Preface xv
List of Figures xvii
1 Introduction
Why do we process images? . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is an image? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the brightness of an image at a pixel position? . . . . . . . . . . . .
Why are images often quoted as being 512 X 512. 256 X 256. 128 X 128 etc?
How many bits do we need to store an image? . . . . . . . . . . . . . . . . .
What is meant by image resolution? . . . . . . . . . . . . . . . . . . . . . .
How do we do Image Processing? . . . . . . . . . . . . . . . . . . . . . . . .
What is a linear operator? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How are operators defined? . . . . . . . . . . . . . . . . . . . . . . . . . . .
How does an operator transform an image? . . . . . . . . . . . . . . . . . .
What is the meaning of the point spread function? . . . . . . . . . . . . . .
How can we express in practice the effect of a linear operator on an image?
What is the implication of the separability assumption on the structure of
matrix H? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can a separable transform be written in matrix form? . . . . . . . . .
What is the meaning of the separability assumption? . . . . . . . . . . . . .
What is the “take home” message of this chapter? . . . . . . . . . . . . . .
What is the purpose of Image Processing? . . . . . . . . . . . . . . . . . . .
What is this book about? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
What is this chapter about? . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
How can we define an elementary image? . . . . . . . . . . . . . . . . . . . 21
What is the outer product of two vectors? . . . . . . . . . . . . . . . . . . . 21
How can we expand an image in terms of vector outer products? . . . . . . 21
What is a unitary transform? . . . . . . . . . . . . . . . . . . . . . . . . . . 23
What is a unitary matrix? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
What is the inverse of a unitary transform? . . . . . . . . . . . . . . . . . . 24
How can we construct a unitary matrix? . . . . . . . . . . . . . . . . . . . . 24
vii
viii ImFaugned aPmr oecnetsaTslishn eg :
How should we choose matrices U and V so that g can be represented by
fewer bits than f? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can we diagonalize a matrix? . . . . . . . . . . . . . . . . . . . . . . .
How can we compute matrices U. V and A+ needed for the image
What is the singular value decomposition of an image? . . . . . . . . . . . .
How can we approximate an image using SVD? . . . . . . . . . . . . . . . .
What is the error of the approximation of an image by SVD? . . . . . . . .
How can we minimize the error of the reconstruction? . . . . . . . . . . . .
What are the elementary images in terms of which SVD expands an image?
Are there any sets of elementary images in terms of which ANY image can
be expanded? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is a complete and orthonormal set of functions? . . . . . . . . . . . .
Are there any complete sets of orthonormal discrete valued functions? . . .
How are the Haar functions defined? . . . . . . . . . . . . . . . . . . . . . .
How are the Walsh functions defined? . . . . . . . . . . . . . . . . . . . . .
How can we create the image transformation matrices from the Haar and
Walsh functions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What do the elementary images of the Haar transform look like? . . . . . .
Can we define an orthogonal matrix with entries only +l or -l? . . . . . .
What do the basis images of the Hadamard/Walsh transform look like? . .
What are the advantages and disadvantages of the Walsh and the Haar
transforms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the Haar wavelet? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the discrete version of the Fourier transform? . . . . . . . . . . . .
How can we write the discrete Fourier transform in matrix form? . . . . . .
Is matrix U used for DFT unitary? . . . . . . . . . . . . . . . . . . . . . . .
Which are the elementary images in terms of which DFT expands an image?
Why is the discrete Fourier transform more commonly used than the other
transforms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What does the convolution theorem state? . . . . . . . . . . . . . . . . . . .
How can we display the discrete Fourier transform of an image? . . . . . . .
What happens to the discrete Fourier transform of an image if the image is
rotated? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What happens to the discrete Fourier transform of an image if the image is
shifted? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the relationship between the average value of a function and its DFT?
What happens to the DFT of an image if the image is scaled? . . . . . . . .
What is the discrete cosine transform? . . . . . . . . . . . . . . . . . . . . .
What is the “take home” message of this chapter? . . . . . . . . . . . . . .
diagonalization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Statistical Description of Images
What is this chapter about? . . . . . . . . . . . . . . . . . . . . . . . . . . .
Why do we need the statistical description of images? . . . . . . . . . . . .
