Instructor Solution Manual Image Processing, Analysis, and Machine Vision, Fourth Edition Milan Sonk

SOLUTION MANUAL

Image Processing, Analysis, and Machine Vision Answer Manual Fourth Edition

Milan Sonka The University of Iowa, Iowa City Vaclav Hlavac Czech Technical University, Prague Roger Boyle Prifysgol Aberystwyth, Aberystwyth

January 5, 2014

Contents 1 Introduction

1

2 The image, its representations and properties

2

3 The image, its mathematical and physical background

4

4 Data structures for image analysis

6

5 Image pre-processing

8

6 Segmentation I

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7 Segmentation II

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8 Shape representation and description

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9 Object recognition

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10 Image understanding

28

11 3D geometry, correspondence, 3D from intensities

31

12 Use of 3D vision

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13 Mathematical morphology

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14 Image data compression

39

15 Texture

42

16 Motion analysis

45

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

1

Introduction Short-answer questions S1.1 Vision uses computers to analyze images; graphics uses computers to create images.

S1.2 This is a practical exercise in opening your eyes!

S1.3 This is a moving target, but computational cost plummets and the ‘cleverness’ of algorithms climbs.

S1.4 No simple ‘right’ answer exists, but something like Moore’s Law will apply.

S1.5 Earlier editions of this book are indicative; new areas of vision, and new application areas, continue to appear.

S1.6 There are many: transport scheduling is a very visible one.

S1.7 Swathes of AI and statistics will use a priori reasoning.

S1.8 The best example is natural language.

S1.9 The best example is natural language.

S1.10 The purpose of this question is to illustrate that there is now a rich and accessible vision literature, but that 10, or certainly 20 years ago, this was not true.

S1.11 The purpose of this question is to illustrate that there is massive activity in vision – there has been for some time and some of the older references are still very pertinent.

S1.12 The purpose of this question is to illustrate that vision as an academic area of study as always been present, but ‘took off’ during the 70s and 80s.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

2

The image, its representations and properties Short-answer questions S2.1 The diagram should make clear that perspective projection implies a cone emanating from the focal point of the camera, while orthographic implies parallel projection lines.

S2.2 These definitions may be taken directly from the text.

S2.3 As a cheap way of achieving the illusion of continuous motion through rapid frame change., without having to recast the whole frame.

S2.4 This is a practical exercise, demonstrated by experiment.

S2.5 These definitions may be taken directly from the text.

S2.6

• Ri is the spectral sensitivity of the sensor, I is the spectral density of the illumination and S describes how the surface patch reflects each wavelength.

• qi is the spectral response of the ith sensor.

S2.7 This is illustrated in Figures 2.29 and 2.31.

S2.8 Constancy refers to the ability to interpret the same color regardless of illumination and shadow effects. This is important but difficult in interpreting the wide range of, e.g., RGB triples that might represent a single real-world color.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 2: The image, its representations and properties

3

• Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

3

The image, its mathematical and physical background Short-answer questions S3.1 This is given explicitly in the text.

S3.2 This is given explicitly in the text.

S3.3 These are given explicitly in the text.

S3.4 This is given explicitly in the text.

S3.5 Let us show the R ∞1D case for simplicity. The Dirac function is defined as δ(t) = 0 for all t 6= 0, −∞ δ(t)dt = 1.

The Fourier transform of the Dirac function is calculated by substituting into the Fourier transform definition integral Z ∞ F{δ(t)} = δ(t) e−2πiξt = e−2πiξt =1. −∞

t=0

S3.6 Explicit access to particular frequencies is trivial in the Fourier domain - the noise frequency would have its amplitude zeroed or attenuated.

S3.7 This is given explicitly in the text.

S3.8 Wavelets give immediate access to scale information while the Fourier transform does not.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 3: The image, its mathematical and physical background

5

• Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

4

Data structures for image analysis Short-answer questions S4.1 Determine the 4- and 8-neighborhood chain codes of the regions shown in Figure 4.1.

Figure 4.1: Binary image examples. © Cengage Learning 2015.

Let the reference pixel at which the chain code starts be the top-left pixel.

The direction 0 is in the East direction. The coding number increases in the counterclockwise order, i.e. codes 0 . . . 3 for 4-neighborhood and codes 0 . . . 7 for 8-neighborhood. The object is traversed in clockwise direction.

The 4-neighborhood chain code for the left image is: 000223330303321112232321001111.

The 8-neighborhood chain code for the left image is: 00045677664223554201222.

The 4-neighborhood chain code for the right image is: 003000032223303211212121.

The 8-neighborhood chain code for the right image is: 007000644576423332.

S4.2 (a) This definition is given explicitly in the text. (b) Algorithm 4.2 gives a single-pass solution to this task.

S4.3 This definition is given explicitly in the text.

S4.4 Determine the run length encoding of the images shown in Figure 4.1.

The left image run length code is: ((045)(10145)(20245)(324)(423)(523)(625)).

The right image run length code is: ((145)(245)(334)(427)(517)(613)). © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 4: Data structures for image analysis

7

S4.5 This definition is given explicitly in the text.

S4.6 Let us assume a continuous case. In such a case, if a region “touches in a corner” with another region then it is assumed connected. This makes regions 0, 1, 2, 6, 7 connected and also regions 0, 2, 3, 7 connected. The region adjacency graph is given in Figure 4.2.

0

4

3 2

5

5

4

1 1

3 0

7 6

8

2

7

6

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Figure 4.2: Regions (left) and their corresponding region adjacency graph (right). © Cengage Learning 2015.

S4.7 This definition is given explicitly in the text.

S4.8 This definition is given explicitly in the text.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

5

Image pre-processing Short-answer questions S5.1 Pre-processing readies an image for more ‘intelligent’ later processing; it may address, for example, issues of noise, or focus.

S5.2 Many applications that involve human inspection of images would deploy any or all of these.

S5.3 Brightness correction modifies the pixel brightness taking into account its original brightness and its position in the image. Gray-scale transformations change brightness without regard to position in the image.

S5.4 The aim is to occupy all, rather than some, of the available gray levels, and as far as possible to make their occupation uniform. For the human eye, this will give the illusion of more information.

S5.5 It would be flat if non-integral gray levels, or non-integral assignation of levels to pixels were possible.

S5.6 There cannot be more information in any scientific sense: the human eye can perceive contrast better after equalization.

S5.7 A pixel co-ordinate transform, followed by brightness interpolation.

S5.8 The bilinear transformation has 8 parameters (Equation 5.10). The affine transformation has 6 parameters (Equation 5.11). When calculating the transformation parameters from matching pixels (also control points) in the image before the transformation and in the image after the transformation, each matching pixel has two coordinates, x and y one. Consequently the minimal number of control points pairs is four for the bilinear transformation. Similarly, the minimal number of control points pairs is three for the affine transformation.

S5.9 (a) Rotation: This can be found explicitly in the text. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 5: Image pre-processing

9

(b) Change of scale: This can be found explicitly in the text. (c) Skewing by an angle: This can be found explicitly in the text.

S5.10 The input image has to be mapped by the geometric transform T into the output image. There are two dual approaches: (a) Forward mapping: (x0 , y 0 ) = T (x, y); (b) Backward mapping: (x, y) = T−1 (x0 , y 0 ) as illustrated in Figure 5.1.

T

T-1 Figure 5.1: Forward and backward mappings for the interpolation in geometric transformations. © Cengage Learning 2015.

