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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_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;
}
// 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
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;
}
// 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_AV1_ENCODER_MATHUTILS_H_