Contents ix
Is there an image transformation that allows its representation in terms of
uncorrelated data that can be used to approximate the image in the
least mean square error sense? . . . . . . . . . . . . . . . . . . . . . . 89
What is a random field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
What is a random variable? . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
How do we describe random variables? . . . . . . . . . . . . . . . . . . . . . 90
What is the probability of an event? . . . . . . . . . . . . . . . . . . . . . . 90
What is the distribution function of a random variable? . . . . . . . . . . . 90
What is the probability of a random variable taking a specific value? . . . . 91
What is the probability density function of a random variable? . . . . . . . 91
How do we describe many random variables? . . . . . . . . . . . . . . . . . 92
What relationships may n random variables have with each other? . . . . . 92
How do we then define a random field? . . . . . . . . . . . . . . . . . . . . . 93
How can we relate two random variables that appear in the same random
field? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
How can we relate two random variables that belong to two different random
fields? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Since we always have just one version of an image how do we calculate the
expectation values that appear in all previous definitions? . . . . . . . 96
When is a random field homogeneous? . . . . . . . . . . . . . . . . . . . . . 96
How can we calculate the spatial statistics of a random field? . . . . . . . . 97
When is a random field ergodic? . . . . . . . . . . . . . . . . . . . . . . . . 97
When is a random field ergodic with respect to the mean? . . . . . . . . . . 97
When is a random field ergodic with respect to the autocorrelation
function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
What is the implication of ergodicity? . . . . . . . . . . . . . . . . . . . . . 102
How can we exploit ergodicity to reduce the number of bits needed for
representing an image? . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
What is the form of the autocorrelation function of a random field with
uncorrelated random variables? . . . . . . . . . . . . . . . . . . . . . . 103
How can we transform the image so that its autocorrelation matrix is
diagonal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Is the assumption of ergodicity realistic? . . . . . . . . . . . . . . . . . . . . 104
How can we approximate an image using its K-L transform? . . . . . . . . . 110
What is the error with which we approximate an image when we truncate
its K-L expansion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
What are theb asis images in termso f which the Karhunen-Loeve transform
expands an image? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
What is the “take home” message of this chapter? . . . . . . . . . . . . . . 124
4 125
What is image enhancement? . . . . . . . . . . . . . . . . . . . . . . . . . . 125
How can we enhance an image? . . . . . . . . . . . . . . . . . . . . . . . . . 125
Which methods of the image enhancement reason about the grey level
statistics of an image? . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
X Image Processing: The Fundamentals
What is the histogram of an image? . . . . . . . . . . . . . . . . . . . . . .
When is it necessary to modify the histogram of an image? . . . . . . . . .
How can we modify the histogram of an image? . . . . . . . . . . . . . . . .
What is histogram equalization? . . . . . . . . . . . . . . . . . . . . . . . .
Why do histogram equalization programs usually not produce images with
Is it possible to enhance an image to have an absolutely flat histogram? . .
What if we do not wish to have an image with a flat histogram? . . . . . .
Why should one wish to perform something other than histogram
equalization? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What if the image has inhomogeneous contrast? . . . . . . . . . . . . . . .
Is there an alternative to histogram manipulation? . . . . . . . . . . . . . .
How can we improve the contrast of a multispectral image? . . . . . . . . .
What is principal component analysis? . . . . . . . . . . . . . . . . . . . . .
What is the relationship of the Karhunen-Loeve transformation discussed
here and the one discussed in Chapter 3? . . . . . . . . . . . . . . . .
How can we perform principal component analysis? . . . . . . . . . . . . . .
What are the advantages of using principal components to express an
image? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What are the disadvantages of principal component analysis? . . . . . . . .
Some of the images with enhanced contrast appear very noisy . Can we do
anything about that? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What are the types of noise present in an image? . . . . . . . . . . . . . . .
What is a rank order filter? . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is median filtering? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What if the noise in an image is not impulse? . . . . . . . . . . . . . . . . .
Why does lowpass filtering reduce noise? . . . . . . . . . . . . . . . . . . . .
What if we are interested in the high frequencies of an image? . . . . . . . .
What is the ideal highpass filter? . . . . . . . . . . . . . . . . . . . . . . . .
How can we improve an image which suffers from variable illumination? . .
Can any of the objectives of image enhancement be achieved by the linear
methods we learned in Chapter 2? . . . . . . . . . . . . . . . . . . . .
What is the “take home” message of this chapter? . . . . . . . . . . . . . .
flat histograms? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Two-Dimensional Filters 155
What is this chapter about? . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
How do we define a 2D filter? . . . . . . . . . . . . . . . . . . . . . . . . . . 155
How are the system function and the unit sample response of the filter
related? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Why are we interested in the filter function in the real domain? . . . . . . . 157
Are there any conditions which h(k. 1) must fulfil so that it can be used as
a convolution filter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
What is the relationship between the 1D and the 2D ideal lowpass filters? . 161
How can we implement a filter of infinite extent? . . . . . . . . . . . . . . . 161
How is the z-transform of a digital 1D filter defined? . . . . . . . . . . . . . 161
Contents xi
Why do we use z-transforms? . . . . . . . . . . . . . . . . . . . . . . . . . .
