1 /*
2 * Copyright (c) 2017, Alliance for Open Media. All rights reserved
3 *
4 * This source code is subject to the terms of the BSD 2 Clause License and
5 * the Alliance for Open Media Patent License 1.0. If the BSD 2 Clause License
6 * was not distributed with this source code in the LICENSE file, you can
7 * obtain it at www.aomedia.org/license/software. If the Alliance for Open
8 * Media Patent License 1.0 was not distributed with this source code in the
9 * PATENTS file, you can obtain it at www.aomedia.org/license/patent.
10 */
11
12 #ifndef AOM_AV1_ENCODER_MATHUTILS_H_
13 #define AOM_AV1_ENCODER_MATHUTILS_H_
14
15 #include <memory.h>
16 #include <math.h>
17 #include <stdio.h>
18 #include <stdlib.h>
19 #include <assert.h>
20
21 static const double TINY_NEAR_ZERO = 1.0E-16;
22
23 // Solves Ax = b, where x and b are column vectors of size nx1 and A is nxn
linsolve(int n,double * A,int stride,double * b,double * x)24 static INLINE int linsolve(int n, double *A, int stride, double *b, double *x) {
25 int i, j, k;
26 double c;
27 // Forward elimination
28 for (k = 0; k < n - 1; k++) {
29 // Bring the largest magnitude to the diagonal position
30 for (i = n - 1; i > k; i--) {
31 if (fabs(A[(i - 1) * stride + k]) < fabs(A[i * stride + k])) {
32 for (j = 0; j < n; j++) {
33 c = A[i * stride + j];
34 A[i * stride + j] = A[(i - 1) * stride + j];
35 A[(i - 1) * stride + j] = c;
36 }
37 c = b[i];
38 b[i] = b[i - 1];
39 b[i - 1] = c;
40 }
41 }
42 for (i = k; i < n - 1; i++) {
43 if (fabs(A[k * stride + k]) < TINY_NEAR_ZERO) return 0;
44 c = A[(i + 1) * stride + k] / A[k * stride + k];
45 for (j = 0; j < n; j++) A[(i + 1) * stride + j] -= c * A[k * stride + j];
46 b[i + 1] -= c * b[k];
47 }
48 }
49 // Backward substitution
50 for (i = n - 1; i >= 0; i--) {
51 if (fabs(A[i * stride + i]) < TINY_NEAR_ZERO) return 0;
52 c = 0;
53 for (j = i + 1; j <= n - 1; j++) c += A[i * stride + j] * x[j];
54 x[i] = (b[i] - c) / A[i * stride + i];
55 }
56
57 return 1;
58 }
59
60 ////////////////////////////////////////////////////////////////////////////////
61 // Least-squares
62 // Solves for n-dim x in a least squares sense to minimize |Ax - b|^2
63 // The solution is simply x = (A'A)^-1 A'b or simply the solution for
64 // the system: A'A x = A'b
least_squares(int n,double * A,int rows,int stride,double * b,double * scratch,double * x)65 static INLINE int least_squares(int n, double *A, int rows, int stride,
66 double *b, double *scratch, double *x) {
67 int i, j, k;
68 double *scratch_ = NULL;
69 double *AtA, *Atb;
70 if (!scratch) {
71 scratch_ = (double *)aom_malloc(sizeof(*scratch) * n * (n + 1));
72 scratch = scratch_;
73 }
74 AtA = scratch;
75 Atb = scratch + n * n;
76
77 for (i = 0; i < n; ++i) {
78 for (j = i; j < n; ++j) {
79 AtA[i * n + j] = 0.0;
80 for (k = 0; k < rows; ++k)
81 AtA[i * n + j] += A[k * stride + i] * A[k * stride + j];
82 AtA[j * n + i] = AtA[i * n + j];
83 }
84 Atb[i] = 0;
85 for (k = 0; k < rows; ++k) Atb[i] += A[k * stride + i] * b[k];
86 }
87 int ret = linsolve(n, AtA, n, Atb, x);
88 if (scratch_) aom_free(scratch_);
89 return ret;
90 }
91
92 // Matrix multiply
multiply_mat(const double * m1,const double * m2,double * res,const int m1_rows,const int inner_dim,const int m2_cols)93 static INLINE void multiply_mat(const double *m1, const double *m2, double *res,
94 const int m1_rows, const int inner_dim,
95 const int m2_cols) {
96 double sum;
97
98 int row, col, inner;
99 for (row = 0; row < m1_rows; ++row) {
100 for (col = 0; col < m2_cols; ++col) {
101 sum = 0;
102 for (inner = 0; inner < inner_dim; ++inner)
103 sum += m1[row * inner_dim + inner] * m2[inner * m2_cols + col];
104 *(res++) = sum;
105 }
106 }
107 }
108
109 //
110 // The functions below are needed only for homography computation
111 // Remove if the homography models are not used.