The difference between the forward and backward mappings is in the use of image data while performing the interpolation. In the forward mapping, output coordinates (x0 , y 0 ) can lie off raster, two or more input pixels can be mapped to the same output pixel, some output pixels might not get a value assigned.

Consequently, gaps may occur. In the backward mapping, for each pixel in the output image (x0 , y 0 ) the position in the input image (x, y) is determined by T−1 . The appropriate value of the image function f (x, y) is obtained by the interpolation from neighboring pixels. No gaps appear. The backward mapping approach is superior to forward mapping one.

S5.11 The coordinates x, y of the pixel after the geometric transformation lies of the (integer) image raster. The value of the image function f (x, y) has to be determined in this position. It is approximated (more specifically interpolated) locally from the neighboring known image function f values.

The simplest nearest-neighbor method seeks the closest known pixel and takes its value. The linear interpolation takes into account four nearest pixels and approximates the image function value in-between the known values by a plane.

The bi-cubic interpolation takes into account 16 neighboring pixels and approximates the image function by a bi-cubic polynomial surface.

S5.12 Edge detection seeks high-frequency areas of the image, while smoothing seeks to suppress such features.

S5.13 Gaussian filter obtained its name from the shape of its impulse function which 2 2 +y 2 is a Gaussian function; in 1D: f (t) = exp(− σt 2 ); in 2D: f (x, y) = exp(− x 2σ 2 ), Equation 5.47; or an approximation to a Gaussian function. Gaussian filter is optimal in terms of the smallest duration-bandwidth product as was discussed in Section 3.2.3 and illustrated in Figure 3.4.

S5.14 As S5.12. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

10

Chapter 5: Image pre-processing

S5.15 Averaging with a rotating mask, Median filtering, Rank filtering. These are defined in the text.

S5.16 Isolated impulses, high or low, will be removed entirely by the median operation.

S5.17 Convolution masks for listed edge operators are given explicitly in the text.

S5.18 Let us explain first what is meant by a compass operator. The edge is given by the gradient of the image function. The direction of the gradient is of the interest here. Edge detectors (operators) can estimate the gradient direction from the image data. Here is the analogy to magnetic compass.

Laplace edge operator does not depend on the direction and thus cannot serve as the compass operator. Prewit, Sobel and Kirsch operators estimate gradient direction from the 8-neighborhood and can serve as compass operators. Roberts edge detector can estimate the direction only more coarsely in four direction. It is used as compass operator less often.

S5.19 This is an instance of a more general effect called image sharpening which aims at making edges steeper (having a higher gradient magnitude) which provides humans perception of a more sharp image. The sharpened image is obtained by subtracting a value in each pixel proportional to the sheerness of the image function at the pixel position, see Equation 5.36. Laplacian filter is often used as the estimator of image function sheerness because it provides the edge magnitude not taking into account the edge orientation. The sharpening image operation is rather popular in photo editors where more sophisticated versions of it are available.

S5.20 LoG operator stands for the Laplacian of Gaussian operator, Equation 5.48, which provides a second derivative of an image function. DoG (Difference of Gaussians) serves as an effective approximation to LoG operator. The image is blurred by two Gaussian filters with different standard deviations σ. The two resulting images are subtracted.

S5.21 The idea (E. Hildreth & D. Marr) comes from end of 1970th and it is supported by the neurophysiological evidence in human seeing. It can be observed that significant step edges are preserved after the original noisy signal is filtered by a low-pass Gaussian filter. The order of differentiation and convolution can be interchanged due to linearity, cf. Equations 5.48 and 5.49. The outcome of the derivation is the Laplacian of Gaussian (LoG) filter, i.e. the second derivative of the Gaussian filter is pre-calculated analytically. The LoG filter is also called the “Mexican hat” filter. This filter is convolved with the image. The process can be performed at multiple scales controlled by the standard deviation σ of the Gaussian filter.

S5.22 The answer follows from the answer to the previous question S5.21. Laplace edge detector is extremely sensitive to noise. LoG operator is not so sensitive to noise because the Laplacian of Gaussian was calculated analytically which is noise free. Only this “Mexican hat” filter is convolved with the potentially noisy image. The filtration effect of the LoG filter causes superior performance as compared to noise sensitive Laplacian edge detector. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 5: Image pre-processing

11

S5.23 The scale matches to the domain which the image operator takes. Consider the analogy to the scale in topographic maps. There is some very detailed base-map measured by surveyors. Coarser maps are created from this base-map by mapmakers. While making a course map, the map-maker has to generalize, i.e. to omit unimportant features in the map an retain important ones.

Analogical processes can be found in image processing. The captured image is analogous to the base-map. The image processing steps should deal with events in the image important from the application point of view. Unlike in topographic maps, it is often not known what is the size of interesting objects. The objects should be found by segmentation for which the image processing including the knowledge of scale is needed. This creates a chicken-and-egg problem.

The way out from it can be processing the image in various scales, i.e. by increasing the size of the image operators domain gradually and observe the events in the images. Some candidates for “objects” exist in the image only for certain interval of scales. This can serve as a hint about the interesting range of object sizes without the need to perform segmentation.

S5.24 The hysteresis in thresholding sets a strict and weak threshold: everything passing the strict threshold is accepted, and evidence abutting known evidence that passes the weak threshold is too.

Non-maximal suppression retains only evidence that is maximal in the local direction.

S5.25 We assumed for most image preprocessing operators explained in Chapter 5 that the domain of the operator is given and fixed, usually a small rectangular window. The performance of the operator can be enhanced if the operator domain is adapted to current available image data, still without knowing the interpretation (≈ image segmentation). We explained the idea of one such method called Averaging with rotating mask used for noise suppression in Section 5.3.1. A similar idea can be used for histogram modifications or other contrast enhancement in fixed size subimages or adaptively sought neighborhood.

S5.26 If a moving line is seen through a small aperture, only the motion vector perpendicular to the line can be observed, and the component collinear with the line remains invisible: thus, corners are better to track.

S5.27 Stochastic image restoration methods assume a stochastic image distortion model which is linear and global for the whole image. The model can be described in the Fourier domain as G(u, v) = F (u, v) H(u, v) + N (u, v) , where u, v are spatial frequencies, G is the distorted image frequency spectrum, F is the ideal undistorted image frequency spectrum (which is not available and should be estimated by the image restoration process), H is the linear distortion frequency spectrum and N is the noise frequency spectrum.

The inverse convolution called also the inverse filtration estimates the original undistorted image by inverting the formula above, i.e.

F (u, v) = G(u, v) H −1 (u, v) − N (u, v) H −1 (u, v) . © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

12

Chapter 5: Image pre-processing The method works well for images that are not much contaminated by noise.

If noise is not negligible, the additive error appears in the formula pronounced for frequencies, in which the the inverse filter has a small amplitude (analogy to dividing by zero). This effect appears usually for higher frequencies. Consequently, the image restored by the inverse filter has often blurred originally sharp edges.

The above described disadvantage is bypassed by Wiener filtration to certain degree. It works for non-negligible noise, the stochastic properties of which can be estimated. It is assumed that the noise is independent on the signal and that the stochastic process is stationary in the wider sense. The problem is expressed as the optimization task. The undegraded image is the solution of the overdetermined system of linear stochastic equations which minimize the mean square error.

S5.28 Individual degradation will be expressed as the convolution kernel H(u, v): (a) Relative camera motion.

We express the simplest special case for illustration. We consider a constant speed V of the moving object in direction of the axis x with respect to the camera which happened while the shutter was opened for time T .