How is the z-transform defined in 2D? . . . . . . . . . . . . . . . . . . . . .
Is there any fundamental difference between 1D and 2D recursive filters? . .
How do we know that a filter does not amplify noise? . . . . . . . . . . . .
Is there an alternative to using infinite impulse response filters? . . . . . . .
Why do we need approximation theory? . . . . . . . . . . . . . . . . . . . .
How do we know how good an approximate filter is? . . . . . . . . . . . . .
What is the best approximation to an ideal given system function? . . . . .
Why do we judge an approximation according to the Chebyshev norm
instead of the square error? . . . . . . . . . . . . . . . . . . . . . . . .
How can we obtain an approximation to a system function? . . . . . . . . .
What is windowing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is wrong with windowing? . . . . . . . . . . . . . . . . . . . . . . . .
How can we improve the result of the windowing process? . . . . . . . . . .
Can we make use of the windowing functions that have been developed for
1D signals. to define a windowing function for images? . . . . . . . . .
What is the formal definition of the approximation problem we are trying
tosolve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is linear programming? . . . . . . . . . . . . . . . . . . . . . . . . . .
How can we formulate the filter design problem as a linear programming
problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Is there any way by which we can reduce the computational intensity of the
linear programming solution? . . . . . . . . . . . . . . . . . . . . . . .
What is the philosophy of the iterative approach? . . . . . . . . . . . . . . .
Are there any algorithms that work by decreasing the upper limit of the
fitting error? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How does the maximizing algorithm work? . . . . . . . . . . . . . . . . . .
What is a limiting set of equations? . . . . . . . . . . . . . . . . . . . . . .
What does the La Vallee Poussin theorem say? . . . . . . . . . . . . . . . .
What is the proof of the La Vallee Poussin theorem? . . . . . . . . . . . . .
What are the steps of the iterative algorithm? . . . . . . . . . . . . . . . . .
Can we approximate a filter by working fully in the frequency domain? . . .
How can we express the system function of a filter at some frequencies as a
function of its values at other frequencies? . . . . . . . . . . . . . . . .
What exactly are we trying to do when we design the filter in the frequency
domain only? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can we solve for the unknown values H(k. Z)? . . . . . . . . . . . . . .
Does the frequency sampling method yield optimal solutions according to
the Chebyshev criterion? . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the “take home” message of this chapter? . . . . . . . . . . . . . .
6 Image Restoration
What is image restoration? . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the difference between image enhancement and image restoration?
Why may an image require restoration? . . . . . . . . . . . . . . . . . . . .
xii Image Processing: The Fundamentals
How may geometric distortion arise? . . . . . . . . . . . . . . . . . . . . . . 193
How can a geometrically distorted image be restored? . . . . . . . . . . . . 193
How do we perform the spatial transformation? . . . . . . . . . . . . . . . . 194
Why is grey level interpolation needed? . . . . . . . . . . . . . . . . . . . . 195
How does the degraded image depend on the undegraded image and the
point spread function of a linear degradation process? . . . . . . . . . 198
How does the degraded image depend on the undegraded image and the
point spread function of a linear shift invariant degradation process? . 198
What form does equation (6.5) take for the case of discrete images? . . . . 199
What is the problem of image restoration? . . . . . . . . . . . . . . . . . . . 199
How can the problem of image restoration be solved? . . . . . . . . . . . . . 199
How can we obtain information on the transfer function k(u. v) of the
degradation process? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
If we know the transfer function of the degradation process. isn’t the solution
to the problem of image restoration trivial? . . . . . . . . . . . . . . . 209
What happens at points (U. v) where k(u. v) = O? . . . . . . . . . . . . . . 209
Will the zeroes of k(u. v) and G(u. v) always coincide? . . . . . . . . . . . . 209
How can we take noise into consideration when writing the linear degradation
equation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
How can we avoid the amplification of noise? . . . . . . . . . . . . . . . . . 210
How can we express the problem of image restoration in a formal way? . . . 217
What is the solution of equation (6.37)? . . . . . . . . . . . . . . . . . . . . 217
Can we find a linear solution to equation (6.37)? . . . . . . . . . . . . . . . 217
What is the linear least mean square error solution of the image restoration
problem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Since the original image f(r) is unknown. how can we use equation (6.41)
which relies on its cross-spectral density with the degraded image. to
derive the filter we need? . . . . . . . . . . . . . . . . . . . . . . . . . 218
How can we possibly use equation (6.47) if we know nothing about the
statistical properties of the unknown image f(r)? . . . . . . . . . . . . 219
What is the relationship of the Wiener filter (6.47) and the inverse filter of
equation (6.25)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Assuming that we know the statistical propertieosf the unknown imagef (r).
