diff options
Diffstat (limited to 'converter/other/fiasco/codec/approx.c')
| -rw-r--r-- | converter/other/fiasco/codec/approx.c | 1025 |
1 files changed, 505 insertions, 520 deletions
diff --git a/converter/other/fiasco/codec/approx.c b/converter/other/fiasco/codec/approx.c index 5072fae3..d8fefcaa 100644 --- a/converter/other/fiasco/codec/approx.c +++ b/converter/other/fiasco/codec/approx.c @@ -1,10 +1,10 @@ /* - * approx.c: Approximation of range images with matching pursuit + * approx.c: Approximation of range images with matching pursuit * - * Written by: Ullrich Hafner - * - * This file is part of FIASCO («F»ractal «I»mage «A»nd «S»equence «CO»dec) - * Copyright (C) 1994-2000 Ullrich Hafner <hafner@bigfoot.de> + * Written by: Ullrich Hafner + * + * This file is part of FIASCO (Fractal Image And Sequence COdec) + * Copyright (C) 1994-2000 Ullrich Hafner */ /* @@ -34,7 +34,7 @@ /***************************************************************************** - local variables + local variables *****************************************************************************/ @@ -52,227 +52,205 @@ typedef struct mp /***************************************************************************** - prototypes + prototypes *****************************************************************************/ static void orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm, - const word_t *domain_blocks, const coding_t *c); + const word_t *domain_blocks, const coding_t *c); static void matching_pursuit (mp_t *mp, bool_t full_search, real_t price, - unsigned max_edges, int y_state, const range_t *range, - const domain_pool_t *domain_pool, const coeff_t *coeff, - const wfa_t *wfa, const coding_t *c); + unsigned max_edges, int y_state, const range_t *range, + const domain_pool_t *domain_pool, const coeff_t *coeff, + const wfa_t *wfa, const coding_t *c); /***************************************************************************** - public code + public code *****************************************************************************/ real_t approximate_range (real_t max_costs, real_t price, int max_edges, - int y_state, range_t *range, domain_pool_t *domain_pool, - coeff_t *coeff, const wfa_t *wfa, const coding_t *c) -/* - * Approximate image block 'range' by matching pursuit. This functions - * calls the matching pursuit algorithm several times (with different - * parameters) in order to find the best approximation. Refer to function - * 'matching_pursuit()' for more details about parameters. - * - * Return value: - * approximation costs - */ -{ - mp_t mp; - bool_t success = NO; - - /* - * First approximation attempt: default matching pursuit algorithm. - */ - mp.exclude [0] = NO_EDGE; - matching_pursuit (&mp, c->options.full_search, price, max_edges, - y_state, range, domain_pool, coeff, wfa, c); - - /* - * Next approximation attempt: remove domain block mp->indices [0] - * from domain pool (vector with smallest costs) and run the - * matching pursuit again. - */ - if (c->options.second_domain_block) - { - mp_t tmp_mp = mp; + int y_state, range_t *range, domain_pool_t *domain_pool, + coeff_t *coeff, const wfa_t *wfa, const coding_t *c) { +/*---------------------------------------------------------------------------- + Approximate image block 'range' by matching pursuit. This functions + calls the matching pursuit algorithm several times (with different + parameters) in order to find the best approximation. Refer to function + 'matching_pursuit()' for more details about parameters. + + Return value: approximation costs +-----------------------------------------------------------------------------*/ + mp_t mp; + + /* + * First approximation attempt: default matching pursuit algorithm. + */ + mp.exclude [0] = NO_EDGE; + matching_pursuit(&mp, c->options.full_search, price, max_edges, + y_state, range, domain_pool, coeff, wfa, c); + + /* + * Next approximation attempt: remove domain block mp->indices [0] + * from domain pool (vector with smallest costs) and run the + * matching pursuit again. + */ + if (c->options.second_domain_block) { + mp_t