The model of the degradation is H(u, v) =

sin(πV T u) . πV u

(b) Out-of-focus lens The blur caused by the out-of-focus thin lens for small depth of focus is expressed as J1 (a r) H(u, v) = , ar where J1 je Bessel function of the first order, r2 = u2 + v 2 , a is a shift in the image. The last parameter a shows that the model is not space invariant. (c) Atmospheric turbulence The distortions are caused by heat inhomogeneities in the atmosphere (air shimmer) causing a slight light rays diffraction.

The mathematical model was determined experimentally 2

2 5

H(u, v) = e−c(u +v ) 6 , where c is a constant given by the type of the turbulence. The constant c is often determined experimentally for the particular case.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 5: Image pre-processing

13

• Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

6

Segmentation I Short-answer questions S6.1 If objects’ gray-levels are clearly distinct from background gray-levels, thresholding is a suitable segmentation method.

S6.2 Image-function discontinuities form edges and edges form boundaries, edges are easy to compute, and prior knowledge of how to combine edges into boundaries and how to do that (methods) are available as well. Disadvantages include sensitivity to image noise or unsuitable information in an image resulting in edge presence in locations where there is no border, and no edge presence where a real border exists.

S6.3 Follow Fig. 6.12.

S6.4 Advantages: Inner border – always part of a region.

Outer border – never part of the region, useful for deriving properties such as perimeter, compactness, etc., and is consequently often used.

Extended border – defines a single common border between adjacent regions; maintaining other good properties of outer border definition.

Disadvantages: Inner border – may not be useful for deriving region-based properties, especially for small objects; if two regions are adjacent, they never have a common border.

Outer border – some border elements may be repeated in the outer border up to three times; if two regions are adjacent, they only have some parts common.

Extended border – some boundary portions are and some are not part of the region.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 6: Segmentation I

15

S6.5

(a)

(b) S6.6 The lower bound can be considered heuristic information when expanding nodes and assigning costs to subpaths in the graph. By summing the minimum value for each profile, the total is an estimate of the minimum-cost path through the graph. This total cost will be the lower limit of the cost of any path through the graph. This result allows the heuristic to be admissible, thus guaranteeing the success of the algorithm in finding the optimal path.

S6.7

• If h̃(ni ) = 0, the algorithm produces a minimum-cost search.

• If h̃(ni ) > h(ni ), the algorithm may run faster, but the minimum-cost result is not guaranteed.

• If h̃(ni ) ≤ h(ni ), the search will produce the minimum-cost path if and only if c(np , nq ) ≥ h̃(np ) − h̃(nq ) for any p, q, where c(np , nq ) is the true minimum cost of getting from np to nq , which is not easy to fulfill for a specific f (x).

• If h(ni ) = h̃(ni ), the search will always produce the minimum-cost path with a minimum number of expanded nodes.

• The better the estimate of h(n), the smaller the number of nodes that must be expanded.

S6.8 The main idea of the principle of optimality is: Whatever the path to the node E was, there exists an optimal path between E and the end point. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

16

Chapter 6: Segmentation I In other words, if the optimal path start point–endpoint goes through E, then both its parts start point–E and E–end point, are also optimal.

This principle allows to search for optimal predecessors and thus perform the search in an orderly fashion.

S6.9 The time complexity of A-algorithm depends on the heuristic. In the worst case, the number of nodes expanded is exponential in the length of the solution (the shortest path), but it is polynomial when proper heuristics are available (see answer to Question S6.7).

The complexity of DP, assuming 8-connected border and n nodes xm i in each graph layer m, 3n cost combinations must be computed for each layer, 3n(M − 1) + n being the total number of cost combination computations. Note that the number of successors does not have to be limited to “3” as is the case in 8-connectivity.

S6.10 DP requires a complete graph to be constructed, a subset may suffice for Aalgorithm.

DP presents an efficient way of simultaneously searching for optimal paths from multiple starting and ending points. If these points are not known, DP is probably a better choice, especially if computation of the partial costs g m (i, k) is simple.

Which approach is more efficient for a particular problem depends on evaluation functions and on the quality of heuristics for an A-algorithm.

S6.11 Follow Fig. 6.33.

S6.12 Using the standard representation, gradient k is unbounded (infinite) for vertical lines while all parameters are bounded for polar co-ordinate representation.

S6.13 Dimensionality is 3.

• x-coordinate of center of the circle, • y-coordinate of center of the circle, • circle radius.

S6.14 As a result, only a subset of all non-directional contributions must be considered and only a small number of the accumulator cells need be incremented. For example, considering a circle segmentation, if edge directions are quantized into four possible values, only one-fourth of the circle need take part in incrementing of accumulator cells.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 6: Segmentation I

17

S6.15 Let’s assume that outer object boundary definition is used to form the R-table — this corresponds well with the stated 90o increments of boundary directions.

Let’s not consider scaling.

N

2 1 Following Fig. 6.35 and Table 6.1, the first R-table entry (associated with the p boundary point marked “1” above) populates the 0o entry chain by adding ( (32 + 32 ), 135o ) = (4.24,135); the second R-table entry (associated with the boundary point marked “2” above) populates the 0o entry chain by adding (3.61,124); another R-table entry (associated with the boundary point marked “N” above) populates the 90o entry chain by adding (3.00,0); etc.

R-table is constructed: 0o : (4.24,135),(3.61,124),(2.24,117),(3.00,0),(3.16,72),(3.61,56),(3.00,180).

90o : (3.61,123),(3.61,34),(3.16,18),(3.00,0),(1.00,270).

180o : (2.24,333),(1.41,315),(1.00,270),(2.24,243),(2.83,225),(3.61,214),(3.00,180).

270o : (2.00,90),(4.12,194),(3.00,180),(4.12,176),(4.47,153).

S6.16 In general, these approaches are not dual = the edge based segmentation does not always produce regions that are identical to those obtained by region-based segmentation approaches.

Edge-based segmentations rely mainly on identification of contiguous locations with high-likelihoods of object boundary information (like high image gradient).

Region-based methods are attempting to satisfy Equations (6.22) and (6.23) according to some criteria of regional homogeneity.

S6.17 Equations (6.22) and (6.23).

S6.18 Principles are explained in Section 6.3, specifically in Algorithms 6.16, 6.18 and Section 6.3.2.

Splitting starts with an undersegmented image and only allows splits.

Merging starts with an oversegmented image and only allows merges.

Split-an-merge strategy allows both split and merge steps.

S6.19 Main principles are explained in Algorithm 6.21, compare computational efficiency with operations given in Chapter 13 and Section 13.7. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

18

Chapter 6: Segmentation I

S6.20 This is caused mainly by local variations of the image function and image noise.

S6.21 This strategy is based on the typical scenario, in which there are many more mismatches than matches in the template-based segmentation and so rejecting mismatches quickly pays off.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

7

Segmentation II Short-answer questions S7.1 The final location of mean-shift convergence process identifying the local density maximum is called the local probability density mode.

S7.2 Local density maximum of the probability density function results from each initialization.

S7.3 Follow Algorithm 7.3.

S7.4 Eqs. (7.2), (7.3), profile of gN (x) is defined by the same exponential expression as kN (x).

S7.5 Eqs. (7.4), (7.5), profile gE (x) is uniform.

S7.6 The set of all locations that converge to the same mode ycon defines the basin of attraction associated with this mode.

S7.7 This is an open-ended question – derive from text, Section 7.2.1.