112 //
113 ///////////////////////////////////////////////////////////////////////////////
114 // svdcmp
115 // Adopted from Numerical Recipes in C
116
sign(double a,double b)117 static INLINE double sign(double a, double b) {
118 return ((b) >= 0 ? fabs(a) : -fabs(a));
119 }
120
pythag(double a,double b)121 static INLINE double pythag(double a, double b) {
122 double ct;
123 const double absa = fabs(a);
124 const double absb = fabs(b);
125
126 if (absa > absb) {
127 ct = absb / absa;
128 return absa * sqrt(1.0 + ct * ct);
129 } else {
130 ct = absa / absb;
131 return (absb == 0) ? 0 : absb * sqrt(1.0 + ct * ct);
132 }
133 }
134
svdcmp(double ** u,int m,int n,double w[],double ** v)135 static INLINE int svdcmp(double **u, int m, int n, double w[], double **v) {
136 const int max_its = 30;
137 int flag, i, its, j, jj, k, l, nm;
138 double anorm, c, f, g, h, s, scale, x, y, z;
139 double *rv1 = (double *)aom_malloc(sizeof(*rv1) * (n + 1));
140 g = scale = anorm = 0.0;
141 for (i = 0; i < n; i++) {
142 l = i + 1;
143 rv1[i] = scale * g;
144 g = s = scale = 0.0;
145 if (i < m) {
146 for (k = i; k < m; k++) scale += fabs(u[k][i]);
147 if (scale != 0.) {
148 for (k = i; k < m; k++) {
149 u[k][i] /= scale;
150 s += u[k][i] * u[k][i];
151 }
152 f = u[i][i];
153 g = -sign(sqrt(s), f);
154 h = f * g - s;
155 u[i][i] = f - g;
156 for (j = l; j < n; j++) {
157 for (s = 0.0, k = i; k < m; k++) s += u[k][i] * u[k][j];
158 f = s / h;
159 for (k = i; k < m; k++) u[k][j] += f * u[k][i];
160 }
161 for (k = i; k < m; k++) u[k][i] *= scale;
162 }
163 }
164 w[i] = scale * g;
165 g = s = scale = 0.0;
166 if (i < m && i != n - 1) {
167 for (k = l; k < n; k++) scale += fabs(u[i][k]);
168 if (scale != 0.) {
169 for (k = l; k < n; k++) {
170 u[i][k] /= scale;
171 s += u[i][k] * u[i][k];
172 }
173 f = u[i][l];
174 g = -sign(sqrt(s), f);
175 h = f * g - s;
176 u[i][l] = f - g;
177 for (k = l; k < n; k++) rv1[k] = u[i][k] / h;
178 for (j = l; j < m; j++) {
179 for (s = 0.0, k = l; k < n; k++) s += u[j][k] * u[i][k];
180 for (k = l; k < n; k++) u[j][k] += s * rv1[k];
181 }
182 for (k = l; k < n; k++) u[i][k] *= scale;
183 }
184 }
185 anorm = fmax(anorm, (fabs(w[i]) + fabs(rv1[i])));
186 }
187
188 for (i = n - 1; i >= 0; i--) {
189 if (i < n - 1) {
190 if (g != 0.) {
191 for (j = l; j < n; j++) v[j][i] = (u[i][j] / u[i][l]) / g;
192 for (j = l; j < n; j++) {
193 for (s = 0.0, k = l; k < n; k++) s += u[i][k] * v[k][j];
194 for (k = l; k < n; k++) v[k][j] += s * v[k][i];
195 }
196 }
197 for (j = l; j < n; j++) v[i][j] = v[j][i] = 0.0;
198 }
199 v[i][i] = 1.0;
200 g = rv1[i];
201 l = i;
202 }
203 for (i = AOMMIN(m, n) - 1; i >= 0; i--) {
204 l = i + 1;
205 g = w[i];
206 for (j = l; j < n; j++) u[i][j] = 0.0;
207 if (g != 0.) {
208 g = 1.0 / g;
209 for (j = l; j < n; j++) {
210 for (s = 0.0, k = l; k < m; k++) s += u[k][i] * u[k][j];
211 f = (s / u[i][i]) * g;
212 for (k = i; k < m; k++) u[k][j] += f * u[k][i];
213 }
214 for (j = i; j < m; j++) u[j][i] *= g;
215 } else {
216 for (j = i; j < m; j++) u[j][i] = 0.0;
