
Figure 1.1 General image analysis flowchart
Figure 1.2 An example of the car licence plate recognition
Figure 1.3 Image acquisition process with degradations
Chapter 2: Introduction to Object Recognition
Figure 2.1 Two-dimensional feature space with two classes, almost an ideal example. Each class forms a compact cluster (the features are invariant to translation, rotation, scaling, and skewing of the characters) and the clusters are well separated from one another (the features are discriminative although the characters are visually similar to each other)
Figure 2.2 The object and its convex hull
Figure 2.3 The object and its minimum bounding rectangle
Figure 2.4 Radial function of the object
Figure 2.5 Star-shaped object and its radial shape vector
Figure 2.6 The object and its shape matrix
Figure 2.7 Examples of textures. The texture is often a more discriminative property than the shape and the color
Figure 2.8 The original Barabara image (a) and its wavelet decomposition into depth two (b)
Figure 2.9 Semi-differential invariants. The object is divided by inflection points. Both convex and concave cuts can be used for a description by global invariants
Figure 2.10 Partition of the feature space defines a classifier
Figure 2.11 Three different classifiers as the results of three training algorithms on the same training set. (a) Over-simplified classifier, (b) over-trained classifier, and (c) close-to-optimal classifier
Figure 2.12 Decision boundary of the NN classifier depends on the used distance: (a) the nearest distance and (b) the mean distance
Figure 2.13 Robustness of the -NN classifier. The unknown sample “+” is classified as a circle by the NN classifier and as a cross by the 2-NN classifier. The later choice better corresponds to our intuition
Figure 2.14 SVM classifiers. The hard margin (a) and the soft margin (b) constraints
Figure 2.15 Multispectral satellite image. The objects are single pixels, the features are their intensities in the individual spectral bands. This kind of data is ideal for the Bayesian classifier
Figure 2.16 Principal component transformation: (a) unstructured original data in feature space, correlated (b) transformation into new feature space , decorrelated. The first principal component is
Figure 2.17 PCT of data consisting of two classes: (a) the original feature space, (b) new feature space after the PCT. The first principal component is thanks to higher variance but the between-class separability is provided solely by
Chapter 3: 2D Moment Invariants to Translation, Rotation, and Scaling
Figure 3.1 The desired behavior of TRS moment invariants–all instances of a rotated and scaled image have almost the same values of the invariants (depicted for two invariants)
Figure 3.2 Numerical test of the normalized moment . Computer-generated scaling of the test image ranged form to . To show robustness, each image was corrupted by additive Gaussian white noise. Signal-to-noise ratio (SNR) ranged from 50 (low noise) to 10 (heavy noise). Horizontal axes: scaling factor and SNR, respectively. Vertical axis–relative deviation (in %) between of the original and that of the scaled and noisy image. The test proves the invariance of and illustrates its high robustness to noise
Figure 3.3 Numerical test of the aspect-ratio invariant . Computer-generated scaling of the test image ranged form 0.5 to 2 in both directions independently. Horizontal axes: scaling factors and , respectively. Vertical axis–relative deviation (in %) between of the original and that of the scaled image. The test illustrates the invariance of . Higher relative errors for low scaling factors and typical jagged surface of the graph are the consequences of the image resampling
Figure 3.4 Numerical test of the basic invariant . Computer-generated rotation of the test image ranged form 0 to 360 degrees. To show robustness, each image was corrupted by additive Gaussian white noise. Signal-to-noise ratio (SNR) ranged from 40 (low noise) to 10 (heavy noise). Horizontal axes: rotation angle and SNR, respectively. Vertical axis–relative deviation (in %) between of the original and that of the rotated and noisy image. The test proves the invariance of and illustrates its high robustness to noise
Figure 3.5 The smiles: (a) original and (b) another Figure created from the original according to Eq. (3.34). For the values of the respective invariants see Table 3.1
Figure 3.6 (a) Original image of a pan and (b) a virtual “two-handle” pan. These objects are distinguishable by the basic invariants but not by the Hu invariants
Figure 3.7 The test image and its mirrored version. Basic invariants of the mirrored image are complex conjugates of those of the original
Figure 3.8 Numerical test of the contrast and TRS invariant for and . Computer-generated scaling of the test image ranged from to , and the contrast stretching factor ranged from to . Horizontal axes: scaling factor and contrast stretching factor , respectively. Vertical axis–relative deviation (in %) between of the original and that of the scaled and stretched image. The test proves the invariance of with respect to both factors. However, for down-scaling with and , the resampling effect leads to higher relative errors
Figure 3.9 Sample objects with an -fold rotation symmetry. From (a) to (e): , 3, 5, 4, and 2, respectively. All depicted cases have also an axial symmetry; however this is not a rule
