Image Processing: The Fundamentals,9780471998839

Image Processing: The Fundamentals

by ;
ISBN13: 9780471998839
ISBN10: 0471998834
Format: Hardcover
Pub. Date: 1999-11-01
Publisher(s): WILEY
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Summary

Image Processing The Fundamentals Maria Petrou, University of Surrey, Guildford, UK Panagiota Bosdogianni, Technical University of Crete, Chania, Greece Image processing has been one of the most active areas of research in recent years. The techniques involved have found significant applications in areas as diverse as video-conferencing, image communication, robotics, geoscience and medicine. From intelligent cars that drive themselves to key-hole surgery, this enormous impact on society is expected to change our lives radically. Providing a step by step guide to the basic principles underlying all image processing tasks, this volume is the result of 11 years of teaching experience. * Features numerous worked examples, guiding the reader through the intricacies of reaching the solutions. * Explains the concepts introduced using small sized images that the reader can manipulate without the use of computers. * Allows the reader to appreciate the 'nuts and bolts' of each method, the issues involved and the problems that may be encountered in real applications. * Presents detailed mathematical explanations at two levels - an easy-to-follow narrative with minimum use of mathematics, and a higher level that uses mathematical rigour. Image Processing: The Fundamentals is an ideal self-teaching aide and will prove an invaluable companion for research students in related fields. Alternative techniques are demonstrated for each image allowing the reader to appreciate subtle differences between them. Visit Our Web Page! http://www.wiley.com/

Table of Contents

Preface xv
List of Figures
xvii
Introduction
1(20)
Why do we process images?
1(1)
What is an image?
1(1)
What is the brightness of an image at a pixel position?
2(1)
Why are images often quoted as being 512 x 512, 256 x 256, 128 x 128 etc?
2(1)
How many bits do we need to store an image?
2(1)
What is meant by image resolution?
2(4)
How do we do Image Processing?
6(1)
What is a linear operator?
6(1)
How are operators defined?
6(1)
How does an operator transform an image?
6(1)
What is the meaning of the point spread function?
7(2)
How can we express in practice the effect of a linear operator on an image?
9(5)
What is the implication of the separability assumption on the structure of matrix H?
14(1)
How can a separable transform be written in matrix form?
15(1)
What is the meaning of the separability assumption?
15(3)
What is the ``take home'' message of this chapter?
18(1)
What is the purpose of Image Processing?
18(1)
What is this book about?
18(3)
Image Transformations
21(68)
What is this chapter about?
21(1)
How can we define an elementary image?
21(1)
What is the outer product of two vectors?
21(1)
How can we expand an image in terms of vector outer products?
21(2)
What is a unitary transform?
23(1)
What is a unitary matrix?
23(1)
What is the inverse of a unitary transform?
24(1)
How can we construct a unitary matrix?
24(1)
How should we choose matrices U and V so that g can be represented by fewer bits than f?
24(1)
How can we diagonalize a matrix?
24(6)
How can we compute matrices U, V and A1/2 needed for the image diagonalization?
30(4)
What is the singular value decomposition of an image?
34(1)
How can we approximate an image using SVD?
34(1)
What is the error of the approximation of an image by SVD?
35(1)
How can we minimize the error of the reconstruction?
36(1)
What are the elementary images in terms of which SVD expands an image?
37(8)
Are there any sets of elementary images in terms of which ANY image can be expanded?
45(1)
What is a complete and orthonormal set of functions?
45(1)
Are there any complete sets of orthonormal discrete valued functions?
46(1)
How are the Haar functions defined?
46(1)
How are the Walsh functions defined?
47(1)
How can we create the image transformation matrices from the Haar and Walsh functions?
47(4)
What do the elementary images of the Haar transform look like?
51(6)
Can we define an orthogonal matrix with entries only + 1 or - 1?
57(1)
What do the basis images of the Hadamard/Walsh transform look like?