S7.8 This is an open-ended question – derive from text, Section 7.2.1.

S7.9 Eline = 128 S7.10 Eq. (7.23), the solution is found if ∂v(s, t)/∂t = 0.

Eq. (7.29) when used with GVF snakes.

Unit time steps are typically used (∆t = 1).

Additional details can be found in Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

S7.11 Eq. (7.23), the solution is found if ∂v(s, t)/∂t = 0.

Eq. (7.29) when used with GVF snakes.

External forces are typically normalized to have maximum magnitude equal to © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

20

Chapter 7: Segmentation II 1 (pixel/step). After each step, or after a small number of steps, the snake is resampled to maintain uniform distances between points, usually corresponding to the “step” size of 1 (pixel/step).

Additional details can be found in Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

S7.12 Balloons include an inflation force term.

S7.13 GVF points toward the boundaries when in their proximity and varies smoothly over homogeneous image regions all the way to image borders. Consequently, it can drive a snake toward a border from a large distance and can segment object concavities.

S7.14 For example, level set representations of higher dimensional functions yield seamless treatment of topologic changes, movement of the propagating front is described by understandable speed functions, well formalized mathematical environment is used consisting of partial differential equations. The disadvantages include convergence to local optima, the topologic freedom may be both an advantage and disadvantage, etc.

S7.15 For example, using Eq. (7.49).

S7.16 (a) The speed function controls the behavior of front evolution. (b) Follow Eqs. (7.36) and (7.37). (c) Explanation is provided in text immediately following Eq. (7.37).

S7.17 Fuzzy connectedness µψ is a global fuzzy relationship that assigns every pair of image elements c and d a value in the interval [0,1] based on the affinity values ψ along all possible paths between these two image elements. The elements c and d are not expected to be nearby.

S7.18 For example, follow Fig. 7.31.

S7.19 Follow Fig. 7.31.

S7.20 Follow Fig. 7.35.

S7.21 Direct extension of dynamic programming or A-algorithm graph search to surfaces exponentially increases the numbers of successors that must be considered during the graph optimization.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

8

Shape representation and description Short-answer questions S8.1 Image must be segmented into image into objects/regions, objects/regions must be identified.

S8.2

• Input representation • Object reconstruction • Incomplete shape recognition • Local/global description character • Mathematical/heuristic techniques • Statistical or syntactic object description • Robustness to translation, rotation, and scale

S8.3 Shape description may be affected by (boundary/region-segmentation) noise in high resolution while small (but possibly important) detail may disappear in low resolution.

S8.4 Recognizing 3D objects projected onto a 2D image plane requires projectioninvariant features so that features do not change with a changing viewpoint.

S8.5 Rectangular, polar, tangential. Follow Fig. 8.5.

S8.6 See Fig. 8.6 and associated text in Section 8.2.1.

S8.7 See Fig. 8.6.

S8.8 (a) Boundary length is associated with the length from the beginning to the end of a regional boundary segment (for non-closed boundaries) or the complete length of a closed boundary. Boundary length values differ based on pixel connectivity considered, on image resolution, as well as on boundary definition itself. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

22

Chapter 8: Shape representation and description (b) Curvature is the rate of change of boundary (segment) slope. Follow Figs.

8.7, 8.8. (c) Bending energy is defined according to Eq. (8.2). Follow Fig. 8.9. (d) The signature of a region is obtained as a sequence of normal contour distances. Follow Fig. 8.10. (e) A line joining any two points of the region boundary is a chord and the distribution of lengths and angles of all chords on a contour gives chord distribution. Follow Fig. 8.11. (f) Fourier transform of boundaries using Tn descriptors is defined by Eq. (8.9).

Follow Fig. 8.12a. (g) Fourier transform of boundaries using Sn descriptors is defined by Eq. (8.17). Follow Fig. 8.12b. (h) Polygonal segment representation uses a sequence of vertices of a polygon outlining or approximated the region or segment. (i) Constant curvature representation partitions the region contour to segments that can be represented by constant values of curvature, these segments may be concatenated and used as primitives in syntactic description. (j) Tolerance interval representation uses a maximum allowed difference e to determine vertices of a derived polygonal representation. Follow Fig. 8.13.

S8.9 (a) 01012323. (b) 31113111.

S8.10 See Eqs. (8.3) - (8.6), Fig. 8.11.

S8.11 See Fig. 8.16 and associated text.

S8.12 See Eq. (8.18) and follow Figs. 8.17, 8.18.

S8.13 (a) Cross ratio – see Eq. (8.25). (b) System of four co-planar concurrent lines – dual to cross-ratio, see Eq. (8.25) and Fig. 8.20. (c) System of five co-planar concurrent lines – see Eq. (8.26). (d) System of five co-planar points – dual to five co-planar concurrent lines, see Eq. (8.26). (e) Plane conics – see Eqs. (8.29)-(8.32).

S8.14 (a) 4 collinear points. (b) 4 co-planar concurrent lines. (c) 5 general co-planar lines. (d) 5 co-planar points. (e) 1 conic and 2 co-planar lines; or a pair of plane conics. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 8: Shape representation and description

23

S8.15 Global invariants consider a complete shape of the object or object–surrounding configurations (e.g., all invariants mentioned in Question S8.13 are global). Local invariants are formed from local properties of objects (or their boundaries – e.g., curvature, torsion, Gaussian curvature) which are not affected by projective transforms.

S8.16 0, 0, -1, 0, -1, 1, 0.

S8.17 (a) Area is given by the number of pixels within it, considering the real size of pixels and noting that pixel sizes may differ across the image. (b) Euler’s number – see Eq. (8.34). (c) Horizontal and vertical projections – see Eq. (8.35). (d) Eccentricity can be defined in several ways, for example as a ratio of major and minor axes of an object, a ratio of main region axes of inertia, etc., see Fig. 8.24. (e) Elongatedness can be defined in several ways, for example as the aspect ratio of the region bounding rectangle, ratio of the region area and the square of its thickness, etc. Follow also Fig. 8.25. (f) Rectangularity – see Eq. (8.38). (g) Direction is defined as the direction of the longer side of a minimum bounding rectangle, see also Eq. (8.39). (h) Compactness – see Eq. (8.40). (i) Statistical moments – see Eqs. (8.41) - (8.65). (j) Convex hull of region R – the smallest convex region H which satisfies the condition R ⊆ H – see also Fig. 8.27. (k) Region concavity tree – follow Fig. 8.31.

S8.18 Iterative removal of region boundary pixels.

S8.19 Set of all region points which have the same minimum distance from the region boundary for at least two separate boundary points.

S8.20 See upper left panel of Fig. 8.35.

S8.21

• Homotopy • One-pixel thickness • Mediality • Rotation invariance • Noise immunity Noise immunity and mediality cannot be satisfied simultaneously.

Rotation invariance and one-pixel thickness are mutually contradictory.

S8.22 Graph nodes are constructed from primitives, arcs describe the mutual primitive relations. See Fig. 8.37.

S8.23 Shape classes represent generic shapes of the objects belonging to the same class so that inter-class differences are emphasized and intra-class differences minimized.

Generalization of class shape yields the importance of this concept. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

24

Chapter 8: Shape representation and description

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

9

Object recognition Short-answer questions S9.1 These definitions may be taken directly from the text, Section 9.1.

S9.2 These descriptions may be taken directly from the text, Section 9.1.

S9.3 These definitions may be taken directly from the text, Section 9.2.