Figure 3.10 The matrix of the complex moments of an -fold symmetric object. The gray elements are always zero. The distance between neighboring non-zero diagonals is
Figure 3.11 The test logos (from left to right): Mercedes-Benz, Mitsubishi, Recycling, Fischer, and Woolen product
Figure 3.12 The logo positions in the space of two invariants and showing good discrimination power. The symbols: –Mercedes-Benz, –Mitsubishi, –Recycling, *–Fischer, and –Woolen product. Each logo was randomly rotated ten times
Figure 3.13 The logo positions in the space of two invariants and introduced in Theorem 3.2. These invariants have no discrimination power with respect to this logo set. The symbols: –Mercedes-Benz, –Mitsubishi, –Recycling, *–Fischer, and –Woolen product
Figure 3.14 The test patterns: capital L, rectangle, equilateral triangle, circle, capital F, diamond, tripod, cross, and ring
Figure 3.15 The space of two invariants and introduced in Theorem 3.2. The symbols: ×–rectangle, –diamond, –equilateral triangle, –tripod +–cross, •–circle, and –ring. The discriminability is very poor
Figure 3.16 The space of two invariants and introduced in Theorem 3.2, . The symbols: ×–rectangle, –diamond, –equilateral triangle, –tripod +–cross, •–circle, and –ring, *–capital F, and –capital L. Some clusters are well separated
Figure 3.17 The space of two invariants and introduced in Theorem 3.2, (logarithmic scale). The symbols: ×–rectangle, –diamond, –equilateral triangle, –tripod +–cross, •–circle, and –ring, *–capital F, and –capital L. All clusters except the circle and the ring are separated
Figure 3.18 The space of two invariants and . The symbols: ×–rectangle, –diamond, –equilateral triangle, –tripod +–cross, •–circle, and –ring, *–capital F, and –capital L. All clusters except the circle and the ring are separated. Comparing to Figure 3.17, note less correlation of the invariants and a lower dynamic range
Figure 3.19 The toy set used in the experiment
Figure 3.21 Ambiguity of the principal axis normalization. These four positions of the object satisfy . Additional constraints and make the normalization unique
Figure 3.20 Principal axis normalization to rotation–an object in the normalized position along with its reference ellipse superimposed
Figure 3.22 An example of the ambiguity of the normalization by complex moments. In all these six positions, is real and positive as required
Figure 3.23 Turbulence in a fluid (the graylevels show the local velocity of the flow, the direction is not displayed)
Figure 3.24 Image gradient as a vector field. For visualization purposes, the field is depicted by arrows on a sparse grid and laid over the original Lena image
Figure 3.25 The wind velocity forecast for the Czech Republic (courtesy of the Czech Hydrometeorological Institute, the numerical model Aladin). The longer dash is constant part of the wind, the shorter dash expresses perpendicular squalls. The Figure actually represents two vector fields
Figure 3.26 Various rotations of a vector field: (a) the original vector field, (b) the inner rotation, (c) the outer rotation, (d) the total rotation. The graylevels corresponds to the vector sizes
Figure 3.27 Optical flow as a vector field: (a) the original field, (b) the optical flow computed from the video sequence rotated by , (c) the original optical flow field after total rotation by , (d) the optical flow computed from the video sequence rotated by . All rotations are counterclockwise. The arrows show the direction and velocity of the movement between two consecutive frames of the video sequence
Chapter 4: 3D Moment Invariants to Translation, Rotation, and Scaling
Figure 4.1 3D images of various nature: (a) real volumetric data – CT of a human head, (b) real binary volumetric data measured by Kinect (the bag), (c) binary object with a triangulated surface (the pig) from the PSB, and (d) artificial binary object with a triangulated surface (decagonal prism)
Figure 4.2 Yaw, pitch, and roll angles in the Tait-Bryan convention
Figure 4.3 The generating graphs of (a) both and , (b) both and and (c) both and
Figure 4.4 The spherical harmonics. (a) , , (b) , , (c) , , (d) , . Imaginary parts are displayed for and real parts for
Figure 4.5 The microscope: (a) the original position, (b) the first random rotation, (c) the second random rotation, (d) the standard position by the covariance matrix, (e) the standard position by the matrix of inertia and (f) the standard position by the 3D complex moments
Figure 4.6 The symmetric bodies with the symmetry groups (a) (the repeated tetrahedrons), (b) (the repeated tetrahedrons), (c) (the repeated tetrahedrons), (d) (the pentagonal pyramid), (e) (the repeated pyramids), (f) (the repeated tetrahedrons), (g) (the repeated pyramids), (h) (the pentagonal prism), (i) (the repeated tetrahedrons), (j) (the regular tetrahedron), (k) (the cube) and (l) (the octahedron)
Figure 4.7 The symmetric bodies with the symmetry groups: (a) (the dodecahedron), (b) (the icosahedron), (c) (the simplest canonical polyhedron), (d) (the pyritohedral cube), (e) (the propello-tetrahedron), (f) (the snub dodecahedron), (g) (the snub cube), (h) (the chiral cube), (i) (the conic), (j) (the cylinder) and (k) (the sphere)
Figure 4.8 Ancient Greek amphoras: (a) photo of A1, (b) photo of A2, (c) wire model of the triangulation of A2
Figure 4.9 The archeological findings used for the experiment
