Numerical Algorithms Methods for Computer Vision Machine Learning and Graphics 1st Solomon Solution Manual

Product details:

• ISBN-10 ‏ : ‎ 1482251884
• ISBN-13 ‏ : ‎ 978-1482251883
• Author: Justin Solomon

Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic design from a practical standpoint and provides insight into the theoretical tools needed to support these skills.

Table contents:

1. CHAPTER 1 Mathematics Review
2. 1.1 PRELIMINARIES: NUMBERS AND SETS
3. 1.2 VECTOR SPACES
4. 1.2.1 Defining Vector Spaces
5. Definition 1.1
6. Example 1.1
7. Figure 1.1 (a) Vectors (b) their span is the plane ℝ2; (c) because is a linear combination of and
8. Example 1.2
9. 1.2.2 Span, Linear Independence, and Bases
10. Definition 1.2
11. Example 1.3
12. Example 1.4
13. Definition 1.3
14. Definition 1.4
15. Example 1.5
16. Example 1.6
17. 1.2.3 Our Focus: ℝn
18. Definition 1.5
19. Example 1.7
20. Definition 1.6
21. Aside 1.1.
22. 1.3 LINEARITY
23. Definition 1.7
24. Example 1.8
25. Example 1.9
26. Example 1.10
27. 1.3.1 Matrices
28. Example 1.11
29. Example 1.12
30. Example 1.13
31. Example 1.14
32. 1.3.2 Scalars, Vectors, and Matrices
33. Definition 1.8
34. Example 1.15
35. Example 1.16
36. Figure 1.2 Two implementations of matrix-vector multiplication with different loop ordering.
37. 1.3.3 Matrix Storage and Multiplication Methods
38. Figure 1.3 Two possible ways to store (a) a matrix in memory: (b) row-major ordering and (c) column-major ordering.
39. 1.3.4 Model Problem:
40. Definition 1.9
41. 1.4 NON-LINEARITY: DIFFERENTIAL CALCULUS
42. Figure 1.4 The closer we zoom into f(x) = x3 + x2 − 8x + 4, the more it looks like a line.
43. 1.4.1 Differentiation in One Variable
44. Figure 1.5 Big-O notation; in the ε neighborhood of the origin, f(x) is dominated by Cg(x); outside this neighborhood, Cg(x) can dip back down.
45. Definition 1.10
46. 1.4.2 Differentiation in Multiple Variables
47. Definition 1.11
48. Figure 1.6 We can visualize a function f(x1, x2) as a three-dimensional graph; then is the direction on the (x1, x2) plane corresponding to the steepest ascent of f. Alternatively, we can think of f(x1, x2) as the brightness at (x1, x2) (dark indicates a low value of f), in which case ▿f points perpendicular to level sets in the direction where f is increasing and the image gets lighter.
49. Example 1.17
50. Example 1.18
51. Example 1.19
52. Example 1.20
53. Definition 1.12
54. Example 1.21
55. Example 1.22
56. 1.4.3 Optimization
57. Example 1.23
58. Example 1.24
59. Figure 1.7 Three rectangles with the same perimeter 2w + 2h but unequal areas wh; the square on the right with w = h maximizes wh over all possible choices with prescribed 2w + 2h = 1.
60. Example 1.25
61. Figure 1.8 (a) An equality-constrained optimization. Without constraints, is minimized at the star; solid lines show isocontours for increasing c. Minimizing subject to forces to be on the dashed curve, (b) The point is suboptimal since moving in the direction decreases while maintaining (c) The point is optimal since decreasing f from would require moving in the −▿f direction, which is perpendicular to the curve
62. Theorem 1.1
63. Example 1.26
64. Example 1.27
65. 1.5 EXERCISES
66. CHAPTER 2 Numerics and Error Analysis
67. 2.1 STORING NUMBERS WITH FRACTIONAL PARTS
68. 2.1.1 Fixed-Point Representations
69. 2.1.2 Floating-Point Representations
70. Figure 2.1 The values from Example 2.1 plotted on a number line; typical for floating-point number systems, they are unevenly spaced between the minimum (0.5) and the maximum (3.5).
71. Example 2.1
72. 2.1.3 More Exotic Options
73. 2.2 UNDERSTANDING ERROR
74. Example 2.2
75. 2.2.1 Classifying Error
76. Definition 2.1
77. Definition 2.2
78. Example 2.3
79. Example 2.4
80. Figure 2.2 Values of f(x) from Example 2.5, computed using IEEE floating-point arithmetic.
81. Example 2.5
82. Definition 2.3