57(5)
What are the advantages and disadvantages of the Walsh and the Haar transforms?
62(1)
What is the Haar wavelet?
62(1)
What is the discrete version of the Fourier transform?
63(2)
How can we write the discrete Fourier transform in matrix form?
65(1)
Is matrix U used for DFT unitary?
66(2)
Which are the elementary images in terms of which DFT expands an image?
68(4)
Why is the discrete Fourier transform more commonly used than the other transforms?
72(1)
What does the convolution theorem state?
72(7)
How can we display the discrete Fourier transform of an image?
79(1)
What happens to the discrete Fourier transform of an image if the image is rotated?
79(2)
What happens to the discrete Fourier transform of an image if the image is shifted?
81(1)
What is the relationship between the average value of a function and its DFT?
82(1)
What happens to the DFT of an image if the image is scaled?
83(3)
What is the discrete cosine transform?
86(1)
What is the ``take home'' message of this chapter?
86(3)
Statistical Description of Images
89(36)
What is this chapter about?
89(1)
Why do we need the statistical description of images?
89(1)
Is there an image transformation that allows its representation in terms of uncorrelated date that can be used to approximate the image in the least mean square error sense?
89(1)
What is a random field?
90(1)
What is a random variable?
90(1)
How do we describe random variables?
90(1)
What is the probability of an event?
90(1)
What is the distribution function of a random variable?
90(1)
What is the probability of a random variable taking a specific value?
91(1)
What is the probability density function of a random variable?
91(1)
How do we describe many random variables?
92(1)
What relationships may n random variables have with each other?
92(1)
How do we then define a random field?
93(1)
How can we relate two random variables that appear in the same random field?
94(1)
How can we relate two random variables that belong to two different random fields?
95(1)
Since we always have just one version of an image how do we calculate the expectation values that appear in all previous definitions?
96(1)
When is a random field homogeneous?
96(1)
How can we calculate the spatial statistics of a random field?
97(1)
When is a random field ergodic?
97(1)
When is a random field ergodic with respect to the mean?
97(1)
When is a random field ergodic with respect to the autocorrelation function?
97(5)
What is the implication of ergodicity?
102(1)
How can we exploit ergodicity to reduce the number of bits needed for representing an image?
102(1)
What is the form of the autocorrelation function of a random field with uncorrelated random variables?
103(1)
How can we transform the image so that its autocorrelation matrix is diagonal?
103(1)
Is the assumption of ergodicity realistic?
104(6)
How can we approximate an image using its K-L transform?
110(1)
What is the error with which we approximate an image when we truncate its K-L expansion?
110(1)
What are the basis images in terms of which the Karhunen-Loeve transform expands an image?
111(13)
What is the ``take home'' message of this chapter?
124(1)
Image Enhancement
125(30)
What is image enhancement?
125(1)
How can we enhance an image?
125(1)
Which methods of the image enhancement reason about the grey level statistics of an image?
125(1)
What is the histogram of an image?
125(1)
When is it necessary to modify the histogram of an image?
126(1)
How can we modify the histogram of an image?
126(1)
What is histogram equalization?
127(1)
Why do histogram equalization programs usually not produce images with flat histograms?
127(1)
Is it possible to enhance an image to have an absolutely flat histogram?
127(2)
What if we do not wish to have an image with a flat histogram?
129(1)
Why should one wish to perform something other than histogram equalization?
130(1)
What is the image has inhomogeneous contrast?
131(1)
Is there an alternative to histogram manipulation?
132(3)
How can we improve the contrast of a multispectral image?
135(1)
What is principal component analysis?
136(1)
What is the relationship of the Karhunen-Loeve transformation discussed here and the one discussed in Chapter 3?
136(1)
How can we perform principal component analysis?
137(1)
What are the advantages of using principal components to express an image?
138(1)
What are the disadvantages of principal component analysis?
138(6)
Some of the images with enhanced contrast appear very noisy. Can we do anything about that?
