1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
|
/*
* Copyright (c) 2017, Alliance for Open Media. All rights reserved
*
* This source code is subject to the terms of the BSD 2 Clause License and
* the Alliance for Open Media Patent License 1.0. If the BSD 2 Clause License
* was not distributed with this source code in the LICENSE file, you can
* obtain it at www.aomedia.org/license/software. If the Alliance for Open
* Media Patent License 1.0 was not distributed with this source code in the
* PATENTS file, you can obtain it at www.aomedia.org/license/patent.
*/
#ifndef AOM_AOM_DSP_MATHUTILS_H_
#define AOM_AOM_DSP_MATHUTILS_H_
#include <assert.h>
#include <math.h>
#include <string.h>
#include "aom_dsp/aom_dsp_common.h"
static const double TINY_NEAR_ZERO = 1.0E-16;
// Solves Ax = b, where x and b are column vectors of size nx1 and A is nxn
static INLINE int linsolve(int n, double *A, int stride, double *b, double *x) {
int i, j, k;
double c;
// Forward elimination
for (k = 0; k < n - 1; k++) {
// Bring the largest magnitude to the diagonal position
for (i = n - 1; i > k; i--) {
if (fabs(A[(i - 1) * stride + k]) < fabs(A[i * stride + k])) {
for (j = 0; j < n; j++) {
c = A[i * stride + j];
A[i * stride + j] = A[(i - 1) * stride + j];
A[(i - 1) * stride + j] = c;
}
c = b[i];
b[i] = b[i - 1];
b[i - 1] = c;
}
}
for (i = k; i < n - 1; i++) {
if (fabs(A[k * stride + k]) < TINY_NEAR_ZERO) return 0;
c = A[(i + 1) * stride + k] / A[k * stride + k];
for (j = 0; j < n; j++) A[(i + 1) * stride + j] -= c * A[k * stride + j];
b[i + 1] -= c * b[k];
}
}
// Backward substitution
for (i = n - 1; i >= 0; i--) {
if (fabs(A[i * stride + i]) < TINY_NEAR_ZERO) return 0;
c = 0;
for (j = i + 1; j <= n - 1; j++) c += A[i * stride + j] * x[j];
x[i] = (b[i] - c) / A[i * stride + i];
}
return 1;
}
////////////////////////////////////////////////////////////////////////////////
// Least-squares
// Solves for n-dim x in a least squares sense to minimize |Ax - b|^2
// The solution is simply x = (A'A)^-1 A'b or simply the solution for
// the system: A'A x = A'b
//
// This process is split into three steps in order to avoid needing to
// explicitly allocate the A matrix, which may be very large if there
// are many equations to solve.
//
// The process for using this is (in pseudocode):
//
// Allocate mat (size n*n), y (size n), a (size n), x (size n)
// least_squares_init(mat, y, n)
// for each equation a . x = b {
// least_squares_accumulate(mat, y, a, b, n)
// }
// least_squares_solve(mat, y, x, n)
//
// where:
// * mat, y are accumulators for the values A'A and A'b respectively,
// * a, b are the coefficients of each individual equation,
// * x is the result vector
// * and n is the problem size
static INLINE void least_squares_init(double *mat, double *y, int n) {
memset(mat, 0, n * n * sizeof(double));
memset(y, 0, n * sizeof(double));
}
// Round the given positive value to nearest integer
static AOM_FORCE_INLINE int iroundpf(float x) {
assert(x >= 0.0);
return (int)(x + 0.5f);
}
static INLINE void least_squares_accumulate(double *mat, double *y,
const double *a, double b, int n) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
mat[i * n + j] += a[i] * a[j];
}
}
for (int i = 0; i < n; i++) {
y[i] += a[i] * b;
}
}
static INLINE int least_squares_solve(double *mat, double *y, double *x,
int n) {
return linsolve(n, mat, n, y, x);
}
// Matrix multiply
static INLINE void multiply_mat(const double *m1, const double *m2, double *res,
const int m1_rows, const int inner_dim,
const int m2_cols) {
double sum;
int row, col, inner;
for (row = 0; row < m1_rows; ++row) {
for (col = 0; col < m2_cols; ++col) {
sum = 0;
for (inner = 0; inner < inner_dim; ++inner)
sum += m1[row * inner_dim + inner] * m2[inner * m2_cols + col];
*(res++) = sum;
}
}
}
static AOM_INLINE float approx_exp(float y) {
#define A ((1 << 23) / 0.69314718056f) // (1 << 23) / ln(2)
#define B \
127 // Offset for the exponent according to IEEE floating point standard.
#define C 60801 // Magic number controls the accuracy of approximation
union {
float as_float;
int32_t as_int32;
} container;
container.as_int32 = ((int32_t)(y * A)) + ((B << 23) - C);
return container.as_float;
#undef A
#undef B
#undef C
}
#endif // AOM_AOM_DSP_MATHUTILS_H_
|
