#include <math.h>
#include <stdio.h>
#include <algorithm>
+#include <Eigen/Sparse>
+#include <Eigen/SparseQR>
+#include <Eigen/OrderingMethods>
#include "effect_chain.h"
#include "effect_util.h"
#include "resample_effect.h"
#include "util.h"
+using namespace Eigen;
using namespace std;
namespace movit {
return num_samples_saved;
}
+// Normalize so that the sum becomes one. Note that we do it twice;
+// this sometimes helps a tiny little bit when we have many samples.
+template<class T>
+void normalize_sum(Tap<T>* vals, unsigned num)
+{
+ for (int normalize_pass = 0; normalize_pass < 2; ++normalize_pass) {
+ double sum = 0.0;
+ for (unsigned i = 0; i < num; ++i) {
+ sum += to_fp64(vals[i].weight);
+ }
+ for (unsigned i = 0; i < num; ++i) {
+ vals[i].weight = from_fp64<T>(to_fp64(vals[i].weight) / sum);
+ }
+ }
+}
+
// Make use of the bilinear filtering in the GPU to reduce the number of samples
// we need to make. This is a bit more complex than BlurEffect since we cannot combine
// two neighboring samples if their weights have differing signs, so we first need to
src_samples,
src_samples - src_bilinear_samples);
assert(int(src_samples) - int(num_samples_saved) == src_bilinear_samples);
-
- // Normalize so that the sum becomes one. Note that we do it twice;
- // this sometimes helps a tiny little bit when we have many samples.
- for (int normalize_pass = 0; normalize_pass < 2; ++normalize_pass) {
- double sum = 0.0;
- for (int i = 0; i < src_bilinear_samples; ++i) {
- sum += to_fp64(bilinear_weights_ptr[i].weight);
- }
- for (int i = 0; i < src_bilinear_samples; ++i) {
- bilinear_weights_ptr[i].weight = from_fp64<DestFloat>(
- to_fp64(bilinear_weights_ptr[i].weight) / sum);
- }
- }
+ normalize_sum(bilinear_weights_ptr, src_bilinear_samples);
}
return src_bilinear_samples;
}
return sum_sq_error;
}
+// Given a predefined, fixed set of bilinear weight positions, try to optimize
+// their weights through some linear algebra. This can do a better job than
+// the weight calculation in combine_samples() because it can look at the entire
+// picture (an effective weight can sometimes be affected by multiple samples).
+// It will also optimize weights for non-combined samples, which is useful when
+// a sample happens in-between texels for numerical reasons.
+//
+// The math goes as follows: The desired result is a weighted sum, where the
+// weights are the coefficients in <weights>:
+//
+// y = sum(c_j x_j, j)
+//
+// We try to approximate this by a different set of coefficients, which have
+// weights d_i and are placed at some fraction to the right of a source texel x_j.
+// This means it will influence two texels (x_j and x_{j+1}); generalizing this,
+// let us define that w_ij means the amount texel <j> influences bilinear weight
+// <i> (keeping in mind that w_ij = 0 for all but at most two different j).
+// This means the actually computed result is:
+//
+// y' = sum(d_i w_ij x_j, j)
+//
+// We assume w_ij fixed and wish to find {d_i} so that y' gets as close to y
+// as possible. Specifically, let us consider the sum of squred errors of the
+// coefficients:
+//
+// ε² = sum((sum( d_i w_ij, i ) - c_j)², j)
+//
+// The standard trick, which also applies just fine here, is to differentiate
+// the error with respect to each variable we wish to optimize, and set each
+// such expression to zero. Solving this equation set (which we can do efficiently
+// by letting Eigen invert a sparse matrix for us) yields the minimum possible
+// error. To see the form each such equation takes, pick any value k and
+// differentiate the expression by d_k:
+//
+// ∂(ε²)/∂(d_k) = sum(2(sum( d_i w_ij, i ) - c_j) w_kj, j)
+//
+// Setting this expression equal to zero, dropping the irrelevant factor 2 and
+// rearranging yields:
+//
+// sum(w_kj sum( d_i w_ij, i ), j) = sum(w_kj c_j, j)
+//
+// where again, we remember where the sums over j are over at most two elements,
+// since w_ij is nonzero for at most two values of j.
