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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_AOM_DSP_MATHUTILS_H_ #define AOM_AOM_DSP_MATHUTILS_H_ #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 // // 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)); } // Round the given positive value to nearest integer static AOM_FORCE_INLINE int iroundpf(float x) { assert(x >= 0.0); return (int)(x + 0.5f); } 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); } // 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; } } } static AOM_INLINE float approx_exp(float y) { #define A ((1 << 23) / 0.69314718056f) // (1 << 23) / ln(2) #define B \ 127 // Offset for the exponent according to IEEE floating point standard. #define C 60801 // Magic number controls the accuracy of approximation union { float as_float; int32_t as_int32; } container; container.as_int32 = ((int32_t)(y * A)) + ((B << 23) - C); return container.as_float; #undef A #undef B #undef C } #endif // AOM_AOM_DSP_MATHUTILS_H_