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 /* * Copyright (c) 2021, Alliance for Open Media. All rights reserved * * This source code is subject to the terms of the BSD 3-Clause Clear License * and the Alliance for Open Media Patent License 1.0. If the BSD 3-Clause Clear * License was not distributed with this source code in the LICENSE file, you * can obtain it at aomedia.org/license/software-license/bsd-3-c-c/. 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 * aomedia.org/license/patent-license/. */ #ifndef AOM_AOM_DSP_MATHUTILS_H_ #define AOM_AOM_DSP_MATHUTILS_H_ #include #include #include #include #include #include "aom_dsp/aom_dsp_common.h" #include "aom_mem/aom_mem.h" // Calculate the Euclidean norm of a vector static INLINE double norm(double *x, int len) { double normsq = 0.0; for (int i = 0; i < len; ++i) normsq += x[i] * x[i]; return sqrt(normsq); } 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; } // Solves Ax = b, where x and b are column vectors of size nx1 and A is nxn, // without destroying the contents of matrix A and vector b. static INLINE int linsolve_const(int n, const double *A, int stride, const double *b, double *x) { assert(n > 0); assert(stride > 0); double *A_ = (double *)malloc(sizeof(*A_) * n * n); double *b_ = (double *)malloc(sizeof(*b_) * n); for (int i = 0; i < n; ++i) { memcpy(A_ + i * n, A + i * stride, sizeof(*A_) * n); } memcpy(b_, b, sizeof(*b_) * n); int ret = linsolve(n, A_, n, b_, x); free(A_); free(b_); return ret; } //////////////////////////////////////////////////////////////////////////////// // 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)); } 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); } // All-in-one least squares function // This integrates the other least_squares_* functions into a single call. // However, it requires the caller to allocate a potentially large intermediate // matrix, so the separate functions should be preferred where possible. static INLINE int least_squares(int n, double *A, int rows, int stride, double *b, double *scratch, double *x) { double *scratch_ = NULL; if (!scratch) { scratch_ = (double *)aom_malloc(sizeof(*scratch) * n * (n + 1)); scratch = scratch_; } double *AtA = scratch; double *Atb = scratch + n * n; least_squares_init(AtA, Atb, n); for (int row = 0; row < rows; row++) { least_squares_accumulate(AtA, Atb, &A[row * stride], b[row], n); } int ret = least_squares_solve(AtA, Atb, x, n); 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 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 = -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 = -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 + 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; } // Finds n - dimensional KLT to decorrelate n image components of size // width x height stored in components arrays each with the same stride. // The n x n forward KLT is returned in klt array which is assumed to store n^2 // values in the KLT matrix in row by row order. // Returns 0 for success, 1 for failure. static INLINE int klt_components(int n, const int16_t **components, int width, int height, int stride, double *klt) { const int size = width * height; double one_by_size = 1.0 / size; int64_t *sumsq = (int64_t *)aom_malloc(n * (n + 2) * sizeof(*sumsq)); if (!sumsq) return 1; int64_t *sum = sumsq + n * n; int64_t *vec = sum + n; double *covar = (double *)aom_malloc(2 * n * (n + 1) * sizeof(*covar)); if (!covar) { aom_free(sumsq); return 1; } double *means = covar + n * n; double *V = means + n; double *W = V + n * n; for (int i = 0; i < n; ++i) sum[i] = 0; for (int i = 0; i < n * n; ++i) sumsq[i] = 0; for (int r = 0; r < height; ++r) { for (int c = 0; c < width; ++c) { const int o = r * stride + c; for (int i = 0; i < n; ++i) vec[i] = components[i][o]; for (int i = 0; i < n; ++i) { for (int j = i; j < n; ++j) sumsq[i * n + j] += vec[i] * vec[j]; sum[i] += vec[i]; } } } for (int i = 0; i < n; ++i) means[i] = (double)sum[i] * one_by_size; for (int i = 0; i < n; ++i) for (int j = i; j < n; ++j) covar[i * n + j] = (double)sumsq[i * n + j] * one_by_size - means[i] * means[j]; // Fill up with Symmetry for (int i = 0; i < n; ++i) for (int j = 0; j < i; ++j) covar[i * n + j] = covar[j * n + i]; aom_free(sumsq); int res = SVD(klt, W, V, covar, n, n); if (!res) { // Transpose to get the forward klt for (int i = 0; i < n; ++i) { for (int j = i + 1; j < n; ++j) { double tmp = klt[i * n + j]; klt[i * n + j] = klt[j * n + i]; klt[j * n + i] = tmp; } } // As a convention make the first column of the KLT non-negative for (int i = 0; i < n; ++i) { if (klt[i * n] < 0.0) { for (int j = 0; j < n; ++j) klt[i * n + j] = -klt[i * n + j]; } } } aom_free(covar); return res; } // Variation of the above where filtered versions of the components // are used where the filter kernel is provided as an input. static INLINE int klt_filtered_components(int n, const int16_t **components, int width, int height, int stride, int kernel_size, int *kernel, double *klt) { assert(kernel_size & 1); // must be odd const int half_kernel_size = kernel_size >> 1; assert(width > 2 * half_kernel_size); assert(height > 2 * half_kernel_size); const int size = (width - 2 * half_kernel_size) * (height - 2 * half_kernel_size); double one_by_size = 1.0 / size; int64_t *sumsq = (int64_t *)aom_malloc(n * (n + 2) * sizeof(*sumsq)); if (!sumsq) return 1; int64_t *sum = sumsq + n * n; int64_t *vec = sum + n; double *covar = (double *)aom_malloc(2 * n * (n + 1) * sizeof(*covar)); if (!covar) { aom_free(sumsq); return 1; } double *means = covar + n * n; double *V = means + n; double *W = V + n * n; for (int i = 0; i < n; ++i) sum[i] = 0; for (int i = 0; i < n * n; ++i) sumsq[i] = 0; for (int r = half_kernel_size; r < height - half_kernel_size; ++r) { for (int c = half_kernel_size; c < width - half_kernel_size; ++c) { const int o = r * stride + c; for (int i = 0; i < n; ++i) { vec[i] = 0; int m = 0; for (int k = -half_kernel_size; k <= half_kernel_size; ++k) for (int l = -half_kernel_size; l <= half_kernel_size; ++l) vec[i] += components[i][o + k * stride + l] * kernel[m++]; } for (int i = 0; i < n; ++i) { for (int j = i; j < n; ++j) sumsq[i * n + j] += vec[i] * vec[j]; sum[i] += vec[i]; } } } for (int i = 0; i < n; ++i) means[i] = (double)sum[i] * one_by_size; for (int i = 0; i < n; ++i) for (int j = i; j < n; ++j) covar[i * n + j] = (double)sumsq[i * n + j] * one_by_size - means[i] * means[j]; // Fill up with Symmetry for (int i = 0; i < n; ++i) for (int j = 0; j < i; ++j) covar[i * n + j] = covar[j * n + i]; aom_free(sumsq); int res = SVD(klt, W, V, covar, n, n); if (!res) { // Transpose to get the forward klt for (int i = 0; i < n; ++i) { for (int j = i + 1; j < n; ++j) { double tmp = klt[i * n + j]; klt[i * n + j] = klt[j * n + i]; klt[j * n + i] = tmp; } } // As a convention make the first column of the KLT non-negative for (int i = 0; i < n; ++i) { if (klt[i * n] < 0.0) { for (int j = 0; j < n; ++j) klt[i * n + j] = -klt[i * n + j]; } } } aom_free(covar); return res; } #endif // AOM_AOM_DSP_MATHUTILS_H_