Figure 4.10 The absolute values of the invariants in the experiment with the archeological findings from Figure 4.9. Legend: (a) , (b) ×, (c) +, (d) and (e)
Figure 4.11 Examples of the class representatives from the Princeton Shape Benchmark: (a) sword, (b) two-story home, (c) dining chair, (d) handgun, (e) ship, and (f) car
Figure 4.12 Six submarine classes used in the experiment
Figure 4.13 Example of the rotated and noisy submarine from Figure 4.12c with SNR (a) 26 dB and (b) 3.4 dB
Figure 4.14 The success rate of the invariants from the volume geometric, complex, and normalized moments compared with the spherical harmonic representation and with the surface geometric moment invariants
Figure 4.15 3D models of two teddy bears created by Kinect
Figure 4.16 The values of the invariants of two teddy bears
Figure 4.17 The bodies used in the experiment: (a) two connected tetrahedrons with the symmetry group , (b) the body with the reflection symmetry , (c) the rectangular pyramid with the symmetry group , (d) the triangular pyramid with the symmetry group and (e) the triangular prism with the symmetry group
Figure 4.18 Randomly rotated sampled bodies: (a) the rectangular pyramid, (b) the cube and (c) the sphere
Figure 4.19 The images of the symmetric objects downloaded from the PSB: (a) hot air balloon 1338, (b) hot air balloon 1337, (c) ice cream 760, (d) hot air balloon 1343, (e) hot air balloon 1344, (f) gear 741, (g) vase 527, (h) hot air balloon 1342
Figure 4.20 The objects without texture and color, used as the database templates: (a) hot air balloon 1338, (b) hot air balloon 1337, (c) ice cream 760, (d) hot air balloon 1343, (e) hot air balloon 1344, (f) gear 741, (g) vase 527, (h) hot air balloon 1342
Figure 4.21 Examples of the noisy and rotated query objects: (a) hot air balloon 1338, (b) hot air balloon 1337, (c) ice cream 760, (d) hot air balloon 1343, (e) hot air balloon 1344, (f) gear 741, (g) vase 527, (h) hot air balloon 1342
Chapter 5: Affine Moment Invariants in 2D and 3D
Figure 5.1 The projective deformation of a scene due to a non-perpendicular view. Square tiles appear as quadrilaterals; the transformation preserves straight lines but does not preserve their collinearity
Figure 5.2 The projective transformation maps a square onto a quadrilateral (computer-generated example)
Figure 5.3 The affine transformation maps a square to a parallelogram
Figure 5.4 The affine transformation approximates the perspective projection if the objects are small
Figure 5.5 The graphs corresponding to the invariants (a) (5.14) and (b) (5.15)
Figure 5.6 The graph leading to a vanishing invariant (5.18)
Figure 5.7 The graph corresponding to the invariant from (5.23)
Figure 5.8 The standard positions of the cross with varying thickness : (a) Thin cross, , (b) slightly less than , (c) slightly greater than , and (d) . Note the difference between the two middle positions
Figure 5.9 Numerical test of invariance of . Horizontal axes: horizontal skewing and rotation angle, respectively. Vertical axis – relative deviation (in %) between of the original and that of the transformed image. The test proves the invariance of
Figure 5.10 The comb. The viewing angle increases from (top) to (bottom)
Figure 5.11 The original digits used in the recognition experiment
Figure 5.12 The examples of the deformed digits
Figure 5.13 The digits in the feature space of affine moment invariants and (ten various transformations of each digit, a noise-free case)
Figure 5.14 The test patterns. (a) originals, (b) examples of distorted patterns, and (c) the normalized positions
Figure 5.15 The space of two AMIs and
Figure 5.16 The space of two normalized moments and
Figure 5.17 The part of the mosaic used in the experiment
Figure 5.18 The image of a tile with a slight projective distortion
Figure 5.19 The geometric patterns. The feature space of and : – equilateral triangle, – isosceles triangle, – square, – rhombus, – big rectangle, – small rectangle, – rhomboid, – trapezoid, – pentagon, – regular hexagon, – irregular hexagon, – circle, – ellipse
Figure 5.20 The animal silhouettes. The feature space of and : – bear, – squirrel, – pig, – cat, – bird, – dog, – cock, – hedgehog, – rabbit, – duck, – dolphin, – cow
Figure 5.21 The geometric patterns. The feature space of and : – equilateral triangle, – isosceles triangle, – square, – rhombus, – big rectangle, – small rectangle, – rhomboid, – trapezoid, – pentagon, – regular hexagon, – irregular hexagon, – circle, – ellipse
Figure 5.22 The animal silhouettes. The feature space of and : – bear, – squirrel, – pig, – cat, – bird, – dog, – cock, – hedgehog, – rabbit, – duck, – dolphin, – cow
Figure 5.23 The image of a tile with a heavy projective distortion
Figure 5.24 Scrabble tiles – the templates
Figure 5.25 Scrabble tiles to be recognized – a sample scene
Figure 5.26 The space of invariants and . Even in two dimensions the clustering tendency is evident. Legend: A, B, C, D, E, H, I, J, K, L, M, N, O, P, R, S, T, U, V, Y, Z
Figure 5.27 The mastercards. Top row from the left: Girl, Carnival, Snowtubing, Room-bell, and Fireplace. Bottom row: Winter cottage, Spring cottage, Summer cottage, Bell, and Star
Figure 5.28 The card “Summer cottage” including all its rotations. The rotations are real due to the rotations of the hand-held camera. This acquisition scheme also introduces mild perspective deformations. The third and fourth rows contain the other card from the pair
