Image processing: dealing with texture

Cover......Page 1
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Contents......Page 7
Preface......Page 15
Why are we interested in texture?......Page 17
How does texture give us information about the material of the imaged object?......Page 19
What are the problems of texture which image processing is trying to solve?......Page 20
What are the limitations of image processing in trying to solve the above problems?......Page 21
Box 1.1. An algorithm for the isolation of textured regions......Page 22
What is this chapter about?......Page 27
Which tools are appropriate for each type of texture?......Page 28
Box 2.1. Shape grammars......Page 29
What happens if the placement of the primitive pattern is not regular?......Page 37
What happens if the primitive patterns vary in a continuous way?......Page 38
Box 2.2. How can we draw random numbers according to a given probability density function?......Page 39
Box 2.3. What is a Poisson process?......Page 44
How can we estimate some aggregate parameters of the 2D Boolean model?......Page 47
How can we estimate some individual parameters of the 2D Boolean model?......Page 53
Box 2.4. How can we relate the individual parameters to the aggregate parameters of the 2D Boolean model?......Page 54
What is a ID Boolean model?......Page 60
How can we create ID strings from a 2D image?......Page 61
Box 2.5. Hilbert curves......Page 62
How can we estimate the parameters of the ID Boolean model?......Page 65
Box 2.6. Parameter estimation for the discrete ID Boolean model......Page 68
What happens if the primitive patterns are very irregular?......Page 69
What is dilation?......Page 70
What is opening?......Page 72
How do we do morphological operations if the structuring element is not symmetric about its centre?......Page 73
Is closing a commutative operation?......Page 76
Can we use different structuring elements for the erosion and the dilation parts of the opening and closing operators?......Page 77
Can we apply more than one morphological operator to the same image?......Page 79
Is erosion an associative operation as well?......Page 80
How can we use morphological operations to characterise a texture?......Page 82
Box 2.7. Formal definitions in mathematical morphology......Page 84
What is the \"take home\" message of this chapter?......Page 95
Are any of the methods appropriate for classifying binary textures useful for the analysis of grey textures?......Page 97
How may a grey image be analysed into a set of binary images by bit-slicing?......Page 99
Is there any relationship between the binary planes produced by thresholding and the bit planes?......Page 103
How does mathematical morphology generalise for grey images?......Page 106
What is a non-flat structuring element?......Page 108
What is the relationship between the morphological operations applied to an image and those applied to its complement?......Page 112
What is the purpose of using a non-flat structuring element?......Page 114
How can we perform granulometry with a grey image?......Page 115
Can we extract in one go the details of a signal, peaks or valleys, smaller than a certain size?......Page 116
How can we use the pattern spectrum to classify textures?......Page 120
What is the fractal dimension?......Page 121
Box 3.1. What is self-affine scaling?......Page 133
Box 3.2. What is the relationship between the fractal dimension and exponent HI......Page 134
Box 3.3. What is the range of values of H?......Page 135
What is a fractional Brownian motion?......Page 137
Box 3.4. Prove that the range of values of H for a fractional Brownian motion is (0,1)......Page 141
Box 3.5. What is the correlation between two increments of a fractional Brownian motion?......Page 142
Box 3.6. What is the power spectrum of a fractal?......Page 143
Box 3.8. What is the autocorrelation function of a fractal?......Page 168
Is there a way to enrich the description of textures offered by fractal models?......Page 180
What is lacunarity?......Page 181
What is a Markov random field?......Page 184
What is texture synthesis by analysis?......Page 185
How can we apply the Markov model to create textures?......Page 188
Can we apply the method discussed in the previous section to create images with 256 grey levels?......Page 190
What is the auto-normal Markov random field model?......Page 196
What is maximum likelihood estimation?......Page 198
What is the log-likelihood?......Page 200
How can we apply maximum likelihood estimation to estimate the parameters of a Markov random field?......Page 201
How do we know which parameter values to try when we apply MLE to estimate the Markov parameters?......Page 202
How can we estimate the Markov parameters with the least square error estimation method?......Page 205
Box 3.10. Least square parameter estimation for the MRF parameters......Page 206
What is a clique in a neighbourhood structure?......Page 212
What is a clique potential?......Page 214
Can we have a Markov random field with only singleton cliques?......Page 217
What is the relationship between the clique potentials and the Markov parameters?......Page 227
Box 3.11. Prove the equivalence of Markov random fields and Gibbs distributions (Hammersley-Clifford theorem)......Page 231
How can we use the Gibbs distribution to create textures?......Page 236
How can we create an image compatible with a Gibbs model if we are not interested in fixing the histogram of the image?......Page 242
How does the temperature parameter of the Gibbs distribution determine how distinguishable one configuration is from another?......Page 246
What is the critical temperature of a Markov random field?......Page 254
Can we use the autocorrelation function itself to characterise a texture?......Page 262
How can we use the autocorrelation function directly for texture characterisation?......Page 266
How can we infer the periodicity of a texture from the autocorrelation function?......Page 268
How can we extract parametric features from the autocorrelation function?......Page 269
Box 3.12. Least square fitting in 2D and ID......Page 273
Can we infer the periodicity of a texture directly from its power spectrum?......Page 276
Does the phase of the Fourier transform convey any useful information?......Page 281