S9.4 These definitions may be taken directly from the text, Section 9.2.1.

S9.5 (a) Follow Eq. (9.11), Algorithm 9.1, Fig. 9.6. (b) Follow Eq. (9.16), Section 9.2.3, Figs. 9.7, 9.8.

S9.6 A set of objects, each being accompanied by knowledge of its proper classification.

The larger the training set, and the better its quality with respect to its representativeness, the better the guarantee that the classifier may be set correctly— classification correctness and the size/quality of the training set are closely related. If the statistical properties of patterns are known, the necessary sizes of the training sets can be estimated, but in reality this is rarely the case.

S9.7 The explanation is provided in Section 9.2.5.

S9.8 Classifier learning methods must be inductive in the sense that information obtained from the elements of the training set must be generalized to cover the whole feature space, implying that the learning should be (near) optimal for all feasible patterns, not only for those patterns that were present in the training set. The classifier should be able to recognize even those objects that it has never ‘seen’ before.

The general idea of sequential increase in training set size can be understood as presenting small portions of a large training set to the classifier, whose performance is checked after each portion—the smallest portion size is just one element.

The unavoidable requirements of sequential information presentation necessitate © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

26

Chapter 9: Object recognition the sequential character of learning. Learning is not a one-step process, but rather a step-by-step process of improvement.

S9.9 Supervised learning does and unsupervised learning does not require training sets and in particular information about the class of objects is/is not required in the learning stage.

S9.10 See Figs. 9.14 and 9.16.

S9.11 Back-propagation algorithm allows training of neural networks in cases when the feature space is not linearly separable.

Its steps are explained in Algorithm 9.6.

S9.12 The momentum constant contributes a proportion of the update of the previous iteration into the current one. Thus, in areas of very low gradient, some movement continues.

S9.13 Follow Section 9.3.2 and Algorithm 9.7.

S9.14 Follow Section 9.3.3 and Algorithm 9.8.

S9.15 These definitions may be taken directly from the text, Section 9.4.

S9.16 Follow Algorithm 9.9.

S9.17 Follow Eq. (9.64) and Section 9.4.1.

S9.18 When they generate the same language.

S9.19 Grammar construction from examples – see Fig. 9.20 for block diagram.

S9.20 These definitions may be taken directly from the text, Section 9.5.

S9.21 The smallest number of deletions, insertions, and substitutions necessary to convert one string into the other.

S9.22 Follow Fig. 9.24.

S9.23 These definitions may be taken directly from the text, Section 9.6.1, explanation can be found in Algorithm 9.12.

S9.24 Follow Section 9.6.2 and Algorithm 9.13.

S9.25 GAs and simulated annealing avoid getting stuck in local optima, they do not guarantee that the global optimum is found, but the solution is usually nearoptimal; they search for an extremum of an objective function that represents the fitness of some complex system – for which meaningful derivatives may not even be available. Approaches based on derivatives are likely to converge to local optima (for non-monotonic objective functions), and the objective functions must be differentiable.

S9.26 These definitions may be taken directly from the text, Section 9.7.1.

S9.27 These definitions may be taken directly from the text, Section 9.7.2. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 9: Object recognition

27

S9.28 Follow Section 9.7.3 and Figs. 9.30–9.33.

S9.29 It is rare that a single classifier will solve a problem completely, or indeed ‘well enough’. It is common to combine a number of independent classifiers to improve overall performance. Often, these individual classifiers may be very weak in isolation. The final combination of weak classifiers in a single classification rule that is much more accurate than any of the constituent rules.

S9.30 As a result, the weak classifier Wk+1 in the “later” stage concentrates on the difficult examples misclassified in the previous stage.

S9.31 Follow Section 9.9.1 and Fig. 9.37.

S9.32 See Section 9.9.2. (a) Follow Fig. 9.40b. (b) Follow Fig. 9.40c.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

10

Image understanding Short-answer questions S10.1 Your own thoughts are necessary to provide the answer.

S10.2 Lower levels include pre-processing and simpler segmentation approaches. Upper processing levels mainly include image interpretation and understanding.

Your own thoughts are necessary to come up with examples.

S10.3 These explanations may be taken directly from the text, Section 10.1, which allows drawing the block diagrams.

Your own thoughts are necessary to come up with examples.

S10.4 Conflicts may arise in non-hierarchical control when two or more subsystems disagree in, e.g., region/object labeling (one region can be marked by two or more region detection subsystems at the same time and it is necessary to decide which label is the best).

S10.5 Short-term memory contains information about interpretation of the image.

Long-term memory, the knowledge base, consists of more general information that is valid for (almost) all representations of the problems to be solved.

S10.6 Your own thoughts are necessary to come up with examples.

S10.7 Follow Section 10.2 and Algorithm 10.4.

S10.8 Your own thoughts are necessary to come up with the answer.

S10.9 Information about the average shape from a population of shapes, as well as information about shape variation in the population.

S10.10 Follow Section 10.4, leading to Eqs. (10.4) and (10.6).

S10.11 The answer is “8”. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 10: Image understanding

29

S10.12 AAM’s do and ASM’s do not consider “appearance” (e.g., image intensity information) for the entire object or object group to be identified.

S10.13 (a) Follow Algorithm 10.8. (b) Follow Algorithm 10.9.

S10.14 Band-specific image intensities may serve as feature vectors, provided to supervised (with “truth”) or unsupervised statistical classifiers.

S10.15 Early-stage simple classifiers reject less likely locations while maintaining a very low false negative rate. Later-stage more complex classifiers eliminate more and more false positives, while still set not to reject true positives.

The classifiers are ordered in a cascade sequence starting with simple and fast classifiers used for quickly rejecting object-detection hypotheses, to employing more complex and therefore more powerful but slower classifiers that however are applied only to the remaining hypotheses.

S10.16 The additional stages in the classifier cascade are added until overall requirements on detection performance are met. See also Fig. 9.34.

S10.17 The classification capability of random forests is used for object recognition and the regression capability predicts the object location. Follow Section 10.8 for a more detailed explanation.

S10.18 Discrete labeling allows only one label to be assigned to each object in the final labeling. Effort is directed to achieving a consistent labeling all over the image.

Your own thoughts are necessary to come up with examples.

S10.19 Discrete labeling may be considered a special case of probabilistic labeling with one label probability always being 1 and all the others being 0 for each object.

S10.20 It is based on a hypothesize and verify principle, an objective function evaluating the quality of a segmentation and interpretation is optimized by a genetic algorithm. The method is initialized with an over-segmented image called a primary segmentation, in which starting regions are called primary regions. Primary regions are repeatedly merged into current regions during the segmentation process. The genetic algorithm is responsible for generating new populations of feasible image segmentation and interpretation hypotheses.

The primary region adjacency graph describes the primary image segmentation.

The specific region adjacency graph represents an image after the merging of all adjacent regions of the same interpretation into a single region (collapsing the primary region adjacency graph).

S10.21 Region growing segmentation approaches are sensitive to the split/merge order (Section 6.3) and usually result in an under-segmented or an over-segmented image. It is practically impossible to stop the region growing process with a high confidence that there are neither too many nor too few regions in the image.

Similarly, semantic approaches to segmentation and interpretation start with an over-segmented image in which some merges are not best possible, the semantic process attempts grouping regions which may already be incorrect. Conventional semantic region growing merges regions in an interpretation level only and does © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

30

Chapter 10: Image understanding not evaluate properties of newly merged regions – leading to local optima of region labeling.