Figure 5.29 The mastercards. Legend: – Girl, – Carnival, – Snowtubing, – Room-bell and – Fireplace, – Winter cottage, – Spring cottage, – Summer cottage, – Bell and – Star. A card from each pair is expressed by the black symbol, while the other card is expressed by the gray symbol. Note that some clusters have been split into two sub-clusters. This is because the two cards of the pair may be slightly different. This minor effect does not influence the recognition rate
Figure 5.30 The hypergraphs corresponding to invariants (a) and (b)
Figure 5.31 Nonlinear deformation of the text captured by a fish-eye-lens camera
Figure 5.32 Character deformations due to the print on a flexible surface
Figure 5.33 The six bottle images used in the experiment
Chapter 6: Invariants to Image Blurring
Figure 6.1 Two examples of image blurring. Space-variant out-of-focus blur caused by a narrow depth of field of the camera (a) and camera-shake blur at a long exposure (b)
Figure 6.2 Visualization of the space variant PSF. A photograph of a point grid by a hand-held camera blurred by camera shake. The curves are images of bright points equivalent to the PSF at the particular places
Figure 6.3 Blurred image (top) to be matched against a database (bottom). A typical situation where the convolution invariants may be employed
Figure 6.4 The flowchart of image deconvolution (left) and of the recognition by blur invariants (right)
Figure 6.5 The PSF of non-linear motion blur which exhibits an approximate central symmetry but no axial symmetry. Visualization of the PSF was done by photographing a single bright point
Figure 6.6 Two examples of an out-of-focus blur on a circular aperture: (a) the books, (b) the harbor in the night
Figure 6.7 Real examples of a circularly symmetric out-of-focus PSF: (a) a disk-like PSF on a circular aperture, (b) a ring-shaped PSF of a catadioptric objective
Figure 6.8 A bokeh example. The photo was taken with a mirror-lens camera. The picture (a) is a close-up of a larger scene and shows the out-of-focus background. The estimated PSF of an annular shape, which is characteristic for this kind of lenses, is shown in (b)
Figure 6.9 Airy function, a diffraction PSF on a circular aperture
Figure 6.10 (a) The original image (the Rainer cottage in High Tatra mountains, Slovakia) and (b) its projection
Figure 6.11 Partially open diaphragm with 9 blades forms a polygonal aperture
Figure 6.12 Out-of-focus PSFs on polygonal apertures of three different cameras obtained as photographs of a bright point. A nine-blade diaphragm was used in (a) and (b); a seven-blade diaphragm in (c)
Figure 6.13 Various projections of the original image from Figure 6.10: (a) , (b) , (c) and (d)
Figure 6.14 The structure of the matrix of -FRS blur invariants. The gray elements on the diagonals are identically zero regardless of . The white elements stand for non-trivial invariants
Figure 6.15 The shape of the out-of-focus PSF can be observed in the background. This bokeh effect may serve for estimation of . In this case,
Figure 6.16 Two examples of real out-of-focus PSF on apertures with dihedral symmetry, obtained by photographing a bright point. (a) , (b)
Figure 6.17 Dihedral projections of the original image from Figure 6.10. (a) , (b) , (c) and (d)
Figure 6.18 The structure of the dihedral invariant matrix. The dark gray elements on the main diagonal vanish for any . The light gray elements on the minor diagonals are non-trivial, but their real and imaginary parts are constrained. The white elements stand for non-trivial complex-valued invariants
Figure 6.19 The image blurred by a camera shake blur, which is approximately a directional blur
Figure 6.20 Directional projections of the original image from Figure 6.10. (a) , (b)
Figure 6.21 The image of a bright point (a close-up of Figure 6.1b) can serve for estimation of the blur direction
Figure 6.22 Amplitude spectrum of a motion-blurred image. The zero lines are perpendicular to the motion direction
Figure 6.23 Estimated blur direction from the image spectrum in Figure 6.22
Figure 6.24 Gaussian blur caused by the mist coupled with a contrast decrease (Vltava river, Prague, Czech Republic)
Figure 6.25 Gaussian blur as a result of denoising. The original image corrupted by a heavy noise was smoothed by a Gaussian kernel to suppress the high-frequency noise components
Figure 6.26 2D Gaussian projection of the original image from Figure 6.10
Figure 6.27 The original image and its blurred and affinely deformed version (Liberec, Czech Republic). The values of the combined blur-affine invariants are the same for both images
Figure 6.28 The test image of the size (the statue of Pawlu Boffa in Valletta, Malta): (a) original, (b) the image blurred by circularly symmetric blur of a standard deviation 100 pixels, (c) the same blurred image without margins
Figure 6.29 The values of the invariants, the exact convolution model
Figure 6.30 The values of the invariants violated by the boundary effect, when the realistic convolution model was applied
Figure 6.31 The original high-resolution satellite photograph, the City of Plzeň, Czech Republic, with three selected templates
Figure 6.32 The templates: The confluence of the rivers Mže and Radbuza (a), the apartment block (b), and the road crossing (c)
Figure 6.33 The blurred and noisy image. An example of a frame in which all three templates were localized successfully
