aom_dsp/mathutils.h

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