The genetic image interpretation method solves these basic problems by allowing both region merging and splitting, incorporation of semantics and higher-level knowledge into the main segmentation process, not as post-processing, and by inherently employing genetic algorithms yielding a (near) global maximum of an objective function.

S10.22 A Markov model assumes that the probability of occupying a state is determined solely by recent history. An order-k Markov model assumes these probabilities depend only on the last three preceding states.

Section 10.11 provides more detailed information.

S10.23 An order-k hidden Markov model describes the probabilities of the state at time t = 1 with the current state depending probabilistically (only) on k predecessors in time.

S10.24 These definitions may be taken directly from the text, Section 10.11.

S10.25 Many correct answers exist here — search for problems in papers published in archival literature.

S10.26 You may use MATLAB or other graphing tools to plot such signal mixtures.

Your own thoughts are necessary to come up with observations and derived conclusions.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

11

3D geometry, correspondence, 3D from intensities Short-answer questions S11.1 Bottom-up vision will use low-level algorithms such as edge detectors to construct a solution; top-down will fit a model and somehow ‘require’ low-level features to match.

S11.2 Let start with the quote from J.J. Gibson (1904-1994, US psychologist, one of the pioneers of visual perception) “We not only see but we look, we not only touch we feel”. Active vision stresses seeking new view points from which more information is available to solve the task at hand. E.g., if someone grabs an unknown object, she/he attempts to look at it from different sides to see all what is important for object understanding. Active vision provides algorithms for analogous tasks when a machine is “looking”.

S11.3 Any modern building has parallel straight lines that do not appear parallel in images. Many man-made artifacts, such as railway lines, also demonstrate this.

S11.4 Intrinsic camera parameters are: focal length, two coordinates of the image center (the position in which the optical axis passes the image plane), scaling in x, y directions and the sheer parameter.

Extrinsic parameters are six parameters relating image coordinates and world coordinates, i.e. a translation vector (3 parameters) between the camera center and the origin of world coordinate system and three angles which align camera system coordinate axes and world coordinate axes.

S11.5 Yes, the geometric distortion of zoom lenses is worse than in the fix-focal-length lenses. The reason is a much more complicated optical construction of zoom lenses. Many design parameters have to be compromised. The geometric distortions change significantly for different zooms. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

32

Chapter 11: 3D geometry, correspondence, 3D from intensities

S11.6 It represents a constraint on where points may lie. This constraint says that corresponding point can be sought only along a single epipolar line. The onedimensional search problem is much simpler than the general two-dimensional search.

S11.7 The epipoles move to infinity causing that the epipolar lines are aligned with image rows. Such an image configuration is named the canonical camera configuration and is often used.

S11.8 The fundamental matrix expresses epipolar geometry algebraically in a general case. It tells how the corresponding point in stereo images are related.

The essential matrix captures similar relation. However, it is more specific as it is “metric” because it pertains to calibrated cameras.

S11.9 Solving the correspondence problem is one of the evergreen problems in computer vision. The most common application of the correspondence problem is stereo vision, structure from motion and tracking in video sequences. As the task has no solution in general (e.g., for texture-less scenes), available solutions are pragmatic.

In stereopsis, the relation between the left and right images is given by the epipolar geometry. This constraint is expressed algebraically by the fundamental matrix which has 7 free parameters. Let us assume the 8-point algorithm which needs at least 8 corresponding points in the left and right images to solve for unknown fundamental parameters by solving a system of linear equations. In practice, the overconstraint system of equations is solved. The mismatches are ruled out by robust statistics, usually by the RANSAC algorithm.

S11.10 The epipolar geometry is applied in computer vision mainly for easing correspondence problem is stereopsis and structure from motion.

S11.11 There is an algebraic constraint which relates three views (cf. epipolar constraint relating two views). The three views are related by a 3 × 3 × 3 algebraic structure (a symmetric tensor) called the trifocal tensor. This trifocal tensor can be estimated from known correspondences analogously as it was described in S11.9.

Having a trifocal tensor, allows to reconstruct the configuration of 3D points for which the correspondences are available as if an another camera was looking at the scene from a new view direction. That is a new look at a 3D scene which can be created. This process is named image transfer.

S11.12 A single camera view cannot provide a depth information. Each pixel corresponds to a ray stemming from the projection center. It is not known how far from the camera a 3D point lies on the ray. If the camera translates, we have two or more images of the same scene and we can reconstruct the depth of a 3D point if we have correspondences established.

S11.13 Relation between scene points in panoramic images (given by different view direction from a single point) constitute a 2D homography. The panorama construction counts with it. If the camera moves a more general structure from motion task would have to be solved. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 11: 3D geometry, correspondence, 3D from intensities

33

S11.14 The active illumination is the solution. Several principles can be used: (1) one camera is replaced by a projector which projects a known pattern (or patterns) on a scene, e.g. stripes. The widely used device, the Microsoft Kinect, serves as an interface in computer games. It projects infrared patterns. (2) Time-of-flight rangefinders which measure the delay of a pulse returning from the target. They can use meter or centrimeter electromagnetic waves as in radars or visible or infrared light as in laser range finders. (3) Photometric stereo as described in Section 11.7.2.

S11.15 The solution is to capture different views from different viewpoints. Multiple correspondences allow to disambiguate the structure of the scene. Structure from motion techniques are used. E.g. the Google Street View is the example of a complex system based on this principle.

S11.16 Moiré interferometry is an active vision technique. Two periodic light patterns, typically stripes, are projected on the scene. Due to light interference, the object is covered by a system of closed, non-intersecting curves, each of which lies in a plane of a constant distance from the viewer. Distance measurements obtained are only relative.

S11.17 Let us assume the simplest case. The scene is non-transparent, object surfaces are opaque and have Lambertian reflection properties. We have only a single illuminant, a point source at infinity. The image formation physics teaches us that the image intensity in each pixel depends on the direction to the viewer (two angles), direction to the light source (two angles), orientation of the infinitesimal object surface patch (given by the normal vector, 3 parameters), and the reflection parameter (albedo, one parameter). We like to solve the inverse task in which we intend to estimate all mentioned parameters from the intensity. This inverse task is ill-conditioned and no general solution to it exists. However, there is the photometric stereo which provides the solution. See S11.18 and Section 11.7.2 for more details.

S11.18 The context was described in S11.17. In a special case of a Lambertian surface, the photometric stereo provides the information about surface orientation (up to convex, concave ambiguity). The idea is to illuminate the same scene by at least three light sources in a non-degenerate configuration. The lights are switched on and off one by one. Three images serve as an input. See Section 11.7.2 for more details.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

12

Use of 3D vision Short-answer questions S12.1

• Correspondences are found between points at different times that (similarly to stereo) permit 3D information to be extracted.

• Knowledge of the texture permits an estimate of the plane on which it lies from its varying appearance in the scene.

• Sharp edges may be expected to be in focus,and therefore at a known distance form a variable focus camera.

• Vergence is an adaption of stereo geometry in which the twin heads can adjust their relative directions to assist in point localization.

• Contours in images may correspond to object boundaries; reasoning with these may permit the reconstruction of the objects.

S12.2 The image sequence informs only about intensity changes which are not necessarily related to the motion of points in the 3D scene. The intensity changes constitute the apparent motion only. Consider a classical B. Horn’s counterexample from his 1986 book Robot Vision. Have an opaque textureless sphere. In the first case, the sphere rotates. Now intensity changes are seen in the observer’s image sequence. In the second case, the sphere stays still and a single illuminant moves. The intensity changes and creates the apparent motion.