Figure 6.35 The graph summarizing the results of the experiment. The area below the curve denotes the domain in a “noise-blur space” in which the algorithm works mostly successfully
Figure 6.34 The boundary effect. Under a discrete convolution on a bounded support, the pixels near the template boundary (white square) are affected by the pixels lying outside the template. An impact of this effect depends on the size of the blurring mask (black square)
Figure 6.36 The influence of the boundary effect and noise on the numerical properties of the centrosymmetric convolution invariant . Horizontal axes: the blurring mask size and the signal-to-noise ratio, respectively. Vertical axis: the relative error in %. The invariant corrupted by a boundary effect (a) and the same invariant calculated from a zero-padded template where no boundary effect appeared (b)
Figure 6.37 The house image: (a) the deblurred version, (b) the original blurred version
Figure 6.38 Four images of the sunspot blurred by a real atmospheric turbulence blur of various extent. The images are ordered from the less to the most blurred one. A template is depicted in the first image to illustrate its size
Figure 6.39 Sample “clear” images of the the CASIA HFB database. The database consists of very similar faces
Figure 6.40 Sample test images degraded by heavy blur and noise ( and SNR = 0 dB)
Figure 6.41 The traffic signs used in the experiment. First row: No entry, No entry into a one-way road, Main road, End of the main road. Second row: No stopping, No parking, Give way, Be careful in winter. Third row: Roundabout ahead, Roundabout, Railway crossing, End of all prohibitions. Fourth row: Two-way traffic, Intersection, First aid, Hospital
Figure 6.42 The traffic signs successively blurred by masks with radii 0, 33, 66, and 100 pixels
Figure 6.43 The feature space of two non-invariant moments and . Legend: – No entry, – No entry into a one-way road, – Main road, – End of the main road, – No stopping, – No parking, – Give way, – Be careful in winter, – Roundabout ahead, – Roundabout, – Railway crossing, – End of all prohibitions, – Two-way traffic, – Intersection, – First aid, – Hospital
Figure 6.44 The feature space of real and imaginary parts of the invariant . Legend: – No entry, – No entry into a one-way road, – Main road, – End of the main road, – No stopping, – No parking, – Give way, – Be careful in winter, – Roundabout ahead, – Roundabout, – Railway crossing, – End of all prohibitions, – Two-way traffic, – Intersection, – First aid, – Hospital
Figure 6.45 The feature space of real parts of the invariant and . Legend: – No entry, – No entry into a one way road, – Main road, – End of the main road, – No stopping, – No parking, – Give way, – Be careful in winter, – Roundabout ahead, – Two-way traffic, – Intersection, – First aid, – Hospital
Figure 6.46 The detail of the feature space of real parts of the invariant and around zero. Legend: – No entry, – Main road, – No stopping, – Intersection, – First aid
Figure 6.47 The feature space of real parts of the invariant and – the minimum zoom. Legend: – No entry, – No entry into a one way road, – Main road, – End of the main road, – No stopping, – No parking, – Give way, – Be careful in winter, – Roundabout ahead, – Roundabout, – Railway crossing, – End of all prohibitions, – Two-way traffic, – Intersection, – First aid, – Hospital
Figure 6.49 The feature space of real parts of the invariant and – the maximum zoom. Legend: – No entry, – No stopping, – No parking
Figure 6.48 The feature space of real parts of the invariant and – the medium zoom. Legend: – No entry, – No entry into a one way road, – No stopping, – No parking, – Give way, – Be careful in winter, – Roundabout ahead, – End of all prohibitions
Chapter 7: 2D and 3D Orthogonal Moments
Figure 7.1 The graphs of the Legendre polynomials up to the sixth degree
Figure 7.2 The graphs of the standard powers up to the sixth degree
Figure 7.3 The graphs of 2D kernel functions of the Legendre moments (2D Legendre polynomials) up to the fourth degree. Black color corresponds to , white color to 1
Figure 7.4 The graphs of the Chebyshev polynomials of the first kind up to the sixth degree
Figure 7.5 The graphs of the Chebyshev polynomials of the second kind up to the sixth degree
Figure 7.6 The graphs of 2D kernel functions of the Chebyshev moments of the first kind up to the fourth order. Black color corresponds to , white color to 1
Figure 7.7 The graphs of the Hermite polynomials up to the sixth degree
Figure 7.8 The graphs of the Gaussian-Hermite polynomials with
Figure 7.9 The graphs of the Gegenbauer polynomials for up to the sixth degree
Figure 7.10 The graphs of the weighted Laguerre polynomials up to the sixth degree
Figure 7.11 The graphs of the Krawtchouk polynomials up to the sixth degree
Figure 7.12 The graphs of the weighted Krawtchouk polynomials for up to the second degree
Figure 7.13 The graphs of the weighted Krawtchouk polynomials for up to the second degree
Figure 7.14 The values of the selected invariants computed from the rotated pexeso card
Figure 7.15 The graphs of the Zernike radial functions up to the sixth degree. The graphs of the polynomials of the same degree but different repetition are drawn by the same type of the line
Figure 7.16 The graphs of the Zernike polynomials up to the fourth degree. Black , white . Real parts: 1st row: , , 3rd row: , , 5th row: , , 7th row: , , 9th row: , . Imaginary parts: 2nd, 4th, 6th, 8th and 10th row, respectively. The indices are the same as above
