# Image processing : the fundamentals /

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Author / Creator: Petrou, Maria. Chichester, [England] ; New York : Wiley, c1999. xx, 333 p. : ill. ; 26 cm. English Image processing -- Digital techniques. Image processing -- Digital techniques. Print Book http://pi.lib.uchicago.edu/1001/cat/bib/4040791
Other authors / contributors: Bosdogianni, Panagiota. 0471998834 (alk. paper) Includes bibliographical references (p. -327) and index.
• Preface
• List of Figures
• 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 [times] 512, 256 [times] 256, 128 [times] 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. Image Transformations
• What is this chapter about?
• How can we define an elementary image?
• What is the outer product of two vectors?
• How can we expand an image in terms of vector outer products?
• What is a unitary transform?
• What is a unitary matrix?
• What is the inverse of a unitary transform?
• How can we construct a unitary matrix?
• 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 [Lambda]1/2 needed for the image diagonalization?
• 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 +1 or -1?
• 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?
• 3. Statistical Description of Images
• What is this chapter about?
• Why do we need the statistical description of images?
• 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?
• What is a random field?
• What is a random variable?
• How do we describe random variables?
• What is the probability of an event?
• What is the distribution function of a random variable?
• What is the probability of a random variable taking a specific value?
• What is the probability density function of a random variable?
• How do we describe many random variables?
• What relationships may n random variables have with each other?
• How do we then define a random field?
• How can we relate two random variables that appear in the same random field?
• How can we relate two random variables that belong to two different random fields?
• Since we always have just one version of an image how do we calculate the expectation values that appear in all previous definitions?
• When is a random field homogeneous?
• How can we calculate the spatial statistics of a random field?
• When is a random field ergodic?
• When is a random field ergodic with respect to the mean?
• When is a random field ergodic with respect to the autocorrelation function?
• What is the implication of ergodicity?
• How can we exploit ergodicity to reduce the number of bits needed for representing an image?
• What is the form of the autocorrelation function of a random field with uncorrelated random variables?
• How can we transform the image so that its autocorrelation matrix is diagonal?
• Is the assumption of ergodicity realistic?
• How can we approximate an image using its K-L transform?
• What is the error with which we approximate an image when we truncate its K-L expansion?
• What are the basis images in terms of which the Karhunen-Loeve transform expands an image?
• What is the "take home" message of this chapter?
• 4. Image Enhancement
• What is image enhancement?
• How can we enhance an image?
• Which methods of the image enhancement reason about the grey level statistics of an image?
• 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 flat histograms?
• 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?
• 5. Two-Dimensional Filters
• What is this chapter about?
• How do we define a 2D filter?
• How are the system function and the unit sample response of the filter related?
• Why are we interested in the filter function in the real domain?
• Are there any conditions which h(k,l) must fulfil so that it can be used as a convolution filter?
• What is the relationship between the 1D and the 2D ideal lowpass filters?
• How can we implement a filter of infinite extent?
• How is the z-transform of a digital 1D filter defined?
• 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 Chebyshey 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 to solve?
• 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,l)?
• 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?
• How may geometric distortion arise?
• How can a geometrically distorted image be restored?
• How do we perform the spatial transformation?
• Why is grey level interpolation needed?
• How does the degraded image depend on the undegraded image and the point spread function of a linear shift invariant degradation process?
• What form does equation (6.5) take for the case of discrete images?
• What is the problem of image restoration?
• How can the problem of image restoration be solved?
• How can we obtain information on the transfer function H(u, v) of the degradation process?
• If we know the transfer function of the degradation process, isn't the solution to the problem of image restoration trivial?
• What happens at points (u, v) where H(u, v) = 0?
• Will the zeroes of H(u, v) and G(u, v) always coincide?
• How can we take noise into consideration when writing the linear degradation equation?
• How can we avoid the amplification of noise?
• How can we express the problem of image restoration in a formal way?
• What is the solution of equation (6.37)?
• Can we find a linear solution to equation (6.37)?
• What is the linear least mean square error solution of the image restoration problem?
• 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?
• How can we possibly use equation (6.47) if we know nothing about the statistical properties of the unknown image f(r)?
• What is the relationship of the Wiener filter (6.47) and the inverse filter of equation (6.25)?
• 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)?
• 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?
• Equation (6.76) seems pretty straightforward, why bother with any other approach?
• Is there any way by which matrix H can be inverted?
• When is a matrix block circulant?
• When is a matrix circulant?
• Why can block circulant matrices be inverted easily?
• Which are the eigenvalues and the eigenvectors of a circulant matrix?
• 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 ln f(x, 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 any segmentation methods that take into consideration 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?
• What happens when the image has noise?
• How can we choose the weights of a 3 [times] 3 mask for edge detection?
• What is the best value of parameter K?
• In the general case, how do we decide whether a pixel is an edge pixel or not?
• Are Sobel masks appropriate for all images?
• 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?
• Can we use the optimal filters for edges to detect lines in an image in an optimal way?
• What is the fundamental difference between step edges and lines?
• What is the "take home" message of this chapter?
• Bibliography
• Index