83. Example 2.6
84. Example 2.7
85. 2.2.2 Conditioning, Stability, and Accuracy
86. Example 2.8
87. Definition 2.4
88. Example 2.9
89. Example 2.10
90. 2.3 PRACTICAL ASPECTS
91. 2.3.1 Computing Vector Norms
92. Figure 2.3 (a) A simplistic method for summing the elements of a vector (b) the Kahan summation algorithm.
93. 2.3.2 Larger-Scale Example: Summation
94. 2.4 EXERCISES
95. Figure 2.4 z-fighting, for Exercise 2.6; the overlap region is zoomed on the right.
96. II Linear Algebra
97. CHAPTER 3 Linear Systems and the LU Decomposition
98. 3.1 SOLVABILITY OF LINEAR SYSTEMS
99. 3.2 AD-HOC SOLUTION STRATEGIES
100. 3.3 ENCODING ROW OPERATIONS
101. 3.3.1 Permutation
102. Example 3.1
103. 3.3.2 Row Scaling
104. 3.3.3 Elimination
105. Example 3.2
106. Example 3.3
107. 3.4 GAUSSIAN ELIMINATION
108. 3.4.1 Forward-Substitution
109. Figure 3.1 Forward-substitution without pivoting; see §3.4.3 for pivoting options.
110. 3.4.2 Back-Substitution
111. 3.4.3 Analysis of Gaussian Elimination
112. Figure 3.2 Back-substitution for solving upper-triangular systems; this implementation returns the solution to the system without modifying U.
113. Example 3.4
114. 3.5 LU FACTORIZATION
115. 3.5.1 Constructing the Factorization
116. Proposition 3.1
117. 3.5.2 Using the Factorization
118. 3.5.3 Implementing LU
119. 3.6 EXERCISES
120. Figure 3.3 Pseudocode for computing the LU factorization of A ∈ ℝnxn, stored in the compact n × n format described in §3.5.3. This algorithm will fail if pivoting is needed.
121. CHAPTER 4 Designing and Analyzing Linear Systems
122. 4.1 SOLUTION OF SQUARE SYSTEMS
123. Figure 4.1 (a) The input for regression, a set of (x(k), y(k)) pairs; (b) a set of basis functions {f1, f2, f3, f4}; (c) the output of regression, a set of coefficients c1, …, c4 such that the linear combination goes through the data points.
124. 4.1.1 Regression
125. Example 4.1
126. Example 4.2
127. Example 4.3
128. Figure 4.2 Drawbacks of fitting function values exactly: (a) noisy data might be better represented by a simple function rather than a complex curve that touches every data point and (b) the basis functions might not be tuned to the function being sampled. In (b), we fit a polynomial of degree eight to nine samples from f(x) = |x| but would have been more successful using a basis of line segments.
129. Example 4.4
130. 4.1.2 Least-Squares
131. Theorem 4.1
132. 4.1.3 Tikhonov Regularization
133. Example 4.5
134. 4.1.4 Image Alignment
135. Figure 4.3 (a) The image alignment problem attempts to find the parameters A and of a transformation from one image of a scene to another using labeled keypoints on the first image paired with points on the second. As an example, keypoints marked in white on the two images in (b) are used to create (c) the aligned image.
136. Figure 4.4 Suppose rather than taking (a) the sharp image, we accidentally take (b) a blurry photo; then, deconvolution can be used to recover (c) a sharp approximation of the original image. The difference between (a) and (c) is shown in (d); only high-frequency detail is different between the two images.
137. 4.1.5 Deconvolution
138. Figure 4.5 (a) An example of a triangle mesh, the typical structure used to represent three-dimensional shapes in computer graphics, (b) In mesh parameterization, we seek a map from a three-dimensional mesh (left) to the two-dimensional image plane (right); the right-hand side shown here was computed using the method suggested in §4.1.6. (c) The harmonic condition is that the position of vertex is the average of the positions of its neighbors w1, …, w5.
139. 4.1.6 Harmonic Parameterization
140. 4.2 SPECIAL PROPERTIES OF LINEAR SYSTEMS
141. 4.2.1 Positive Definite Matrices and the Cholesky Factorization
142. Definition 4.1
143. Proposition 4.1
144. Aside 4.1
145. Example 4.6
146. Example 4.7

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