#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 "fp16.h"
+#include "init.h"
#include "resample_effect.h"
#include "util.h"
+using namespace Eigen;
using namespace std;
namespace movit {
namespace {
+template<class T>
+struct Tap {
+ T weight;
+ T pos;
+};
+
float sinc(float x)
{
if (fabs(x) < 1e-6) {
return a;
}
-unsigned combine_samples(float *src, float *dst, unsigned num_src_samples, unsigned max_samples_saved)
+template<class DestFloat>
+unsigned combine_samples(const Tap<float> *src, Tap<DestFloat> *dst, unsigned src_size, unsigned num_src_samples, unsigned max_samples_saved)
{
unsigned num_samples_saved = 0;
for (unsigned i = 0, j = 0; i < num_src_samples; ++i, ++j) {
// Copy the sample directly; it will be overwritten later if we can combine.
if (dst != NULL) {
- dst[j * 2 + 0] = src[i * 2 + 0];
- dst[j * 2 + 1] = src[i * 2 + 1];
+ dst[j].weight = convert_float<float, DestFloat>(src[i].weight);
+ dst[j].pos = convert_float<float, DestFloat>(src[i].pos);
}
if (i == num_src_samples - 1) {
continue;
}
- float w1 = src[i * 2 + 0];
- float w2 = src[(i + 1) * 2 + 0];
+ float w1 = src[i].weight;
+ float w2 = src[i + 1].weight;
if (w1 * w2 < 0.0f) {
// Differing signs; cannot combine.
continue;
}
- float pos1 = src[i * 2 + 1];
- float pos2 = src[(i + 1) * 2 + 1];
+ float pos1 = src[i].pos;
+ float pos2 = src[i + 1].pos;
assert(pos2 > pos1);
- float offset, total_weight, sum_sq_error;
- combine_two_samples(w1, w2, &offset, &total_weight, &sum_sq_error);
+ fp16_int_t pos, total_weight;
+ float sum_sq_error;
+ combine_two_samples(w1, w2, pos1, pos2, src_size, &pos, &total_weight, &sum_sq_error);
// If the interpolation error is larger than that of about sqrt(2) of
// a level at 8-bit precision, don't combine. (You'd think 1.0 was enough,
// but since the artifacts are not really random, they can get quite
// visible. On the other hand, going to 0.25f, I can see no change at
// all with 8-bit output, so it would not seem to be worth it.)
- if (sum_sq_error > 0.5f / (256.0f * 256.0f)) {
+ if (sum_sq_error > 0.5f / (255.0f * 255.0f)) {
continue;
}
// OK, we can combine this and the next sample.
if (dst != NULL) {
- dst[j * 2 + 0] = total_weight;
- dst[j * 2 + 1] = pos1 + offset * (pos2 - pos1);
+ dst[j].weight = total_weight;
+ dst[j].pos = pos;
}
++i; // Skip the next sample.
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
+// figure out the maximum number of samples. Then, we downconvert all the weights to
+// that number -- we could have gone for a variable-length system, but this is simpler,
+// and the gains would probably be offset by the extra cost of checking when to stop.
+//
+// The greedy strategy for combining samples is optimal.
+template<class DestFloat>
+unsigned combine_many_samples(const Tap<float> *weights, unsigned src_size, unsigned src_samples, unsigned dst_samples, Tap<DestFloat> **bilinear_weights)
+{
+ int src_bilinear_samples = 0;
+ for (unsigned y = 0; y < dst_samples; ++y) {
+ unsigned num_samples_saved = combine_samples<DestFloat>(weights + y * src_samples, NULL, src_size, src_samples, UINT_MAX);
+ src_bilinear_samples = max<int>(src_bilinear_samples, src_samples - num_samples_saved);
+ }
+
+ // Now that we know the right width, actually combine the samples.
+ *bilinear_weights = new Tap<DestFloat>[dst_samples * src_bilinear_samples];
+ for (unsigned y = 0; y < dst_samples; ++y) {
+ Tap<DestFloat> *bilinear_weights_ptr = *bilinear_weights + y * src_bilinear_samples;
+ unsigned num_samples_saved = combine_samples(
+ weights + y * src_samples,
+ bilinear_weights_ptr,
+ src_size,
+ src_samples,
+ src_samples - src_bilinear_samples);
+ assert(int(src_samples) - int(num_samples_saved) == src_bilinear_samples);
+ normalize_sum(bilinear_weights_ptr, src_bilinear_samples);
+ }
+ return src_bilinear_samples;
+}
+
+// Compute the sum of squared errors between the ideal weights (which are
+// assumed to fall exactly on pixel centers) and the weights that result
+// from sampling at <bilinear_weights>. The primary reason for the difference
+// is inaccuracy in the sampling positions, both due to limited precision
+// in storing them (already inherent in sending them in as fp16_int_t)
+// and in subtexel sampling precision (which we calculate in this function).