tmp_mp; - tmp_mp.exclude [0] = tmp_mp.indices [0]; - tmp_mp.exclude [1] = NO_EDGE; - - matching_pursuit (&tmp_mp, c->options.full_search, price, max_edges, - y_state, range, domain_pool, coeff, wfa, c); - if (tmp_mp.costs < mp.costs) /* success */ - { - success = YES; - mp = tmp_mp; - } - } - - /* - * Next approximation attempt: check whether some coefficients have - * been quantized to zero. Vectors causing the underflow are - * removed from the domain pool and then the matching pursuit - * algorithm is run again (until underflow doesn't occur anymore). - */ - if (c->options.check_for_underflow) - { - int iteration = -1; - mp_t tmp_mp = mp; + tmp_mp = mp; /* initial value */ + + tmp_mp.exclude[0] = tmp_mp.indices [0]; + tmp_mp.exclude[1] = NO_EDGE; + + matching_pursuit(&tmp_mp, c->options.full_search, price, max_edges, + y_state, range, domain_pool, coeff, wfa, c); + if (tmp_mp.costs < mp.costs) /* success */ + mp = tmp_mp; + } + + /* + * Next approximation attempt: check whether some coefficients have + * been quantized to zero. Vectors causing the underflow are + * removed from the domain pool and then the matching pursuit + * algorithm is run again (until underflow doesn't occur anymore). + */ + if (c->options.check_for_underflow) { + mp_t tmp_mp; + int iteration; + + tmp_mp = mp; /* initial value */ + iteration = -1; /* initial value */ - do - { - int i; + do { + int i; + + ++iteration; + tmp_mp.exclude[iteration] = NO_EDGE; + + for (i = 0; isdomain(tmp_mp.indices[i]); ++i) { + if (tmp_mp.weight [i] == 0) { + tmp_mp.exclude[iteration] = tmp_mp.indices [i]; + break; + } + } + if (isdomain (tmp_mp.exclude [iteration])) { + /* try again */ + tmp_mp.exclude [iteration + 1] = NO_EDGE; + + matching_pursuit(&tmp_mp, c->options.full_search, price, + max_edges, y_state, range, domain_pool, + coeff, wfa, c); + if (tmp_mp.costs < mp.costs) /* success */ + mp = tmp_mp; + } + } while (isdomain (tmp_mp.exclude [iteration]) + && iteration < MAXEDGES - 1); + } + + /* + * Next approximation attempt: check whether some coefficients have + * been quantized to +/- max-value. Vectors causing the overflow are + * removed from the domain pool and then the matching pursuit + * algorithm is run again (until overflow doesn't occur anymore). + */ + if (c->options.check_for_overflow) { + mp_t tmp_mp; + int iteration; + + tmp_mp = mp; /* initial value */ + iteration = -1; /* initial value */ + + do { + int i; - iteration++; - tmp_mp.exclude [iteration] = NO_EDGE; - - for (i = 0; isdomain (tmp_mp.indices [i]); i++) - if (tmp_mp.weight [i] == 0) - { - tmp_mp.exclude [iteration] = tmp_mp.indices [i]; - break; - } + ++iteration; + tmp_mp.exclude[iteration] = NO_EDGE; + + for (i = 0; isdomain (tmp_mp.indices [i]); ++i) { + rpf_t * const rpf = + tmp_mp.indices [i] ? coeff->rpf : coeff->dc_rpf; + + if (tmp_mp.weight [i] == btor (rtob (200, rpf), rpf) + || tmp_mp.weight [i] == btor (rtob (-200, rpf), rpf)) { + tmp_mp.exclude [iteration] = tmp_mp.indices [i]; + break; + } + } - if (isdomain (tmp_mp.exclude [iteration])) /* try again */ - { - tmp_mp.exclude [iteration + 1] = NO_EDGE; - - matching_pursuit (&tmp_mp, c->options.full_search, price, - max_edges, y_state, range, domain_pool, - coeff, wfa, c); - if (tmp_mp.costs < mp.costs) /* success */ - { - success = YES; - mp = tmp_mp; - } - } - } while (isdomain (tmp_mp.exclude [iteration]) - && iteration < MAXEDGES - 1); - } - - /* - * Next approximation attempt: check whether some coefficients have - * been quantized to +/- max-value. Vectors causing the overflow are - * removed from the domain pool and then the matching pursuit - * algorithm is run again (until overflow doesn't occur anymore). - */ - if (c->options.check_for_overflow) - { - int iteration = -1; - mp_t tmp_mp = mp; - - do - { - int i; - - iteration++; - tmp_mp.exclude [iteration] = NO_EDGE; - - for (i = 0; isdomain (tmp_mp.indices [i]); i++) - { - rpf_t *rpf = tmp_mp.indices [i] ? coeff->rpf : coeff->dc_rpf; - - if (tmp_mp.weight [i] == btor (rtob (200, rpf), rpf) - || tmp_mp.weight [i] == btor (rtob (-200, rpf), rpf)) - { - tmp_mp.exclude [iteration] = tmp_mp.indices [i]; - break; - } - } + if (isdomain(tmp_mp.exclude[iteration])) { + /* try again */ + tmp_mp.exclude[iteration + 1] = NO_EDGE; + + matching_pursuit(&tmp_mp, c->options.full_search, price, + max_edges, y_state, range, domain_pool, + coeff, wfa, c); + if (tmp_mp.costs < mp.costs) /* success */ + mp = tmp_mp; + } + } while (isdomain (tmp_mp.exclude [iteration]) + && iteration < MAXEDGES - 1); + } + + /* + * Finally, check whether the best approximation has costs + * smaller than 'max_costs'. + */ + if (mp.costs < max_costs) { + int edge; + int new_index, old_index; + + new_index = 0; + for (old_index = 0; isdomain (mp.indices[old_index]); ++old_index) { + if (mp.weight [old_index] != 0) { + mp.indices [new_index] = mp.indices [old_index]; + mp.into [new_index] = mp.into [old_index]; + mp.weight [new_index] = mp.weight [old_index]; + ++new_index; + } + } + mp.indices [new_index] = NO_EDGE; + mp.into [new_index] = NO_EDGE; + + /* + * Update of probability models + */ + { + word_t * const domain_blocks = + domain_pool->generate(range->level, y_state, + wfa, + domain_pool->model); + domain_pool->update(domain_blocks, mp.indices, + range->level, y_state, wfa, + domain_pool->model); + coeff->update (mp.weight, mp.into, range->level, coeff); + + Free(domain_blocks); + } - if (isdomain (tmp_mp.exclude [iteration])) /* try again */ - { - tmp_mp.exclude [iteration + 1] = NO_EDGE; - - matching_pursuit (&tmp_mp, c->options.full_search, price, - max_edges, y_state, range, domain_pool, - coeff, wfa, c); - if (tmp_mp.costs < mp.costs) /* success */ - { - success = YES; - mp = tmp_mp; - } - } - } while (isdomain (tmp_mp.exclude [iteration]) - && iteration < MAXEDGES - 1); - } - - /* - * Finally, check whether the best approximation has costs - * smaller than 'max_costs'. - */ - if (mp.costs < max_costs) - { - int edge; - bool_t overflow = NO; - bool_t underflow = NO; - int new_index, old_index; - - new_index = 0; - for (old_index = 0; isdomain (mp.indices [old_index]); old_index++) - if (mp.weight [old_index] != 0) - { - rpf_t *rpf = mp.indices [old_index] ? coeff->rpf : coeff->dc_rpf; - - if (mp.weight [old_index] == btor (rtob (200, rpf), rpf) - || mp.weight [old_index] == btor (rtob (-200, rpf), rpf)) - overflow = YES; - - mp.indices [new_index] = mp.indices [old_index]; - mp.into [new_index] = mp.into [old_index]; - mp.weight [new_index] = mp.weight [old_index]; - new_index++; - } - else - underflow = YES; - - mp.indices [new_index] = NO_EDGE; - mp.into [new_index] = NO_EDGE; - - /* - * Update of probability models - */ - { - word_t *domain_blocks = domain_pool->generate (range->level, y_state, - wfa, - domain_pool->model); - domain_pool->update (domain_blocks, mp.indices, - range->level, y_state, wfa, domain_pool->model); - coeff->update (mp.weight, mp.into, range->level, coeff); - - Free (domain_blocks); - } - - for (edge = 0; isedge (mp.indices [edge]); edge++) - { - range->into [edge] = mp.into [edge]; - range->weight [edge] = mp.weight [edge]; - } - range->into [edge] = NO_EDGE; - range->matrix_bits = mp.matrix_bits; - range->weights_bits = mp.weights_bits; - range->err = mp.err; - } - else - { - range->into [0] = NO_EDGE; - mp.costs = MAXCOSTS; - } + for (edge = 0; isedge (mp.indices [edge]); ++edge) { + range->into [edge] = mp.into [edge]; + range->weight [edge] = mp.weight [edge]; + } + range->into [edge] = NO_EDGE; + range->matrix_bits = mp.matrix_bits; + range->weights_bits = mp.weights_bits; + range->err = mp.err; + } else { + range->into [0] = NO_EDGE; + mp.costs = MAXCOSTS; + } - return mp.costs; + return mp.costs; } + + /***************************************************************************** - local variables + local variables *****************************************************************************/ @@ -310,15 +288,15 @@ static bool_t used [MAXSTATES]; /***************************************************************************** - private code + private code *****************************************************************************/ static void