Figure 7.17 The graphs of the radial functions of the orthogonal Fourier-Mellin moments up to the sixth degree
Figure 7.18 The graphs of 2D kernel functions of the orthogonal Fourier-Mellin moments up to the fourth order. Black , white . Real parts: 1st row: , , 3rd row: , , 5th row: , , 7th row: , , 9th row: , . Imaginary parts: 2nd, 4th, 6th, 8th and 10th rows. The indices are the same as above
Figure 7.19 The playing cards: (a) Mole cricket, (b) Cricket, (c) Bumblebee, (d) Heteropter, (e) Poke the Bug, (f) Ferdy the Ant 1, (g) Ferdy the Ant 2, (h) Snail, (i) Ant-lion 1, (j) Ant-lion 2, (k) Butterfly, and (l) Ladybird
Figure 7.20 The feature space of two Zernike normalized moments and . – Ferdy the Ant 1, – Ferdy the Ant 2, – Ladybird, ◊– Poke the Bug, – Ant-lion 1, – Ant-lion 2, ×– Mole cricket, +– Snail, – Butterfly, – Cricket, – Bumblebee, – Heteropter
Figure 7.21 The error rate in dependency on parameter of the Gaussian-Hermite moments. There were used the moments up to: – 3rd order, – 6th order
Figure 7.22 The feature space of two Zernike normalized moments of color images and . – Ferdy the Ant 1, – Ferdy the Ant 2, – Ladybird, ◊– Poke the Bug, – Ant-lion 1, – Ant-lion 2, ×– Mole cricket, +– Snail, – Butterfly, – Cricket, – Bumblebee, – Heteropter
Figure 7.23 The collapse of the image reconstruction from geometric moments by the direct calculation. Top row: original images and , bottom row: the reconstructed images from the moments
Figure 7.24 Image reconstruction from geometric moments in the Fourier domain. The original image and the reconstructed images with maximum moment orders 21, 32, 43, 54, 65, 76, and 87, respectively
Figure 7.25 An example of the polar raster
Figure 7.26 Image reconstruction from the orthogonal moments: (a) Legendre moments, (b) Chebyshev moments of the first kind, (c) Gaussian-Hermite moments, and (d) Zernike moments
Figure 7.27 Image reconstruction from the incomplete set of discrete Chebyshev moments. The maximum moment order is , respectively. The last image (bottom-right) is a precise reconstruction of the original image
Figure 7.28 Image reconstruction from the orthogonal moments: (a) Legendre moments, (b) Chebyshev moments of the first kind, (c) Gaussian-Hermite moments, and (d) Zernike moments
Figure 7.30 Detail of the reconstruction from the Gaussian-Hermite moments: (a) reconstruction and (b) original
Figure 7.29 Image reconstruction from the discrete Chebyshev moments. There are no errors; it is identical with the original Lena image
Figure 7.31 The test image for the reconstruction experiment with discrete Chebyshev moments (Astronomical Clock, Prague, Czech Republic)
Figure 7.32 The reconstruction experiment: (a) the reconstructed cropped image and (b) the error map. The range of errors from to 9 is visualized in the range black – white. The relative mean square error is
Figure 7.33 The close-up of the error map with typical oscillations
Figure 7.34 The reconstruction experiment: (a) the reconstructed cropped image and (b) the error map. The range of errors from to 198 is visualized in the range black – white. The relative mean square error is
Figure 7.35 The close-up of the error map
Figure 7.36 The values of five selected invariants of the teddy bear
Figure 7.37 Archeological findings used for the experiment
Figure 7.38 Volumetric forms of the archeological findings
Figure 7.39 The rotated and noisy version of the object from Figure 7.38c. The noise of SNR = 12 dB was added to the circumscribed sphere
Figure 7.40 The success rates of the recognition of the archeological artifacts by Gaussian-Hermite moments and Zernike moments
Figure 7.41 Two different airplanes from the PSB. (a) Object No. 1245, (b) Object No. 1249
Figure 7.42 The noisy (10 dB) versions of the airplanes. (a) Object No. 1245, (b) Object No. 1249
Figure 7.43 The success rate of the noisy airplane recognition
Figure 7.44 The helicopter: (a) the original converted to volumetric data, (b–d) the reconstruction from the weighted Hermite moments up to the order: (b) 8, (c) 46 and (d) 84
Figure 7.45 The reconstruction of the helicopter from the seventy-first-order geometric moments
Chapter 8: Algorithms for Moment Computation
Figure 8.1 Sampling function with the steps and
Figure 8.2 The concept of a digital image (a) as a sum of Dirac -functions, (b) as a nearest neighbor interpolation of the samples, and (c) as a bilinear interpolation of the samples
Figure 8.3 The spider (PSB No. 19): (a) the original triangulated version, (b) a conversion to the volumetric representation without any preprocessing, and (c) the same with previous detecting and filling-up the holes
Figure 8.4 The delta method. In the basic version the object is decomposed into rows (a). The generalized version unifies the adjacent rows of the same length into a rectangle (b)
Figure 8.5 Generalized delta method: (a) the headless Figure (black=1) (b) the generalized delta method applied row-wise (395 blocks) (c) the generalized delta method applied column-wise (330 blocks). The basic delta method generated 1507 blocks
Figure 8.6 Quadtree decomposition of the image
Figure 8.7 Quadtree decomposition. (a) the headless figure, 3843 square blocks and (b) the moth, 5328 square blocks
Figure 8.8 Partial object decomposition after two outer loops of the morphological method