how can we determine the statistical propertieso f the noise expressed
by Svv(r)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
If the degradation process is assumed linear. why don’t we solve a system of
linear equations to reverse its effect instead of invoking the convolution
theorem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Equation (6.76) seems pretty straightforward. why bother with any other
approach? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Is there any way by which matrix H can be inverted? . . . . . . . . . . . . 232
When is a matrix block circulant? . . . . . . . . . . . . . . . . . . . . . . . 232
When is a matrix circulant? . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Why can block circulant matrices be inverted easily? . . . . . . . . . . . . . 233
Which are the eigenvalues and the eigenvectors of a circulant matrix? . . . 233
Contents xiii
How does the knowledge of the eigenvalues and the eigenvectors of a matrix
help in inverting the matrix? . . . . . . . . . . . . . . . . . . . . . . .
How do we know that matrix H that expresses the linear degradation process
is block circulant? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can we diagonalize a block circulant matrix? . . . . . . . . . . . . . . .
OK. now we know how to overcome the problem of inverting H; however.
how can we overcome the extreme sensitivity of equation (6.76) to
noise? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can we incorporate the constraint in the inversion of the matrix? . . .
What is the relationship between the Wiener filter and the constrained
matrix inversion filter? . . . . . . . . . . . . . . . . . . . . . . . . . . .
What is the “take home” message of this chapter? . . . . . . . . . . . . . .
7 Image Segmentation and Edge Detection
What is this chapter about? . . . . . . . . . . . . . . . . . . . . . . . . . . .
What exactly is the purpose of image segmentation and edge detection? . .
How can we divide an image into uniform regions? . . . . . . . . . . . . . .
What do we mean by “labelling” an image? . . . . . . . . . . . . . . . . . .
What can we do if the valley in the histogram is not very sharply defined? .
How can we minimize the number of misclassified pixels? . . . . . . . . . .
How can we choose the minimum error threshold? . . . . . . . . . . . . . .
What is the minimum error threshold when object and background pixels
are normally distributed? . . . . . . . . . . . . . . . . . . . . . . . . .
What is the meaning of the two solutions of (7.6)? . . . . . . . . . . . . . .
What are the drawbacks of the minimum error threshold method? . . . . .
Is there any method that does not depend on the availability of models for
the distributions of the object and the background pixels? . . . . . . .
Are there any drawbacks to Otsu’s method? . . . . . . . . . . . . . . . . . .
How can we threshold images obtained under variable illumination? . . . .
If we threshold the image according to the histogram of lnf(z.y). are we
thresholding it according to the reflectance properties of the imaged
surfaces? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Since straightforward thresholding methods break down under variable
illumination. how can we cope with it? . . . . . . . . . . . . . . . . . .
Are there any shortcomings of the thresholding methods? . . . . . . . . . .
How can we cope with images that contain regions that are not uniform but
they are perceived as uniform? . . . . . . . . . . . . . . . . . . . . . . .
Are there anys egmentation methods that take intoc onsideration the spatial
proximity of pixels? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How can one choose the seed pixels? . . . . . . . . . . . . . . . . . . . . . .
How does the split and merge method work? . . . . . . . . . . . . . . . . .
Is it possible to segment an image by considering the dissimilarities between
regions. as opposed to considering the similarities between pixels? . . .
How do we measure the dissimilarity between neighbouring pixels? . . . . .
What is the smallest possible window we can choose? . . . . . . . . . . . .
xiv Image Processing: The Fundamentals
What happens when the image has noise? . . . . . . . . . . . . . . . . . . . 291
How can we choose the weights of a 3 X 3 mask for edge detection? . . . . . 294
What is the best value of parameter K? . . . . . . . . . . . . . . . . . . . . 296
In the general case. how do we decide whether a pixel is an edge pixel
ornot? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
Are Sobel masks appropriate for all images? . . . . . . . . . . . . . . . . . . 306
How can we choose the weights of the mask if we need a larger mask owing
to the presence of significant noise in the image? . . . . . . . . . . . . 306
Can we use the optimal filters for edges to detect lines in an image in an
optimal way? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
What is the fundamental difference between step edges and lines? . . . . . . 309
What is the “take home” message of this chapter? . . . . . . . . . . . . . . 322
Bibliography 325
Index 329
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