| /* |
| * 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 <memory.h> |
| #include <math.h> |
| #include <stdio.h> |
| #include <stdlib.h> |
| #include <assert.h> |
| |
| #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_ |