Figure 8.9 Distance transformation decomposition: (a) the headless figure, 421 blocks and (b) the moth, 1682 blocks
Figure 8.10 The first level of the GBD method. (a) The input object. (b) All possible chords connecting the cogrid concave vertices. The crosses indicate the chord intersections. (c) The corresponding bipartite graph with a maximum independent set of three vertices. Other choices are also possible, such as or . (d) The first-level object decomposition
Figure 8.11 The second level of the GBD method. (a) The first-level decomposition (solid line) and a possible second-level decomposition (dashed line). From each concave vertex a single chord of arbitrary direction is constructed. (b) If on the first level the chords , , and were chosen, then both GBD and GDM would yield the same decomposition
Figure 8.12 Graph-based decomposition: (a) the headless figure, 302 blocks and (b) the moth, 1092 blocks. In the case of the moth, the result is very similar to the GDM
Figure 8.13 The time complexity of the moment computation of the headless Figure image. Legend: - definition, - generalized delta method, × - quadtree, - graph-based decomposition, - distance transformation
Figure 8.14 The time complexity of the moment computation of the moth image. Legend: - definition, - generalized delta method, × - quadtree, - graph-based decomposition, - distance transformation
Figure 8.15 The time complexity of the moment computation of the chessboard image. Legend: - definition, - generalized delta method, × - quadtree, - graph-based decomposition, - distance transformation
Figure 8.16 Examples of 3D binary objects in a volumetric representation: (a) Teddy bear – 85068 voxels, (b) Glasses – 3753 voxels
Figure 8.17 Various methods of 3D decomposition: (a) glasses – octree – 6571 blocks, (b) airplane – octree – 4813 blocks, (c) glasses – 3GDM – 776 blocks, (d) airplane – 3GDM – 586 blocks, (e) glasses – suboptimal algorithm – 732 blocks, (f) airplane – suboptimal algorithm – 573 blocks
Figure 8.18 The random cube: (a) original (518 voxels), (b) octree decomposition into 504 blocks, (c) 3GDM decomposition into 227 blocks, (d) suboptimal decomposition into 208 blocks
Figure 8.19 An image containing ten graylevels only
Figure 8.20 Nine intensity slices of the image from Figure 8.19 (the zeroth slice has been omitted)
Figure 8.21 Bit slicing of the Klínovec image: from (a) to (h) the bit planes from 7 to 0
Figure 8.22 (a) The original graylevel photograph of the size (lookout and telecommunication tower at Klínovec, Ore Mountains, Czech Republic), (b) the image after the bit planes 0 and 1 have been removed. To human eyes, the images look the same
Figure 8.23 The computational flow of the Prata method. The initialization is and
Figure 8.24 The computational flow of the Kintner method. The first two values in each sequence, i.e., , and , must be computed directly from equation (8.39)
Figure 8.25 The computational flow of the Chong method. The first two values in each sequence, i.e., , must be computed directly from equation (8.42)
Figure 8.26 The hole filled-up with artificial triangles. (a) The asymmetric method, (b) using the artificial centroid, (c) the iterative “cutting off” method. The rest of the object is not displayed
Chapter 9: Applications
Figure 9.1 Face recognition: Detection step - the person and its face have to be localized in the scene (a). The output is typically a face segmented from the background (b) and bounded by a box, a circle or an ellipse
Figure 9.2 Face recognition: Farokhi et al. [16] proposed using Hermite kernel filters as the local features for face recognition in near infrared domain. (a) Original image, (b)–(d) the output of three directional Hermite kernels
Figure 9.3 Image registration: Satellite example. (a) Landsat image – synthesis of its three principal components, (b) SPOT image – synthesis of its three spectral bands
Figure 9.4 Image registration: Satellite example. (a) Landsat, (b) SPOT image – segmented regions. The regions with a counterpart in the other image are numbered, the matched ones have numbers in a circle
Figure 9.5 Image registration: Satellite example. The superimposed Landsat and the registered SPOT images
Figure 9.7 Image registration: Image fusion example. Examples of several low quality images of the same scene – they are blurred, noisy, and of low resolution
Figure 9.6 Image registration: Image fusion example. Image fusion flowchart
Figure 9.8 Image registration: Image fusion example. Low-quality images of the same scene with detected distinctive points
Figure 9.9 Image registration: Image fusion example. Low-quality images after the registration
Figure 9.10 Image registration: Image fusion example. (a) The output of the image fusion. The resulting image has higher resolution and the blur and noise have been removed or diminished. (b) The scaled version of an input image for comparison. The image does not show comparable quality in terms of edge sharpness, noise level, and details visibility
Figure 9.11 Robot navigation: The examples of navigation marks with detected boundaries. Introduced complex geometric deformations introduced by the fish-eye lens camera are apparent
Figure 9.12 Focus measure: Focus measurement by moments. Four sample frames from the Saturn observation sequence, ordered automatically from the sharpest to the most blurred one. The result matches the visual assessment