144(1)
What are the types of noise present in an image?
144(2)
What is a rank order filter?
146(1)
What is median filtering?
146(1)
What if the noise in an image is not impulse?
146(1)
Why does lowpass filtering reduce noise?
147(1)
What if we are interested in the high frequencies of an image?
148(1)
What is the ideal highpass filter?
148(1)
How can we improve an image which suffers from variable illumination?
148(3)
Can any of the objectives of image enhancement be achieved by the linear methods we learned in Chapter 2?
151(2)
What is the ``take home'' message of this chapter?
153(2)
Two-Dimensional Filters
155(38)
What is this chapter about?
155(1)
How do we define a 2D filter?
155(1)
How are the system function and the unit sample response of the filter related?
155(2)
Why are we interested in the filter function in the real domain?
157(1)
Are there any conditions which h(k,l) must fulfil so that it can be used as a convolution filter?
157(4)
What is the relationship between the 1D and the 2D ideal lowpass filters?
161(1)
How can we implement a filter of infinite extent?
161(1)
How is the z-transform of a digital 1D filter defined?
161(2)
Why do we use z-transforms?
163(1)
How is the z-transform defined in 2D?
163(7)
Is there any fundamental difference between 1D and 2D recursive filters?
170(1)
How do we know that a filter does not amplify noise?
171(1)
Is there an alternative to using infinite impulse response filters?
171(1)
Why do we need approximation theory?
171(1)
How do we know how good an approximate filter is?
171(1)
What is the best approximation to an ideal given system function?
171(1)
Why do we judge an approximation according to the Chebyshev norm instead of the square error?
172(1)
How can we obtain an approximation to a system function?
172(1)
What is windowing?
172(1)
What is wrong with windowing?
173(1)
How can we improve the result of the windowing process?
173(1)
Can we make use of the windowing functions that have been developed for 1D signals, to define a windowing function for images?
173(1)
What is the formal definition of the approximation problem we are trying to solve?
173(1)
What is linear programming?
174(1)
How can we formulate the filter design problem as a linear programming problem?
174(5)
Is there any way by which we can reduce the computational intensity of the linear programming solution?
179(1)
What is the philosophy of the iterative approach?
179(1)
Are there any algorithms that work by decreasing the upper limit of the fitting error?
180(1)
How does the maximizing algorithm work?
180(1)
What is a limiting set of equations?
180(1)
What does the La Vallee Poussin theorem say?
180(1)
What is the proof of the La Vallee Poussin theorem?
181(1)
What are the steps of the iterative algorithm?
181(1)
Can we approximate a filter by working fully in the frequency domain?
182(1)
How can we express the system function of a filter at some frequencies as a function of its values at other frequencies?
182(7)
What exactly are we trying to do when we design the filter in the frequency domain only?
189(1)
How can we solve for the unknown values H(k, l)?
190(1)
Does the frequency sampling method yield optimal solutions according to the Chebyshev criterion?
190(2)
What is the ``take home'' message of this chapter?
192(1)
Image Restoration
193(72)
What is image restoration?
193(1)
What is the difference between image enhancement and image restoration?
193(1)
Why may an image require restoration?
193(1)
How may geometric distortion arise?
193(1)
How can a geometrically distorted image be restored?
193(1)
How do we perform the spatial transformation?
194(1)
Why is grey level interpolation needed?
195(3)
How does the degraded image depend on the undegraded image and the point spread function of a linear degradation process?
198(1)
How does the degraded image depend on the undegraded image and the point spread function of a linear shift invariant degradation process?
198(1)
What form does equation (6.5) take for the case of discrete images?
199(1)
What is the problem of image restoration?
199(1)
How can the problem of image restoration be solved?
199(1)
How can we obtain information on the transfer function H(u, v) of the degradation process?
199(10)
If we know the transfer function of the degradation process, isn't the solution to the problem of image restoration trivial?
