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 /* * 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_DSP_MATHUTILS_H_ #define AOM_DSP_MATHUTILS_H_ #include #include #include #include #include #include "aom_dsp/aom_dsp_common.h" #include "aom_mem/aom_mem.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 static INLINE int least_squares(int n, double *A, int rows, int stride, double *b, double *scratch, double *x) { int i, j, k; double *scratch_ = NULL; double *AtA, *Atb; if (!scratch) { scratch_ = (double *)aom_malloc(sizeof(*scratch) * n * (n + 1)); scratch = scratch_; } AtA = scratch; Atb = scratch + n * n; for (i = 0; i < n; ++i) { for (j = i; j < n; ++j) { AtA[i * n + j] = 0.0; for (k = 0; k < rows; ++k) AtA[i * n + j] += A[k * stride + i] * A[k * stride + j]; AtA[j * n + i] = AtA[i * n + j]; } Atb[i] = 0; for (k = 0; k < rows; ++k) Atb[i] += A[k * stride + i] * b[k]; } int ret = linsolve(n, AtA, n, Atb, x); if (scratch_) aom_free(scratch_); return ret; } // 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; } } } // // The functions below are needed only for homography computation // Remove if the homography models are not used. // /////////////////////////////////////////////////////////////////////////////// // svdcmp // Adopted from Numerical Recipes in C static INLINE double apply_sign(double a, double b) { return ((b) >= 0 ? fabs(a) : -fabs(a)); } static INLINE double pythag(double a, double b) { double ct; const double absa = fabs(a); const double absb = fabs(b); if (absa > absb) { ct = absb / absa; return absa * sqrt(1.0 + ct * ct); } else { ct = absa / absb; return (absb == 0) ? 0 : absb * sqrt(1.0 + ct * ct); } } static INLINE int svdcmp(double **u, int m, int n, double w[], double **v) { const int max_its = 30; int flag, i, its, j, jj, k, l, nm; double anorm, c, f, g, h, s, scale, x, y, z; double *rv1 = (double *)aom_malloc(sizeof(*rv1) * (n + 1)); g = scale = anorm = 0.0; for (i = 0; i < n; i++) { l = i + 1; rv1[i] = scale * g; g = s = scale = 0.0; if (i < m) { for (k = i; k < m; k++) scale += fabs(u[k][i]); if (scale != 0.) { for (k = i; k < m; k++) { u[k][i] /= scale; s += u[k][i] * u[k][i]; } f = u[i][i]; g = -apply_sign(sqrt(s), f); h = f * g - s; u[i][i] = f - g; for (j = l; j < n; j++) { for (s = 0.0, k = i; k < m; k++) s += u[k][i] * u[k][j]; f = s / h; for (k = i; k < m; k++) u[k][j] += f * u[k][i]; } for (k = i; k < m; k++) u[k][i] *= scale; } } w[i] = scale * g; g = s = scale = 0.0; if (i < m && i != n - 1) { for (k = l; k < n; k++) scale += fabs(u[i][k]); if (scale != 0.) { for (k = l; k < n; k++) { u[i][k] /= scale; s += u[i][k] * u[i][k]; } f = u[i][l]; g = -apply_sign(sqrt(s), f); h = f * g - s; u[i][l] = f - g; for (k = l; k < n; k++) rv1[k] = u[i][k] / h; for (j = l; j < m; j++) { for (s = 0.0, k = l; k < n; k++) s += u[j][k] * u[i][k]; for (k = l; k < n; k++) u[j][k] += s * rv1[k]; } for (k = l; k < n; k++) u[i][k] *= scale; } } anorm = fmax(anorm, (fabs(w[i]) + fabs(rv1[i]))); } for (i = n - 1; i >= 0; i--) { if (i < n - 1) { if (g != 0.) { for (j = l; j < n; j++) v[j][i] = (u[i][j] / u[i][l]) / g; for (j = l; j < n; j++) { for (s = 0.0, k = l; k < n; k++) s += u[i][k] * v[k][j]; for (k = l; k < n; k++) v[k][j] += s * v[k][i]; } } for (j = l; j < n; j++) v[i][j] = v[j][i] = 0.0; } v[i][i] = 1.0; g = rv1[i]; l = i; } for (i = AOMMIN(m, n) - 1; i >= 0; i--) { l = i + 1; g = w[i]; for (j = l; j < n; j++) u[i][j] = 0.0; if (g != 0.) { g = 1.0 / g; for (j = l; j < n; j++) { for (s = 0.0, k = l; k < m; k++) s += u[k][i] * u[k][j]; f = (s / u[i][i]) * g; for (k = i; k < m; k++) u[k][j] += f * u[k][i]; } for (j = i; j < m; j++) u[j][i] *= g; } else { for (j = i; j < m; j++) u[j][i] = 0.0; } ++u[i][i]; } for (k = n - 1; k >= 0; k--) { for (its = 0; its < max_its; its++) { flag = 1; for (l = k; l >= 0; l--) { nm = l - 1; if ((double)(fabs(rv1[l]) + anorm) == anorm || nm < 0) { flag = 0; break; } if ((double)(fabs(w[nm]) + anorm) == anorm) break; } if (flag) { c = 0.0; s = 1.0; for (i = l; i <= k; i++) { f = s * rv1[i]; rv1[i] = c * rv1[i]; if ((double)(fabs(f) + anorm) == anorm) break; g = w[i]; h = pythag(f, g); w[i] = h; h = 1.0 / h; c = g * h; s = -f * h; for (j = 0; j < m; j++) { y = u[j][nm]; z = u[j][i]; u[j][nm] = y * c + z * s; u[j][i] = z * c - y * s; } } } z = w[k]; if (l == k) { if (z < 0.0) { w[k] = -z; for (j = 0; j < n; j++) v[j][k] = -v[j][k]; } break; } if (its == max_its - 1) { aom_free(rv1); return 1; } assert(k > 0); x = w[l]; nm = k - 1; y = w[nm]; g = rv1[nm]; h = rv1[k]; f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y); g = pythag(f, 1.0); f = ((x - z) * (x + z) + h * ((y / (f + apply_sign(g, f))) - h)) / x; c = s = 1.0; for (j = l; j <= nm; j++) { i = j + 1; g = rv1[i]; y = w[i]; h = s * g; g = c * g; z = pythag(f, h); rv1[j] = z; c = f / z; s = h / z; f = x * c + g * s; g = g * c - x * s; h = y * s; y *= c; for (jj = 0; jj < n; jj++) { x = v[jj][j]; z = v[jj][i]; v[jj][j] = x * c + z * s; v[jj][i] = z * c - x * s; } z = pythag(f, h); w[j] = z; if (z != 0.) { z = 1.0 / z; c = f * z; s = h * z; } f = c * g + s * y; x = c * y - s * g; for (jj = 0; jj < m; jj++) { y = u[jj][j]; z = u[jj][i]; u[jj][j] = y * c + z * s; u[jj][i] = z * c - y * s; } } rv1[l] = 0.0; rv1[k] = f; w[k] = x; } } aom_free(rv1); return 0; } static INLINE int SVD(double *U, double *W, double *V, double *matx, int M, int N) { // Assumes allocation for U is MxN double **nrU = (double **)aom_malloc((M) * sizeof(*nrU)); double **nrV = (double **)aom_malloc((N) * sizeof(*nrV)); int problem, i; problem = !(nrU && nrV); if (!problem) { for (i = 0; i < M; i++) { nrU[i] = &U[i * N]; } for (i = 0; i < N; i++) { nrV[i] = &V[i * N]; } } else { if (nrU) aom_free(nrU); if (nrV) aom_free(nrV); return 1; } /* copy from given matx into nrU */ for (i = 0; i < M; i++) { memcpy(&(nrU[i][0]), matx + N * i, N * sizeof(*matx)); } /* HERE IT IS: do SVD */ if (svdcmp(nrU, M, N, W, nrV)) { aom_free(nrU); aom_free(nrV); return 1; } /* aom_free Numerical Recipes arrays */ aom_free(nrU); aom_free(nrV); return 0; } #endif // AOM_DSP_MATHUTILS_H_