static real_t
nd_prediction (real_t max_costs, real_t price, unsigned band, int y_state,
range_t *range, wfa_t *wfa, coding_t *c)
{
real_t costs; /* current approximation costs */
range_t lrange = *range;
/*
* Predict 'range' with DC component approximation
*/
{
real_t x = get_ip_image_state (range->image, range->address,
range->level, 0, c);
real_t y = get_ip_state_state (0, 0, range->level, c);
real_t w = btor (rtob (x / y, c->coeff->dc_rpf), c->coeff->dc_rpf);
word_t s [2] = {0, -1};
lrange.into [0] = 0;
lrange.into [1] = NO_EDGE;
lrange.weight [0] = w;
lrange.mv_coord_bits = 0;
lrange.mv_tree_bits = 0;
lrange.nd_tree_bits = tree_bits (LEAF, lrange.level, &c->p_tree);
lrange.nd_weights_bits = 0;
lrange.tree_bits = 0;
lrange.matrix_bits = 0;
lrange.weights_bits = c->coeff->bits (&w, s, range->level, c->coeff);
}
costs = price * (lrange.weights_bits + lrange.nd_tree_bits);
/*
* Recursive aproximation of difference image
*/
if (costs < max_costs)
{
unsigned state;
range_t rrange; /* range: recursive subdivision */
unsigned last_state; /* last WFA state before recursion */
real_t *ipi [MAXSTATES]; /* inner products pointers */
unsigned width = width_of_level (range->level);
unsigned height = height_of_level (range->level);
real_t *pixels;
/*
* Generate difference image original - approximation
*/
{
unsigned n;
real_t *src, *dst; /* pointers to image data */
real_t w = - lrange.weight [0] * c->images_of_state [0][0];
src = c->pixels + range->address * size_of_level (range->level);
dst = c->pixels = pixels = Calloc (width * height, sizeof (real_t));
for (n = width * height; n; n--)
*dst++ = *src++ + w;
}
/*
* Approximate difference recursively.
*/
rrange = *range;
rrange.tree_bits = 0;
rrange.matrix_bits = 0;
rrange.weights_bits = 0;
rrange.mv_coord_bits = 0;
rrange.mv_tree_bits = 0;
rrange.nd_tree_bits = 0;
rrange.nd_weights_bits = 0;
rrange.image = 0;
rrange.address = 0;
last_state = wfa->states - 1;
for (state = 0; state <= last_state; state++)
if (need_image (state, wfa))
{
ipi [state] = c->ip_images_state[state];
c->ip_images_state[state]
= Calloc (size_of_tree (c->products_level), sizeof (real_t));
}
compute_ip_images_state (rrange.image, rrange.address, rrange.level,
1, 0, wfa, c);
costs += subdivide (max_costs - costs, band, y_state, &rrange, wfa, c,
NO, YES);
Free (pixels);
if (costs < max_costs && ischild (rrange.tree)) /* use prediction */
{
unsigned img, adr;
unsigned edge;
img = range->image;
adr = range->address;
*range = rrange;
range->image = img;
range->address = adr;
range->nd_tree_bits += lrange.nd_tree_bits;
range->nd_weights_bits += lrange.weights_bits;
for (edge = 0; isedge (lrange.into [edge]); edge++)
{
range->into [edge] = lrange.into [edge];
range->weight [edge] = lrange.weight [edge];
}
range->into [edge] = NO_EDGE;
range->prediction = edge;
for (state = last_state + 1; state < wfa->states; state++)
if (need_image (state, wfa))
memset (c->ip_images_state [state], 0,
size_of_tree (c->products_level) * sizeof (real_t));
}
else
costs = MAXCOSTS;
for (state = 0; state <= last_state; state++)
if (need_image (state, wfa))
{
Free (c->ip_images_state [state]);
c->ip_images_state [state] = ipi [state];
}
}
else
costs = MAXCOSTS;
return costs;
}
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)
/*
* Find an approximation of the current 'range' with a linear
* combination of vectors of the 'domain_pool'. The linear
* combination is generated step by step with the matching pursuit
* algorithm. If flag 'full_search' is set then compute complete set
* of linear combinations with n = {0, ..., 'max_edges'} vectors and
* return the best one. Otherwise abort the computation as soon as
* costs (LC (n + 1)) exceed costs ( LC (n)) and return the
* sub-optimal solution. 'price' is the langrange multiplier
* weighting rate and distortion. 'band' is the current color band
* and 'y_state' the corresponding state in the Y component at same
* pixel position. 'domain_pool' gives the set of available vectors,
* and 'coeff' the model for the linear factors. The number of
* elements in the linear combination is limited by 'max_edges'. In
* 'mp', vectors may be specified which should be excluded during the
* approximation.
*
* No return value.
*
* Side effects:
* 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);
/*
* 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;
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];
}
}
}
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;
Free (domain_blocks);
}