S12.3 Artful use of clothing stripes (sometimes narrow, sometimes broad) can easily deceive the eye about the true shape of the wearer.

S12.4 The question relates to the shape from contour task. The rim is the set of all points on the object surface where surface normal is perpendicular to the observer’s visual ray. If an observer looks at a sphere then the seen rim is a planar curve. If she/he looks at a curved potato then the rim can be a nonplanar curve. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 12: Use of 3D vision

35

S12.5 A nice counterexample is the Waterfall litograph painting from the Dutch painter M.C. Escher from 1961, see it in Google images. We cannot print it here due to copyright restrictions. We gave simpler counterexamples in Figure 12.10 of the book.

S12.6 Computer graphics has used polyhedral model for several decades. Polyhedral model serve as patterns in 3D model-based recognition. Knowledge acquired in line drawing processing in artificial intelligence in 1960s and 1970s can be explored in 3D model-based recognition. The idea of quickly filtering out impossible configuration speeds up the algorithms.

S12.7 The main procedure allowing to fuse partial range images into a single consistent 3D point cloud is the iterative closest point point algorithm. Its description is given in the book text.

S12.8 The procedure is named structure from motion. Its brief description is given in the book text.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

13

Mathematical morphology Short-answer questions S13.1 The answer is clear from the text.

S13.2 The answer is clear from the text.

S13.3 The answer is given in the text.

S13.4 The erosion is not a commutative operation.

S13.5 A second opening will have no effect. The set is already opened with respect to the structuring element after the first opening. It is an instance of the idempotent operation. Idempotency in general refers to an operation whose application after a certain number of items has no further effect.

S13.6 The answer is given in the text.

S13.7 The top-hat transformation is the gray-scale morphological operation of the set (image) X with the set (structuring element K defined as X (X ◦ K). The top-hat operation ◦ is often used to detect “peaks” in the signal (in 1D) or graylevel image (in 2D) which are “smaller” than the structuring element on a slowly growing or decreasing background. The 1D example is given in Figure 13.19 in the book.

S13.8 The skeleton definition by maximal balls is a formal definition which well illustrates the concept for the continuous image. A ball B(p, r) with center p and radius r, r ≥ 0, is the set of points with distances d from the center less than or equal to r. The ball B included in a set X is said to be maximal if and only if there is no larger ball included in X that contains B, i.e., each ball B 0 , B ⊆ B 0 ⊆ X =⇒ B 0 = B. Balls in 2D means circles. The example for two touching circles of the same diameter is given in Figure 13.1. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 13: Mathematical morphology

37

a Figure 13.1: Example: Skeleton by maximal balls of two touching circles. Left the input image, right the skeleton consisting of three disconnected points of a negligible diameter. © Cengage Learning 2015.

The definition of the skeleton is disadvantageous for digital images. First, inscribing discrete circles for various distance definition is complicated. Second, which is illustrated in Figure 13.1, the resulting skeleton is discontinuous. We would like to have a continuous skeleton for practical reasons. This leads to the definition of a homotopic skeleton. See S13.9.

S13.9 The question “What does the homotopic skeleton of two touching balls look like?” serves pedagogical purpose only. The homotopic skeletons would not be needed in practice. The skeleton would connect three isolated circles from S13.8 by a line which is illustrated in Figure 13.2.

a Figure 13.2: Hypothetical example: Homotopic skeleton by maximal balls of two touching circles. Left input image, right the skeleton is a single line connecting three disconnected points from S13.0 given in Figure 13.1 © Cengage Learning 2015.

S13.10 The role of ultimate erosion for marking particles is described in the book text in Section 13.5.4. as well as in Figure 13.33 of the book.

S13.11 A regional maximum corresponds to a geographical summit: it is a region all of whose boundary is lower than itself.

S13.12 The geodesic distance dX (x, y) is the shortest path between two points x, y which remains entirely contained in the set X. S13.13 Granulometry is the location of image regions/blobs of given size.

S13.14 There are many, especially in medical and biological applications. Analysis of cell images, and pollen counting, are just two such.

S13.15 The granulometric curve (called also granulometric spectrum) serves roughly similar purpose as the Fourier frequency spectrum. The granulometric spectrum tells what is the count of particles of certain size in the image similarly as the Fourier frequency spectrum tells what is the influence of individual harmonic functions of increasing frequency (analogous to increasing size of particles in granulometry) in the resulting linear combination of base functions.

S13.16 A watershed separates ‘catchment basins’. Their location identifies regions in the image. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

38

Chapter 13: Mathematical morphology

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

14

Image data compression Short-answer questions S14.1 Lossless compression permits error-free data reconstruction.

Lossy compression methods do not preserve information completely.

S14.2 (a) Eq. (14.1). (b) Eq. (14.2). (c) Eq. (14.5).

S14.3 Eq. (14.2).

S14.4 Follow Fig. 14.1.

S14.5 Represent (compressed) image data by (an incomplete set of) coefficients of discrete image transforms.

S14.6 Construct an estimate f˜(i, j) of the gray-level value of an image element (i, j) from values of gray-levels in its neighborhood; compressed image data are represented by differences between estimates and reality together with the prediction model parameters.

S14.7 Image is divided into small blocks, these blocks are represented as vectors and are coded using unique codewords from a codeword dictionary; instead of vectors, the vector codes are stored or transmitted with the codeword choice based on the best similarity between the image block represented by a coded vector and the image blocks represented by codewords from the dictionary; the code dictionary is transmitted together with the coded data.

S14.8 Symmetric image compression application — the image is each compressed and decompressed once (or the same number of times), like in video-conferencing.

Asymmetric image compression application — the image is compressed once and decompressed many times, like in web-casting. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

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Chapter 14: Image data compression

S14.9 Image is transmitted in low resolution first, higher resolution appearance stages follow.

S14.10 Basic idea is to break an image into pieces (fractals) and identify self-similar ones. The image is completely partitioned into non-overlapping domain regions of any size and shape; larger range regions are defined that can overlap and need not cover the entire image; these range regions are geometrically transformed using affine transforms to match the domain regions; the set of affine coefficients together with information about the selection of domain regions represents the fractal image encoding; the fractally compressed images are stored and transmitted as recursive algorithms—sets of equations with instructions on how to reproduce the image.

S14.11 To represent data by codes of variable length, with more frequent data being represented by shorter codes.

S14.12 JPEG standard is for still image compression.

MPEG standard is for full-motion video image compression.

S14.13

• Sequential DCT-based compression.

• Progressive DCT-based compression.

• Sequential lossless predictive compression.

• Hierarchical lossy or lossless compression.

Follow Section 14.9.1 for detailed algorithms.

S14.14 While JPEG standard uses primarily DCT, JPEG–2000 predominantly uses wavelelet transforms.

S14.15

• MPEG-1 – low-resolution full-motion video.

• MPEG-2 – higher-resolution standards such as TV and HDTV. • MPEG-4 – small-frame full motion compression with slow refresh needs for video telephony and interactive multimedia such as video conferencing.

S14.16

• I-frames are self-contained and serve as random access frames in MPEG frame streams.

• P-frames are coded using forward predictive coding with reference to the previous I- or P-frame; their compression ratio is substantially higher than that for I-frames.

• B-frames are coded using forward, backward, or bi-directional motioncompensated prediction or interpolation using two reference frames, closest past and future I- or P-frames, and offer the highest compression ratios.