Is it possible to compute from the image phase a function the value of which changes only due to genuine image changes?......Page 286
How do we perform phase unwrapping?......Page 287
What are the drawbacks of the simple phase unwrapping algorithm?......Page 289
Can we use non-parametric descriptions of texture?......Page 291
How is a co-occurrence matrix defined?......Page 293
How can we recognise textures with the help of the co-occurrence matrix?......Page 297
How can we choose the parameters of the co-occurrence matrix?......Page 299
What is the \"take home\" message of this chapter?......Page 310
How can we be sure that the texture inside an image window is stationary?......Page 313
What is the uncertainty principle in signal processing?......Page 314
Box 4.1. Prove the uncertainty principle in signal processing......Page 318
Does the window we choose in order to extract local information influence the result?......Page 321
How can we estimate \"what is happening where\" in a digital signal?......Page 331
How can we deal with the variability of the values of a feature?......Page 334
How do we know which size window we should use?......Page 339
How is the uncertainty principle generalised to 2D?......Page 342
What is a Gabor function?......Page 345
Why are Gabor functions useful in analysing a signal?......Page 346
How can we use the Gabor functions in practice?......Page 352
How is a Gabor function generalised in 2D?......Page 357
How may we use the 2D Gabor functions to analyse an image?......Page 361
Can we have alternative tessellations of the frequency domain?......Page 369
How can we define a Gaussian window in polar coordinates in the frequency domain?......Page 370
How do we express a frequency in octaves?......Page 372
How may we choose the parameters of the Gaussian window in the frequency space?......Page 373
Is it possible to have a window with sharp edges in one domain which has minimal side ripples in the other domain?......Page 396
Box 4.2. Of all the band-limited sequences one can define, which sequence has the maximum energy concentration between a given set of indices?......Page 397
Box 4.3. Do prolate spheroidal wave functions exists in the digital domain?......Page 400
What is the relationship of two band-limited functions, the Fourier transforms of which are given by the real functions $F{\\omega_x,\\omega_y)$, and $F(\227\\omega_x, \227\\omega_y)$, respectively?......Page 409
How can we construct a filter which is band-limited in two bands which are symmetrically placed about the origin of the axes in the frequency domain?......Page 410
Box 4.4. How may we generalise the prolate spheroidal sequence functions to 2D?......Page 419
Could we construct the 2D prolate spheroidal sequence filters as separable filters?......Page 441
What is the advantage of using separable filters?......Page 444
Is there a way other than using Gabor functions to span the whole spatio-frequency space?......Page 452
What is a wavelet?......Page 455
How can we use wavelets to analyse a signal?......Page 456
Box 4.5. How should we choose the mother wavelet?......Page 458
Box 4.6. Does the wavelet function minimise the uncertainty inequality?......Page 464
How is the wavelet transform adapted for digital signals?......Page 476
How do we compute the wavelet coefficients in practice?......Page 479
Why is the continuous wavelet transform invertible and the discrete wavelet transform non-invertible?......Page 490
How can we span the part of the \"what happens when\" space which contains the direct component of the signal?......Page 491
How can we extract the coarse resolution content of a signal from its content at a finer resolution?......Page 493
How can we choose the scaling function?......Page 497
How do we perform the multiresolution analysis of a signal in practice?......Page 501
Why in tree wavelet analysis do we always analyse the part of the signal which contains the low frequencies only?......Page 502
Box 4.7. How do we recover the original signal from its wavelet coefficients in practice?......Page 510
How may we use wavelets to process images?......Page 516
What is the maximum overlap algorithm?......Page 523
What is the relationship between Gabor functions and wavelets?......Page 534
What is feature selection?......Page 537
WThat is the histogram of distances in a feature space?......Page 539
Is it possible that the histogram of distances does not pick up the presence of clusters, even though clusters are present?......Page 541
What is the X-means algorithm?......Page 543
What is deterministic annealing?......Page 544
Box 4.8. Maximum entropy clustering......Page 545
How can we compute the Bhattacharyya distance in practice?......Page 551
How may we assess the quality of a segmentation using a manual segmentation as reference?......Page 552
What is a confusion matrix?......Page 553
What are the over- and under-detection errors?......Page 554
How are Laws' masks defined?......Page 555
Is there a systematic way to construct features that span the \"what looks like where\" space completely?......Page 564
How can we expand a local image neighbourhood in terms of the Walsh elementary images?......Page 572
Can we use convolution to compute the coefficients of the expansion of a sub-image in terms of a set of elementary images?......Page 578
Is there any other way to express the local structure of the image?......Page 589
How can we make this representation rotationally invariant?......Page 590
How can we make this representation appropriate for macro-textures?......Page 591
What is a pseudo-metric?......Page 592
How can we measure the difference between two histograms?......Page 593
How can we use the local binary patterns to segment textures?......Page 595
How can we overcome the shortcomings of the LBP segmentation?......Page 596
What is the Wigner distribution?......Page 599
What is the pseudo-Wigner distribution?......Page 607
What is the Kaiser window?......Page 608
What is the Nyquist frequency?......Page 610
Should we worry about aliasing when we use the pseudo-Wigner distribution for texture analysis?......Page 611
How can the pseudo-Wigner distribution be used for texture segmentation?......Page 613
What is the \"take-home\" message of this chapter?......Page 621
Bibliographical notes......Page 623
References......Page 625
Index......Page 629