+template<class T>
+void optimize_sum_sq_error(const Tap<float>* weights, unsigned num_weights,
+ Tap<T>* bilinear_weights, unsigned num_bilinear_weights,
+ unsigned size)
+{
+ // Find the range of the desired weights.
+ int c_lower_pos = lrintf(weights[0].pos * size - 0.5);
+ int c_upper_pos = lrintf(weights[num_weights - 1].pos * size - 0.5) + 1;
+
+ SparseMatrix<float> A(num_bilinear_weights, num_bilinear_weights);
+ SparseVector<float> b(num_bilinear_weights);
+
+ // Convert each bilinear weight to the (x, frac) form for less junk in the code below.
+ int* pos = new int[num_bilinear_weights];
+ float* fracs = new float[num_bilinear_weights];
+ for (unsigned i = 0; i < num_bilinear_weights; ++i) {
+ const float pixel_pos = to_fp64(bilinear_weights[i].pos) * size - 0.5f;
+ const float f = pixel_pos - floor(pixel_pos);
+ pos[i] = int(floor(pixel_pos));
+ fracs[i] = lrintf(f / movit_texel_subpixel_precision) * movit_texel_subpixel_precision;
+ }
+
+ // The index ordering is a bit unusual to fit better with the
+ // notation in the derivation above.
+ for (unsigned k = 0; k < num_bilinear_weights; ++k) {
+ for (int j = pos[k]; j <= pos[k] + 1; ++j) {
+ const float f_kj = (j == pos[k]) ? (1.0f - fracs[k]) : fracs[k];
+ for (unsigned i = 0; i < num_bilinear_weights; ++i) {
+ float f_ij;
+ if (j == pos[i]) {
+ f_ij = 1.0f - fracs[i];
+ } else if (j == pos[i] + 1) {
+ f_ij = fracs[i];
+ } else {
+ // f_ij = 0
+ continue;
+ }
+ A.coeffRef(i, k) += f_kj * f_ij;
+ }
+ float c_j;
+ if (j >= c_lower_pos && j < c_upper_pos) {
+ c_j = weights[j - c_lower_pos].weight;
+ } else {
+ c_j = 0.0f;
+ }
+ b.coeffRef(k) += f_kj * c_j;
+ }
+ }
+ delete[] pos;
+ delete[] fracs;
+
+ A.makeCompressed();
+ SparseQR<SparseMatrix<float>, COLAMDOrdering<int> > qr(A);
+ assert(qr.info() == Success);
+ SparseMatrix<float> new_weights = qr.solve(b);
+ assert(qr.info() == Success);
+
+ for (unsigned i = 0; i < num_bilinear_weights; ++i) {
+ bilinear_weights[i].weight = from_fp64<T>(new_weights.coeff(i, 0));
+ }
+ normalize_sum(bilinear_weights, num_bilinear_weights);
+}
+
} // namespace
ResampleEffect::ResampleEffect()
bool fallback_to_fp32 = false;
double max_sum_sq_error_fp16 = 0.0;
for (unsigned y = 0; y < dst_samples; ++y) {
+ optimize_sum_sq_error(
+ weights + y * src_samples, src_samples,
+ bilinear_weights_fp16 + y * src_bilinear_samples, src_bilinear_samples,
+ src_size);
double sum_sq_error_fp16 = compute_sum_sq_error(
weights + y * src_samples, src_samples,
bilinear_weights_fp16 + y * src_bilinear_samples, src_bilinear_samples,
if (max_sum_sq_error_fp16 > 2.0f / (255.0f * 255.0f)) {
fallback_to_fp32 = true;
src_bilinear_samples = combine_many_samples(weights, src_size, src_samples, dst_samples, &bilinear_weights_fp32);
+ for (unsigned y = 0; y < dst_samples; ++y) {
+ optimize_sum_sq_error(
+ weights + y * src_samples, src_samples,
+ bilinear_weights_fp32 + y * src_bilinear_samples, src_bilinear_samples,
+ src_size);
+ }
}
// Encode as a two-component texture. Note the GL_REPEAT.
projects / movit / commitdiff