matching_pursuit (mp_t *mp, bool_t full_search, real_t price, - unsigned max_edges, int y_state, const range_t *range, - const domain_pool_t *domain_pool, const coeff_t *coeff, - const wfa_t *wfa, const coding_t *c) + unsigned max_edges, int y_state, const range_t *range, + const domain_pool_t *domain_pool, const coeff_t *coeff, + const wfa_t *wfa, const coding_t *c) /* * Find an approximation of the current 'range' with a linear * combination of vectors of the 'domain_pool'. The linear @@ -339,311 +317,318 @@ matching_pursuit (mp_t *mp, bool_t full_search, real_t price, * No return value. * * Side effects: - * vectors, factors, rate, distortion and costs are stored in 'mp' + * vectors, factors, rate, distortion and costs are stored in 'mp' */ { - unsigned n; /* current vector of the OB */ - int index; /* best fitting domain image */ - unsigned domain; /* counter */ - real_t norm; /* norm of range image */ - real_t additional_bits; /* bits for mc, nd, and tree */ - word_t *domain_blocks; /* current set of domain images */ - const real_t min_norm = 2e-3; /* lower bound of norm */ - unsigned best_n = 0; - unsigned size = size_of_level (range->level); + unsigned n; /* current vector of the OB */ + int index; /* best fitting domain image */ + unsigned domain; /* counter */ + real_t norm; /* norm of range image */ + real_t additional_bits; /* bits for mc, nd, and tree */ + word_t *domain_blocks; /* current set of domain images */ + const real_t min_norm = 2e-3; /* lower bound of norm */ + unsigned best_n = 0; + unsigned size = size_of_level (range->level); - /* - * Initialize domain pool and inner product arrays - */ - domain_blocks = domain_pool->generate (range->level, y_state, wfa, - domain_pool->model); - for (domain = 0; domain_blocks [domain] >= 0; domain++) - { - used [domain] = NO; - rem_denominator [domain] /* norm of domain */ - = get_ip_state_state (domain_blocks [domain], domain_blocks [domain], - range->level, c); - if (rem_denominator [domain] / size < min_norm) - used [domain] = YES; /* don't use domains with small norm */ - else - rem_numerator [domain] /* inner product <s_domain, b> */ - = get_ip_image_state (range->image, range->address, - range->level, domain_blocks [domain], c); - if (!used [domain] && fabs (rem_numerator [domain]) < min_norm) - used [domain] = YES; - } - - /* - * Exclude all domain blocks given in array 'mp->exclude' - */ - for (n = 0; isdomain (mp->exclude [n]); n++) - used [mp->exclude [n]] = YES; - - /* - * Compute the approximation costs if 'range' is approximated with - * no linear combination, i.e. the error is equal to the square - * of the image norm and the size of the automaton is determined by - * storing only zero elements in the current matrix row - */ - for (norm = 0, n = 0; n < size; n++) - norm += square (c->pixels [range->address * size + n]); - - additional_bits = range->tree_bits + range->mv_tree_bits - + range->mv_coord_bits + range->nd_tree_bits - + range->nd_weights_bits; - - mp->err = norm; - mp->weights_bits = 0; - mp->matrix_bits = domain_pool->bits (domain_blocks, NULL, range->level, - y_state, wfa, domain_pool->model); - mp->costs = (mp->matrix_bits + mp->weights_bits - + additional_bits) * price + mp->err; - - n = 0; - do - { - /* - * Current approximation is: b = d_0 o_0 + ... + d_(n-1) o_(n-1) - * with corresponding costs 'range->err + range->bits * p'. - * For all remaining state images s_i (used[s_i] == NO) set - * o_n : = s_i - \sum(k = 0, ... , n-1) {(<s_i, o_k> / ||o_k||^2) o_k} - * and try to beat current costs. - * Choose that vector for the next orthogonalization step, - * which has minimal costs: s_index. - * (No progress is indicated by index == -1) - */ + /* + * Initialize domain pool and inner product arrays + */ + domain_blocks = domain_pool->generate (range->level, y_state, wfa, + domain_pool->model); + for (domain = 0; domain_blocks [domain] >= 0; domain++) + { + used [domain] = NO; + rem_denominator [domain] = /* norm of domain */ + get_ip_state_state (domain_blocks [domain], domain_blocks [domain], + range->level, c); + if (rem_denominator [domain] / size < min_norm) + used [domain] = YES; /* don't use domains with small norm */ + else + rem_numerator [domain] /* inner product <s_domain, b> */ + = get_ip_image_state (range->image, range->address, + range->level, domain_blocks [domain], c); + if (!used [domain] && fabs (rem_numerator [domain]) < min_norm) + used [domain] = YES; + } + + /* + * Exclude all domain blocks given in array 'mp->exclude' + */ + for (n = 0; isdomain (mp->exclude [n]); n++) + used [mp->exclude [n]] = YES; + + /* + * Compute the approximation costs if 'range' is approximated with + * no linear combination, i.e. the error is equal to the square + * of the image norm and the size of the automaton is determined by + * storing only zero elements in the current matrix row + */ + for (norm = 0, n = 0; n < size; n++) + norm += square (c->pixels [range->address * size + n]); + + additional_bits = range->tree_bits + range->mv_tree_bits + + range->mv_coord_bits + range->nd_tree_bits + + range->nd_weights_bits; + + mp->err = norm; + mp->weights_bits = 0; + mp->matrix_bits = domain_pool->bits (domain_blocks, NULL, range->level, + y_state, wfa, domain_pool->model); + mp->costs = (mp->matrix_bits + mp->weights_bits + + additional_bits) * price + mp->err; + + n = 0; + do + { + /* + * Current approximation is: b = d_0 o_0 + ... + d_(n-1) o_(n-1) + * with corresponding costs 'range->err + range->bits * p'. + * For all remaining state images s_i (used[s_i] == NO) set + * o_n : = s_i - \sum(k = 0, ... , n-1) {(<s_i, o_k> / ||o_k||^2) o_k} + * and try to beat current costs. + * Choose that vector for the next orthogonalization step, + * which has minimal costs: s_index. + * (No progress is indicated by index == -1) + */ - real_t min_matrix_bits = 0; - real_t min_weights_bits = 0; - real_t min_error = 0; - real_t min_weight [MAXEDGES]; - real_t min_costs = full_search ? MAXCOSTS : mp->costs; + real_t min_matrix_bits = 0; + real_t min_weights_bits = 0; + real_t min_error = 0; + real_t min_weight [MAXEDGES]; + real_t min_costs = full_search ? MAXCOSTS : mp->costs; - for (index = -1, domain = 0; domain_blocks [domain] >= 0; domain++) - if (!used [domain]) - { - real_t matrix_bits, weights_bits; - /* - * To speed up the search through the domain images, - * the costs of using domain image 'domain' as next vector - * can be approximated in a first step: - * improvement of image quality - * <= square (rem_numerator[domain]) / rem_denominator[domain] - */ - { - word_t vectors [MAXEDGES + 1]; - word_t states [MAXEDGES + 1]; - real_t weights [MAXEDGES + 1]; - unsigned i, k; - - for (i = 0, k = 0; k < n; k++) - if (mp->weight [k] != 0) - { - vectors [i] = mp->indices [k]; - states [i] = domain_blocks [vectors [i]]; - weights [i] = mp->weight [k]; - i++; - } - vectors [i] = domain; - states [i] = domain_blocks [domain]; - weights [i] = 0.5; - vectors [i + 1] = -1; - states [i + 1] = -1; - - weights_bits = coeff->bits (weights, states, range->level, - coeff); - matrix_bits = domain_pool->bits (domain_blocks, vectors, - range->level, y_state, - wfa, domain_pool->model); - } - if (((matrix_bits + weights_bits + additional_bits) * price + - mp->err - - square (rem_numerator [domain]) / rem_denominator [domain]) - < min_costs) - { - /* - * 1.) Compute the weights (linear factors) c_i of the - * linear combination - * b = c_0 v_0 + ... + c_(n-1) v_(n-1) + c_n v_'domain' - * Use backward substitution to obtain c_i from the linear - * factors of the lin. comb. b = d_0 o_0 + ... + d_n o_n - * of the corresponding orthogonal vectors {o_0, ..., o_n}. - * Vector o_n of the orthogonal basis is obtained by using - * vector 'v_domain' in step n of the Gram Schmidt - * orthogonalization (see above for definition of o_n). - * Recursive formula for the coefficients c_i: - * c_n := <b, o_n> / ||o_n||^2 - * for i = n - 1, ... , 0: - * c_i := <b, o_i> / ||o_i||^2 + - * \sum (k = i + 1, ... , n){ c_k <v_k, o_i> - * / ||o_i||^2 } - * 2.) Because linear factors are stored with reduced precision - * factor c_i is rounded with the given precision in step i - * of the recursive