217 }
218 ++u[i][i];
219 }
220 for (k = n - 1; k >= 0; k--) {
221 for (its = 0; its < max_its; its++) {
222 flag = 1;
223 for (l = k; l >= 0; l--) {
224 nm = l - 1;
225 if ((double)(fabs(rv1[l]) + anorm) == anorm || nm < 0) {
226 flag = 0;
227 break;
228 }
229 if ((double)(fabs(w[nm]) + anorm) == anorm) break;
230 }
231 if (flag) {
232 c = 0.0;
233 s = 1.0;
234 for (i = l; i <= k; i++) {
235 f = s * rv1[i];
236 rv1[i] = c * rv1[i];
237 if ((double)(fabs(f) + anorm) == anorm) break;
238 g = w[i];
239 h = pythag(f, g);
240 w[i] = h;
241 h = 1.0 / h;
242 c = g * h;
243 s = -f * h;
244 for (j = 0; j < m; j++) {
245 y = u[j][nm];
246 z = u[j][i];
247 u[j][nm] = y * c + z * s;
248 u[j][i] = z * c - y * s;
249 }
250 }
251 }
252 z = w[k];
253 if (l == k) {
254 if (z < 0.0) {
255 w[k] = -z;
256 for (j = 0; j < n; j++) v[j][k] = -v[j][k];
257 }
258 break;
259 }
260 if (its == max_its - 1) {
261 aom_free(rv1);
262 return 1;
263 }
264 assert(k > 0);
265 x = w[l];
266 nm = k - 1;
267 y = w[nm];
268 g = rv1[nm];
269 h = rv1[k];
270 f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y);
271 g = pythag(f, 1.0);
272 f = ((x - z) * (x + z) + h * ((y / (f + sign(g, f))) - h)) / x;
273 c = s = 1.0;
274 for (j = l; j <= nm; j++) {
275 i = j + 1;
276 g = rv1[i];
277 y = w[i];
278 h = s * g;
279 g = c * g;
280 z = pythag(f, h);
281 rv1[j] = z;
282 c = f / z;
283 s = h / z;
284 f = x * c + g * s;
285 g = g * c - x * s;
286 h = y * s;
287 y *= c;
288 for (jj = 0; jj < n; jj++) {
289 x = v[jj][j];
290 z = v[jj][i];
291 v[jj][j] = x * c + z * s;
292 v[jj][i] = z * c - x * s;
293 }
294 z = pythag(f, h);
295 w[j] = z;
296 if (z != 0.) {
297 z = 1.0 / z;
298 c = f * z;
299 s = h * z;
300 }
301 f = c * g + s * y;
302 x = c * y - s * g;
303 for (jj = 0; jj < m; jj++) {
304 y = u[jj][j];
305 z = u[jj][i];
306 u[jj][j] = y * c + z * s;
307 u[jj][i] = z * c - y * s;
308 }
309 }
310 rv1[l] = 0.0;
311 rv1[k] = f;
312 w[k] = x;
313 }
314 }
315 aom_free(rv1);
316 return 0;
317 }
318
SVD(double * U,double * W,double * V,double * matx,int M,int N)319 static INLINE int SVD(double *U, double *W, double *V, double *matx, int M,
320 int N) {
321 // Assumes allocation for U is MxN
322 double **nrU = (double **)aom_malloc((M) * sizeof(*nrU));
323 double **nrV = (double **)aom_malloc((N) * sizeof(*nrV));
324 int problem, i;
325
326 problem = !(nrU && nrV);
327 if (!problem) {
328 for (i = 0; i < M; i++) {
329 nrU[i] = &U[i * N];
330 }
331 for (i = 0; i < N; i++) {
332 nrV[i] = &V[i * N];
333 }
334 } else {
335 if (nrU) aom_free(nrU);
336 if (nrV) aom_free(nrV);
337 return 1;
338 }
339
340 /* copy from given matx into nrU */
341 for (i = 0; i < M; i++) {
342 memcpy(&(nrU[i][0]), matx + N * i, N * sizeof(*matx));
343 }
344
345 /* HERE IT IS: do SVD */
346 if (svdcmp(nrU, M, N, W, nrV)) {
347 aom_free(nrU);
348 aom_free(nrV);
349 return 1;
350 }
351
352 /* aom_free Numerical Recipes arrays */
353 aom_free(nrU);
354 aom_free(nrV);
355
356 return 0;
357 }
358
359 #endif // AOM_AV1_ENCODER_MATHUTILS_H_
360