+template<class T>
+double compute_sum_sq_error(const Tap<float>* weights, unsigned num_weights,
+ const Tap<T>* bilinear_weights, unsigned num_bilinear_weights,
+ unsigned size)
+{
+ // Find the effective range of the bilinear-optimized kernel.
+ // Due to rounding of the positions, this is not necessarily the same
+ // as the intended range (ie., the range of the original weights).
+ int lower_pos = int(floor(to_fp64(bilinear_weights[0].pos) * size - 0.5));
+ int upper_pos = int(ceil(to_fp64(bilinear_weights[num_bilinear_weights - 1].pos) * size - 0.5)) + 2;
+ lower_pos = min<int>(lower_pos, lrintf(weights[0].pos * size - 0.5));
+ upper_pos = max<int>(upper_pos, lrintf(weights[num_weights - 1].pos * size - 0.5));
+
+ float* effective_weights = new float[upper_pos - lower_pos];
+ for (int i = 0; i < upper_pos - lower_pos; ++i) {
+ effective_weights[i] = 0.0f;
+ }
+
+ // Now find the effective weights that result from this sampling.
+ for (unsigned i = 0; i < num_bilinear_weights; ++i) {
+ const float pixel_pos = to_fp64(bilinear_weights[i].pos) * size - 0.5f;
+ const int x0 = int(floor(pixel_pos)) - lower_pos;
+ const int x1 = x0 + 1;
+ const float f = lrintf((pixel_pos - (x0 + lower_pos)) / movit_texel_subpixel_precision) * movit_texel_subpixel_precision;
+
+ assert(x0 >= 0);
+ assert(x1 >= 0);
+ assert(x0 < upper_pos - lower_pos);
+ assert(x1 < upper_pos - lower_pos);
+
+ effective_weights[x0] += to_fp64(bilinear_weights[i].weight) * (1.0 - f);
+ effective_weights[x1] += to_fp64(bilinear_weights[i].weight) * f;
+ }
+
+ // Subtract the desired weights to get the error.
+ for (unsigned i = 0; i < num_weights; ++i) {
+ const int x = lrintf(weights[i].pos * size - 0.5f) - lower_pos;
+ assert(x >= 0);
+ assert(x < upper_pos - lower_pos);
+
+ effective_weights[x] -= weights[i].weight;
+ }
+
+ double sum_sq_error = 0.0;
+ for (unsigned i = 0; i < num_weights; ++i) {
+ sum_sq_error += effective_weights[i] * effective_weights[i];
+ }
+
+ delete[] effective_weights;
+ 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 w_kj = (j == pos[k]) ? (1.0f - fracs[k]) : fracs[k];
+ for (unsigned i = 0; i < num_bilinear_weights; ++i) {
+ float w_ij;
+ if (j == pos[i]) {
+ w_ij = 1.0f - fracs[i];
+ } else if (j == pos[i] + 1) {
+ w_ij = fracs[i];
+ } else {
+ // w_ij = 0
+ continue;
+ }
+ A.coeffRef(i, k) += w_kj * w_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) += w_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()
float radius_scaling_factor = min(scaling_factor, 1.0f);
int int_radius = lrintf(LANCZOS_RADIUS / radius_scaling_factor);
int src_samples = int_radius * 2 + 1;
- float *weights = new float[dst_samples * src_samples * 2];
+ Tap<float> *weights = new Tap<float>[dst_samples * src_samples];
float subpixel_offset = offset - lrintf(offset); // The part not covered by whole_pixel_offset.
assert(subpixel_offset >= -0.5f && subpixel_offset <= 0.5f);
for (unsigned y = 0; y < dst_samples; ++y) {
for (int i = 0; i < src_samples; ++i) {
int src_y = base_src_y + i - int_radius;
float weight = lanczos_weight(radius_scaling_factor * (src_y - center_src_y - subpixel_offset), LANCZOS_RADIUS);
- weights[(y * src_samples + i) * 2 + 0] = weight * radius_scaling_factor;
- weights[(y * src_samples + i) * 2 + 1] = (src_y + 0.5) / float(src_size);
+ weights[y * src_samples + i].weight = weight * radius_scaling_factor;
+ weights[y * src_samples + i].pos = (src_y + 0.5) / float(src_size);
}
}
// Now 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
- // figure out the maximum number of samples. Then, we downconvert all the weights to
- // that number -- we could have gone for a variable-length system, but this is simpler,
- // and the gains would probably be offset by the extra cost of checking when to stop.