formula. - */ - - unsigned k; /* counter */ - int l; /* counter */ - real_t m_bits; /* number of matrix bits to store */ - real_t w_bits; /* number of weights bits to store */ - real_t r [MAXEDGES]; /* rounded linear factors */ - real_t f [MAXEDGES]; /* linear factors */ - int v [MAXEDGES]; /* mapping of domains to vectors */ - real_t costs; /* current approximation costs */ - real_t m_err; /* current approximation error */ - - f [n] = rem_numerator [domain] / rem_denominator [domain]; - v [n] = domain; /* corresponding mapping */ - for (k = 0; k < n; k++) - { - f [k] = ip_image_ortho_vector [k] / norm_ortho_vector [k]; - v [k] = mp->indices [k]; - } - - for (l = n; l >= 0; l--) - { - rpf_t *rpf = domain_blocks [v [l]] - ? coeff->rpf : coeff->dc_rpf; - - r [l] = f [l] = btor (rtob (f [l], rpf), rpf); - - for (k = 0; k < (unsigned) l; k++) - f [k] -= f [l] * ip_domain_ortho_vector [v [l]][k] - / norm_ortho_vector [k] ; - } - - /* - * Compute the number of output bits of the linear combination - * and store the weights with reduced precision. The - * resulting linear combination is - * b = r_0 v_0 + ... + r_(n-1) v_(n-1) + r_n v_'domain' - */ - { - word_t vectors [MAXEDGES + 1]; - word_t states [MAXEDGES + 1]; - real_t weights [MAXEDGES + 1]; - int i; - - for (i = 0, k = 0; k <= n; k++) - if (f [k] != 0) - { - vectors [i] = v [k]; - states [i] = domain_blocks [v [k]]; - weights [i] = f [k]; - i++; - } - vectors [i] = -1; - states [i] = -1; - - w_bits = coeff->bits (weights, states, range->level, coeff); - m_bits = domain_pool->bits (domain_blocks, vectors, - range->level, y_state, - wfa, domain_pool->model); - } - - /* - * To compute the approximation error, the corresponding - * linear factors of the linear combination - * b = r_0 o_0 + ... + r_(n-1) o_(n-1) + r_n o_'domain' - * with orthogonal vectors must be computed with following - * formula: - * r_i := r_i + - * \sum (k = i + 1, ... , n) { r_k <v_k, o_i> - * / ||o_i||^2 } - */ - for (l = 0; (unsigned) l <= n; l++) - { - /* - * compute <v_n, o_n> - */ - real_t a; - - a = get_ip_state_state (domain_blocks [v [l]], - domain_blocks [domain], - range->level, c); - for (k = 0; k < n; k++) - a -= ip_domain_ortho_vector [v [l]][k] - / norm_ortho_vector [k] - * ip_domain_ortho_vector [domain][k]; - ip_domain_ortho_vector [v [l]][n] = a; - } - norm_ortho_vector [n] = rem_denominator [domain]; - ip_image_ortho_vector [n] = rem_numerator [domain]; - - for (k = 0; k <= n; k++) - for (l = k + 1; (unsigned) l <= n; l++) - r [k] += ip_domain_ortho_vector [v [l]][k] * r [l] - / norm_ortho_vector [k]; - /* - * Compute approximation error: - * error := ||b||^2 + - * \sum (k = 0, ... , n){r_k^2 ||o_k||^2 - 2 r_k <b, o_k>} - */ - m_err = norm; - for (k = 0; k <= n; k++) - m_err += square (r [k]) * norm_ortho_vector [k] - - 2 * r [k] * ip_image_ortho_vector [k]; - if (m_err < 0) /* TODO: return MAXCOSTS */ - warning ("Negative image norm: %f" - " (current domain: %d, level = %d)", - (double) m_err, domain, range->level); - - costs = (m_bits + w_bits + additional_bits) * price + m_err; - if (costs < min_costs) /* found a better approximation */ - { - index = domain; - min_costs = costs; - min_matrix_bits = m_bits; - min_weights_bits = w_bits; - min_error = m_err; - for (k = 0; k <= n; k++) - min_weight [k] = f [k]; - } - } - } + for (index = -1, domain = 0; domain_blocks [domain] >= 0; domain++) + if (!used [domain]) + { + real_t matrix_bits, weights_bits; + /* + * To speed up the search through the domain images, + * the costs of using domain image 'domain' as next vector + * can be approximated in a first step: + * improvement of image quality + * <= square (rem_numerator[domain]) / rem_denominator[domain] + */ + { + word_t vectors [MAXEDGES + 1]; + word_t states [MAXEDGES + 1]; + real_t weights [MAXEDGES + 1]; + unsigned i, k; + + for (i = 0, k = 0; k < n; k++) + if (mp->weight [k] != 0) + { + vectors [i] = mp->indices [k]; + states [i] = domain_blocks [vectors [i]]; + weights [i] = mp->weight [k]; + i++; + } + vectors [i] = domain; + states [i] = domain_blocks [domain]; + weights [i] = 0.5; + vectors [i + 1] = -1; + states [i + 1] = -1; + + weights_bits = coeff->bits (weights, states, range->level, + coeff); + matrix_bits = domain_pool->bits (domain_blocks, vectors, + range->level, y_state, + wfa, domain_pool->model); + } + if (((matrix_bits + weights_bits + additional_bits) * price + + mp->err - + square (rem_numerator[domain]) / rem_denominator[domain]) + < min_costs) + { + /* + * 1.) Compute the weights (linear factors) c_i of the + * linear combination + * b = c_0 v_0 + ... + c_(n-1) v_(n-1) + c_n v_'domain' + * Use backward substitution to obtain c_i from the linear + * factors of the lin. comb. b = d_0 o_0 + ... + d_n o_n + * of the corresponding orthogonal vectors {o_0, ..., o_n}. + * Vector o_n of the orthogonal basis is obtained by using + * vector 'v_domain' in step n of the Gram Schmidt + * orthogonalization (see above for definition of o_n). + * Recursive formula for the coefficients c_i: + * c_n := <b, o_n> / ||o_n||^2 + * for i = n - 1, ... , 0: + * c_i := <b, o_i> / ||o_i||^2 + + * \sum (k = i + 1, ... , n){ c_k <v_k, o_i> + * / ||o_i||^2 } + * 2.) Because linear factors are stored with reduced + * precision factor c_i is rounded with the given + * precision in step i of the recursive formula. + */ + + unsigned k; /* counter */ + int l; /* counter */ + real_t m_bits; /* number of matrix bits to store */ + real_t w_bits; /* number of weights bits to store */ + real_t r [MAXEDGES]; /* rounded linear factors */ + real_t f [MAXEDGES]; /* linear factors */ + int v [MAXEDGES]; /* mapping of domains to vectors*/ + real_t costs; /* current approximation costs */ + real_t m_err; /* current approximation error */ + + f [n] = rem_numerator [domain] / rem_denominator [domain]; + v [n] = domain; /* corresponding mapping */ + for (k = 0; k < n; k++) + { + f[k] = ip_image_ortho_vector[k] / norm_ortho_vector[k]; + v [k] = mp->indices [k]; + } + + for (l = n; l >= 0; --l) { + rpf_t * const rpf = domain_blocks[v[l]] + ? coeff->rpf : coeff->dc_rpf; + + unsigned int k; + + r[l] = f[l] = btor(rtob(f[l], rpf), rpf); + + { + real_t const fl = f[l]; + + for (k = 0; k < l; ++k) { + f[k] -= fl * ip_domain_ortho_vector[v[l]][k] + / norm_ortho_vector[k]; + } + } + } + + /* + * Compute the number of output bits of the linear + * combination and store the weights with reduced + * precision. The resulting linear combination is + * b = r_0 v_0 + ... + r_(n-1) v_(n-1) + r_n v_'domain' + */ + { + word_t vectors [MAXEDGES + 1]; + word_t states [MAXEDGES + 1]; + real_t weights [MAXEDGES + 1]; + int i; + + for (i = 0, k = 0; k <= n; k++) + if (f [k] != 0) + { + vectors [i] = v [k]; + states [i] = domain_blocks [v [k]]; + weights [i] = f [k]; + i++; + } + vectors [i] = -1; + states [i] = -1; + + w_bits = + coeff->bits(weights, states, range->level, coeff); + m_bits = domain_pool->bits (domain_blocks, vectors, + range->level, y_state, + wfa, domain_pool->model); + } + + /* + * To compute the approximation error, the corresponding + * linear factors of the linear combination + * b = r_0 o_0 + ... + r_(n-1) o_(n-1) + r_n o_'domain' + * with orthogonal vectors must be computed with following + * formula: + * r_i := r_i + + * \sum (k = i + 1, ... , n) { r_k <v_k, o_i> + * / ||o_i||^2 } + */ + for (l = 0; (unsigned) l <= n; l++) + { + /* + * compute <v_n, o_n> + */ + real_t a; + + a = get_ip_state_state (domain_blocks [v [l]], + domain_blocks [domain], + range->level, c); + for (k = 0; k < n; k++) + a -= ip_domain_ortho_vector [v [l]][k] + / norm_ortho_vector [k] + * ip_domain_ortho_vector [domain][k]; + ip_domain_ortho_vector [v [l]][n] = a; + } + norm_ortho_vector [n] = rem_denominator [domain]; + ip_image_ortho_vector [n] = rem_numerator [domain]; + + for (k = 0; k <= n; k++) + for (l = k + 1; (unsigned) l <= n; l++) + r [k] += ip_domain_ortho_vector [v [l]][k] * r [l] + / norm_ortho_vector [k]; + /* + * Compute approximation error: + * error := ||b||^2 + + * \sum (k = 0, ... , n){r_k^2 ||o_k||^2 - 2 r_k <b, o_k>} + */ + m_err = norm; + for (k = 0; k <= n; k++) + m_err += square (r [k]) * norm_ortho_vector [k] + - 2 * r [k] * ip_image_ortho_vector [k]; + if (m_err < 0) /* TODO: return MAXCOSTS */ + warning ("Negative image norm: %f" + " (current domain: %d, level = %d)", + (double) m_err, domain, range->level); + + costs = (m_bits + w_bits + additional_bits) * price + m_err; + if (costs < min_costs) /* found a better approximation */ + { + index = domain; + min_costs = costs; + min_matrix_bits = m_bits; + min_weights_bits = w_bits; + min_error = m_err; + for (k = 0; k <= n; k++) + min_weight [k] = f [k]; + } + } + } - if (index >= 0) /* found a better approximation */ - { - if (min_costs < mp->costs) - { - unsigned k; - - mp->costs = min_costs; - mp->err = min_error; - mp->matrix_bits = min_matrix_bits; - mp->weights_bits = min_weights_bits; - - for (k = 0; k <= n; k++) - mp->weight [k] = min_weight [k]; - - best_n = n + 1; - } - - mp->indices [n] = index; - mp->into [n] = domain_blocks [index]; - - used [index] = YES; - - /* - * Gram-Schmidt orthogonalization step n - */ - orthogonalize (index, n, range->level, min_norm, domain_blocks, c); - n++; - } - } - while (n < max_edges && index >= 0); - - mp->indices [best_n] = NO_EDGE; + if (index >= 0) /* found a better approximation */ + { + if (min_costs < mp->costs) + { + unsigned k; + + mp->costs = min_costs; + mp->err = min_error; + mp->matrix_bits = min_matrix_bits; + mp->weights_bits = min_weights_bits; + + for (k = 0; k <= n; k++) + mp->weight [k] = min_weight [k]; + + best_n = n + 1; + } + + mp->indices [n] = index; + mp->into [n] = domain_blocks [index]; + + used [index] = YES; + + /* + * Gram-Schmidt orthogonalization step n + */ + orthogonalize (index, n, range->level, min_norm, domain_blocks, c); + n++; + } + } + while (n < max_edges && index >= 0); + + mp->indices [best_n] = NO_EDGE; - mp->costs = (mp->matrix_bits + mp->weights_bits + additional_bits) * price - + mp->err; + mp->costs = (mp->matrix_bits + mp->weights_bits + additional_bits) * price + + mp->err; - Free (domain_blocks); + Free (domain_blocks); } static void orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm, - const word_t *domain_blocks, const coding_t *c) + const word_t *domain_blocks, const coding_t *c) /* * Step 'n' of the Gram-Schmidt orthogonalization procedure: * vector 'index' is orthogonalized with respect to the set @@ -655,8 +640,8 @@ orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm, * No return value. * * Side effects: - * The remainder values (numerator and denominator) of - * all 'domain_blocks' are updated. + * The remainder values (numerator and denominator) of + * all 'domain_blocks' are updated. */ { unsigned domain; @@ -676,25 +661,25 @@ orthogonalize (unsigned index, unsigned n, unsigned level, real_t min_norm, for (domain = 0; domain_blocks [domain] >= 0; domain++) if (!used [domain]) { - unsigned k; - real_t tmp = get_ip_state_state (domain_blocks [index], - domain_blocks [domain], level, c); - - for (k = 0; k < n; k++) - tmp -= ip_domain_ortho_vector [domain][k] / norm_ortho_vector [k] - * ip_domain_ortho_vector [index][k]; - ip_domain_ortho_vector [domain][n] = tmp; - rem_denominator [domain] -= square (tmp) / norm_ortho_vector [n]; - rem_numerator [domain] -= ip_image_ortho_vector [n] - / norm_ortho_vector [n] - * ip_domain_ortho_vector [domain][n] ; - - /* - * Exclude vectors with small denominator - */ - if (!used [domain]) - if (rem_denominator [domain] / size_of_level (level) < min_norm) - used [domain] = YES; + unsigned k; + real_t tmp = get_ip_state_state (domain_blocks [index], + domain_blocks [domain], level, c); + + for (k = 0; k < n; k++) + tmp -= ip_domain_ortho_vector [domain][k] / norm_ortho_vector [k] + * ip_domain_ortho_vector [index][k]; + ip_domain_ortho_vector [domain][n] = tmp; + rem_denominator [domain] -= square (tmp) / norm_ortho_vector [n]; + rem_numerator [domain] -= ip_image_ortho_vector [n] + / norm_ortho_vector [n] + * ip_domain_ortho_vector [domain][n] ; + + /* + * Exclude vectors with small denominator + */ + if (!used [domain]) + if (rem_denominator [domain] / size_of_level (level) < min_norm) + used [domain] = YES; } } |