209(1)
What happens at points (u, v) where H(u, v) = 0?
209(1)
Will the zeroes of H(u, v) and G(u, v) always coincide?
209(1)
How can we take noise into consideration when writing the linear degradation equation?
210(1)
How can we avoid the amplification of noise?
210(7)
How can we express the problem of image restoration in a formal way?
217(1)
What is the solution of equation (6.37)?
217(1)
Can we find a linear solution to equation (6.37)?
217(1)
What is the linear least mean square error solution of the image restoration problem?
218(1)
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(1)
How can we possibly use equation (6.47) if we know nothing about the statistical properties of the unknown image f(r)?
219(1)
What is the relationship of the Wiener filter (6.47) and the inverse filter of equation (6.25)?
219(1)
Assuming that we know the statistical properties of the unknown image f(r), how can we determine the statistical properties of the noise expressed by Svv(r)?
220(10)
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(1)
Equation (6.76) seems pretty straightforward, why bother with any other approach?
231(1)
Is there any way by which matrix H can be inverted?
232(1)
When is a matrix block circulant?
232(1)
When is a matrix circulant?
232(1)
Why can block circulant matrices be inverted easily?
233(1)
Which are the eigenvalues and the eigenvectors of a circulant matrix?
233(1)
How does the knowledge of the eigenvalues and the eigenvectors of a matrix help in inverting the matrix?
234(5)
How do we know that matrix H that expresses the linear degradation process is block circulant?
239(1)
How can we diagonalize a block circulant matrix?
240(10)
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?
250(1)
How can we incorporate the constraint in the inversion of the matrix?
251(2)
What is the relationship between the Wiener filter and the constrained matrix inversion filter?
253(9)
What is the ``take home'' message of this chapter?
262(3)
Image Segmentation and Edge Detection
265(60)
What is this chapter about?
265(1)
What exactly is the purpose of image segmentation and edge detection?
265(1)
How can we divide an image into uniform regions?
265(1)
What do we mean by ``labelling'' an image?
266(1)
What can we do if the valley in the histogram is not very sharply defined?
266(1)
How can we minimize the number of misclassified pixels?
267(1)
How can we choose the minimum error threshold?
268(5)
What is the minimum error threshold when object and background pixels are normally distributed?
273(1)
What is the meaning of the two solution of (7.6)?
274(4)
What are the drawbacks of the minimum error threshold method?
278(1)
Is there any method that does not depend on the availability of models for the distribution of the object and the background pixels?
278(4)
Are there any drawbacks to Otsu's method?
282(1)
How can we threshold images obtained under variable illumination?
282(1)
If we threshold the image according to the histogram of In f(x, y), are we thresholding it according to the refectance properties of the imaged surfaces?
283(2)
Since straghtforward thresholding methods break down under variable illumination, how can we cope with it?
285(2)
Are there any shortcomings of the thresholding methods?
287(1)
How can we cope with images that contain regions that are not uniform but they are perceived as uniform?
288(1)
Are there any segmentation methods that take into consideration the spatial proximity of pixels?
288(1)
How can one choose the seed pixels?
288(1)
How does the split and merge method work?
288(1)
Is it possible to segment an image by considering the dissimilarities between regions, as opposed to considering the similarities between pixels?
289(1)
How do we measure the dissimilarity between neighbouring pixels?
289(1)
What is the smallest possible window we can choose?
290(1)
What happens when the image has noise?
291(3)
How can we choose the weights of a 3 x 3 mask for edge detection?
294(2)
What is the best value of parameter K?
296(7)
In the general case, how do we decide whether a pixel is an edge pixel or not?
303(3)
Are Sobel masks appropriate for all images?
306(1)
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(3)
Can we use the optimal filters for edges to detect lines in an image in an optimal way?
309(1)
What is the fundamental difference between step edges and lines?
309(13)
What is the ``take home'' message of this chapter?
322(3)
Bibliography 325(4)
Index 329

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