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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_AV1_ENCODER_MATHUTILS_H_
+#define AOM_AV1_ENCODER_MATHUTILS_H_
+
+#include <memory.h>
+#include <math.h>
+#include <stdio.h>
+#include <stdlib.h>
+#include <assert.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 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;
+}
+
+#endif // AOM_AV1_ENCODER_MATHUTILS_H_
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