S14.17 To offer different image quality and inversely corresponding compression ratios.

Associating different access rights with individual frames yields high compression ratios overall while it also allows (mostly via I-frames) random access, fast forward/reverse searches, reverse playback, audio-visual synchronization, robustness to error, editability, format flexibility, and cost trade-offs. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 14: Image data compression

41

Problems P14.1 (a) He = 0.33 + 0.52 + 0.5 + 0.33 = 1.68 (b) r = 2 − 1.68 = 0.32 (c) K = 2/1.68 = 1.19 (d) size = 512×512×1.68 = 440,402 bits = 55,051 bytes P14.2 He = 2, r = 0, K = 1, size = 65.5 kbytes P14.3 He (A) = He (B)

Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

15 Texture

Short-answer questions S15.1 Texture element, also called texture primitive.

S15.2 Weak textures have small spatial interactions between primitives.

Spatial interactions between primitives are somewhat regular for strong textures.

S15.3 Coarse textures are built from larger primitives, fine textures from smaller primitives. Fine textures are characterized by higher spatial frequencies, coarse textures by lower spatial frequencies.

S15.4 Texture description is scale dependent. Image resolution (scale) must be a consistent part of texture description; if the resolution is appropriate, texture character does not change for any position in texture-characterization window.

S15.5 Thre main texture description approaches exist—statistical, syntactic, and hybrid.

Statistical methods are suitable if primitive sizes are comparable with pixel sizes.

Description features are computed and statistical classifiers usually used for texture recognition.

Syntactic methods are more suitable for textures where primitives can be assigned a label—the primitive type—meaning that primitives can be described using a larger variety of properties than just tonal properties; for example, shape description. Syntactic texture description is based on an analogy between the texture primitive spatial relations and the structure of a formal language. Descriptions of textures from one class form a language that can be represented by its grammar. The recognition process is then a syntactic analysis of the texture description word.

Hybrid methods of texture description combine the statistical and syntactic approaches; they are partly syntactic because primitives are exactly defined, and partly statistical because spatial relations between primitives are based on probabilities. Statistical classifiers are typically used for recognition. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 15: Texture

43

S15.6 (a) Lower in fine textures. (b) Higher in fine textures. (c) Higher in fine textures. (d) Lower in fine textures. (e) Lower in fine textures. (f) Lower in fine textures.

S15.7 For example, the following features include directional properties: (a) Features evaluated from wedge slices of the Fourier transform. (b) Directional features derived from co-occurrence matrices. (c) Features derived from directionally-calculated primitive length descriptions. (d) Directional features derived from first-order and second-order statistics of edge distributions. (e) Directionally sensitive LBP features.

S15.8 The presence of information noise is highly probable and deterministic grammars are too restrictive to deal with resulting irregularities.

S15.9

S15.10 Rotational invariance is achieved by normalizing the circular chain code in a way that minimizes the resulting LBPri value — e.g., by rotating the circular neighbor set clockwise so that the resulting LBP value is minimized.

S15.11 Introduction of threshold t – while decreasing noise sensitivity – is the reason for the loss of strict gray-level transform invariance. (So-called ULTP’s add gray-transform invariance.)

S15.12 Fractal dimension D is specified by Eq. (15.30). Fractal dimension has been shown to correlate well with the function’s intuitive roughness.

Lacunarity measures describe characteristics of textures of different visual appearance that have the same fractal dimension. Follow Eq. (15.36). Lacunarity represents a second-order statistic and is small for fine textures and large for coarse ones. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

44

Chapter 15: Texture

S15.13 If the substitution rules are accompanied by probabilities, the grammar is called stochastic.

S15.14 Follow Section 15.2.3.

S15.15 Recognition of roads, road crossings, buildings, agricultural regions, and natural objects, classification of trees into classes, industrial and biomedical activities, quality inspection in the motor or textile industries, workpiece surface monitoring, road surface skidding estimation, micro-electronics, remote sensing, mammography, MR brain imaging, pulmonary parenchyma characterization, threedimensional texture analysis, content-based data retrieval from image databases, etc.

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter

16

Motion analysis Short-answer questions S16.1 Motion detection registers any motion present in the scene.

Moving-object detection identifies a specific moving object (i.e., includes localization, possibly also motion prediction).

S16.2 For example: (a) Maximum velocity (b) Small acceleration (c) Common motion (d) Mutual correspondence S16.3 It provides information about motion direction and other time-related motion properties, and about slow motion and small object motion.

S16.4 An aperture problem may cause ambiguity of contained motion information if only part of an object boundary is visible and it is therefore impossible to determine the motion completely. See Fig. 16.4c.

S16.5 The two basic assumptions are: (a) The observed brightness of any object point is constant over time. (b) Nearby points in the image plane move in a similar manner (velocity smoothness constraint).

S16.6 See Section 16.2.2: (a) Local optical flow estimation is based on standard brightness and smoothness assumptions, however the approach identifies image regions in which these assumptions do and do not hold. Identifying small regions where the assumptions hold solves the error propagation problem. Unfortunately, © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

46

Chapter 16: Motion analysis in regions where spatial gradients change slowly, optical flow estimation becomes ill-conditioned due to lack of motion information — global approaches need to be applied to such regions to overcome the lack of local information. (b) Continuous adaptation to errors determines optical flow iteratively by combining the local average flow vector with the gradient constraint equation.

Confidence is assigned to each flow vector based on heuristic judgments of correctness, and the local average flow is computed as a weighted average by confidence. Thus, the propagation of error-free estimates is inhibited.

S16.7 (a) Eq. (16.22). (b) Eq. (16.23). (c) Eqs. (16.24) – (16.26). (d) Eqs. (16.27) – (16.29), collision occurs if dmin < r, where radius r specifies size of a round observer.

S16.8 Optical flow approaches require high frame rates – intervals between image acquisitions must be very short.

Motion detection based on correspondence of interest points can be applied for lower frame rates (interest point correspondence must be established).

S16.9 The first step of the method is to find significant points in all images of the sequence—points least similar to their surrounding representing object corners, borders, or any other characteristic features in an image that can be tracked over time. Point detection is followed by a matching procedure, which looks for correspondences between these points. The process results in a sparse velocity field construction.

S16.10 The path coherence function represents a measure of agreement between the derived object trajectory and the motion constraints. (a) The function value is always positive. (b) The function reflects local absolute angular deviations of the trajectory. (c) The function should respond equally to positive and negative velocity changes. (d) The function should be normalized.

If several independently moving objects are tracked, the solution methods often rely on motion constraints and minimize a path coherence function (Eq. 16.78) that represents a measure of agreement between the derived object trajectory and the motion constraints.

S16.11 In object tracking, occlusion results in incomplete trajectories. To deal efficiently with incomplete trajectories, phantom points may be introduced and used as substitutes for those which are missing. These hypothetical points allow each potential trajectory to be treated as complete and thus permit consistent application of object tracking approaches. © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Chapter 16: Motion analysis

47

Problems Solutions to many relevant tasks, and Matlab implementations of selected algorithms that can help solve problems associated with this chapter, are provided in the Matlab Companion to this text: • Svoboda T., Kybic J., and Hlavac V. Image Processing, Analysis, and Machine Vision: A MATLAB Companion. Thomson Engineering, 2008.

The Matlab Companion homepage http://visionbook.felk.cvut.cz offers images used in the problems. The Matlab code is well commented and is provided for educational purposes.

© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.