Figure 9.13 Image retrieval: (a) the mountain (Malá Fatra, Slovakia), (b) the city (Rotterdam, Netherlands). The images are similar in terms of simple characteristics such as color distribution, but they have very different content
Figure 9.14 Image retrieval: Suk and Novotný [233] proposed the CBIR system for recognition of woody species in Central Europe, based on the Chebyshev moments and Fourier descriptors
Figure 9.15 Watermarking: The watermarked image. The watermark “COPYRIGHT” is apparent. This is an example of the visible watermark
Figure 9.16 Watermarking: An approach based on the geometric moments. (a) the original host image to be watermarked by the method [251], (b) the corresponding watermarked image. There is no visible inserted pattern into the image, only intensity variations are apparent. These changes are known disadvantages of the method [251]
Figure 9.17 Medical imaging: Landmark recognition in the scoliosis study. An example of the human body with attached landmarks and Moire contour graphs. The aim was to detect the landmarks and find their centers
Figure 9.18 Forensic application: Detection of near-duplicated image regions. An example of the tampered image with near-duplicated regions – (a) the forged image, (b) the original image
Figure 9.19 Forensic application: Detection of near-duplicated image regions. A two-dimensional feature space. Black dots represent overlapping blocks, which have to be analyzed (left image). The method finds all similar blocks to each block and analyzes their neighborhood (right image). In other words, all blocks inside a circle, which has the analyzed block as centroid are found. The radius is determined by the similarity threshold
Figure 9.20 Forensic application: Detection of near-duplicated image regions. The estimated map of duplicated regions
Figure 9.21 Optical flow: Noise resistant estimation. The result of (a) Zernike moments; (b) the standard method [351]. Stabilization of optical flow computation in the case of noisy magnetic resonance images of the heart
Figure 9.22 Solar flare: An example of the solar flare (the arrow-marked bright object). Dark areas correspond to sunspots
Figure 9.23 Solar flare: The time curve of a solar flare – (a) skewness, (b) first principal component. Data from Ondřejov observatory: the scanning began on the 18 December 2003, at 12:25 (time 0 in the graphs)
List of Tables
Chapter 3: 2D Moment Invariants to Translation, Rotation, and Scaling
Table 3.1 The values of the Hu invariants and the basic invariants (3.32) of “The smiles” in Figure 3.5. The only invariant discriminating them is . (The values shown here were calculated after bringing Figure 3.5 into normalized central position and nonlinearly scaled to the range from −10 to 10 for display.)
Chapter 4: 3D Moment Invariants to Translation, Rotation, and Scaling
Table 4.1 The numbers of the 3D irreducible and independent rotation invariants of weight
Table 4.2 The numbers of the 3D irreducible and independent complex moment rotation invariants
Table 4.3 The actual values of the invariants that should vanish due to the circular symmetry of the vases
Table 4.4 The actual values of the invariants that should vanish due to the two fold rotation symmetry of the vases
Table 4.5 The success rates of recognition of the generic classes
Table 4.6 The mean values of the invariants in the experiment with the artificial objects. In the first row, there is the corresponding symmetry group, “c” denotes the cube and “o” the octahedron
Table 4.7 The standard deviations of the invariants in the experiment with the artificial objects. In the first row, there is the corresponding symmetry group, “c” means cube, and “o” means octahedron. In the second row, there are the values of the ideal triangulated bodies, and in the third row the values of the sampled volumetric data
Chapter 5: Affine Moment Invariants in 2D and 3D
Table 5.1 The numbers of the irreducible and independent affine invariants
Table 5.2 The independent graph-generated AMIs up to the sixth order, which were selected thanks to their correspondence with the Hickman's set
Table 5.3 All possible terms of the invariant
Table 5.4 The matrix of the system of linear equations for the coefficients. The empty elements are zero. The solution is in the last two rows
Table 5.5 The values of the affine moment invariants of the comb; is the approximate viewing angle
Table 5.6 The recognition rate (in %) of the AMIs
Table 5.7 The recognition rate (in %) of the limited number of the AMIs
Table 5.8 The numbers of errors in recognition of 200 tile snaps
Table 5.9 The contingency Table of the AMIs up to eighth order
Table 5.10 Classification confidence coefficients of four letters (each having six different instances of distortion) using the invariant distance (top) and the spline-based elastic matching (bottom). MC means a misclassified letter
Chapter 6: Invariants to Image Blurring
Table 6.1 Template matching in astronomical images
Chapter 8: Algorithms for Moment Computation
Table 8.1 Comparison of the numbers of blocks and of the decomposition time
Table 8.2 The numbers of blocks in the individual bit planes, computation time, and the relative mean square error of the moments due to omitting less significant bit planes
2D and 3D Image Analysis by Moments
Jan Flusser, Tomáš Suk, Barbara Zitová
Institute of Information Theory and Automation,Czech Academy of Sciences,Prague,Czech Republic
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