- //
- // The greedy strategy for combining samples is optimal.
- src_bilinear_samples = 0;
+ // we need to make. Try fp16 first; if it's not accurate enough, we go to fp32.
+ Tap<fp16_int_t> *bilinear_weights_fp16;
+ src_bilinear_samples = combine_many_samples(weights, src_size, src_samples, dst_samples, &bilinear_weights_fp16);
+ Tap<float> *bilinear_weights_fp32 = NULL;
+ bool fallback_to_fp32 = false;
+ double max_sum_sq_error_fp16 = 0.0;
for (unsigned y = 0; y < dst_samples; ++y) {
- unsigned num_samples_saved = combine_samples(weights + (y * src_samples) * 2, NULL, src_samples, UINT_MAX);
- src_bilinear_samples = max<int>(src_bilinear_samples, src_samples - num_samples_saved);
+ 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,
+ src_size);
+ max_sum_sq_error_fp16 = std::max(max_sum_sq_error_fp16, sum_sq_error_fp16);
}
- // Now that we know the right width, actually combine the samples.
- float *bilinear_weights = new float[dst_samples * src_bilinear_samples * 2];
- fp16_int_t *bilinear_weights_fp16 = new fp16_int_t[dst_samples * src_bilinear_samples * 2];
- for (unsigned y = 0; y < dst_samples; ++y) {
- float *bilinear_weights_ptr = bilinear_weights + (y * src_bilinear_samples) * 2;
- fp16_int_t *bilinear_weights_fp16_ptr = bilinear_weights_fp16 + (y * src_bilinear_samples) * 2;
- unsigned num_samples_saved = combine_samples(
- weights + (y * src_samples) * 2,
- bilinear_weights_ptr,
- src_samples,
- src_samples - src_bilinear_samples);
- assert(int(src_samples) - int(num_samples_saved) == src_bilinear_samples);
-
- // Convert to fp16.
- for (int i = 0; i < src_bilinear_samples; ++i) {
- bilinear_weights_fp16_ptr[i * 2 + 0] = fp64_to_fp16(bilinear_weights_ptr[i * 2 + 0]);
- bilinear_weights_fp16_ptr[i * 2 + 1] = fp64_to_fp16(bilinear_weights_ptr[i * 2 + 1]);
- }
-
- // 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 += fp16_to_fp64(bilinear_weights_fp16_ptr[i * 2 + 0]);
- }
- for (int i = 0; i < src_bilinear_samples; ++i) {
- bilinear_weights_fp16_ptr[i * 2 + 0] = fp64_to_fp16(
- fp16_to_fp64(bilinear_weights_fp16_ptr[i * 2 + 0]) / sum);
- }
+ // Our tolerance level for total error is a bit higher than the one for invididual
+ // samples, since one would assume overall errors in the shape don't matter as much.
+ 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);
}
}
check_error();
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_T, GL_REPEAT);
check_error();
- glTexImage2D(GL_TEXTURE_2D, 0, GL_RG16F, src_bilinear_samples, dst_samples, 0, GL_RG, GL_HALF_FLOAT, bilinear_weights_fp16);
+ if (fallback_to_fp32) {
+ glTexImage2D(GL_TEXTURE_2D, 0, GL_RG32F, src_bilinear_samples, dst_samples, 0, GL_RG, GL_FLOAT, bilinear_weights_fp32);
+ } else {
+ glTexImage2D(GL_TEXTURE_2D, 0, GL_RG16F, src_bilinear_samples, dst_samples, 0, GL_RG, GL_HALF_FLOAT, bilinear_weights_fp16);
+ }
check_error();
delete[] weights;
- delete[] bilinear_weights;
delete[] bilinear_weights_fp16;
+ delete[] bilinear_weights_fp32;
}
void SingleResamplePassEffect::set_gl_state(GLuint glsl_program_num, const string &prefix, unsigned *sampler_num)
check_error();
set_uniform_int(glsl_program_num, prefix, "sample_tex", *sampler_num);
- ++sampler_num;
+ ++*sampler_num;
set_uniform_int(glsl_program_num, prefix, "num_samples", src_bilinear_samples);
set_uniform_float(glsl_program_num, prefix, "num_loops", num_loops);
set_uniform_float(glsl_program_num, prefix, "slice_height", slice_height);
projects / movit / blobdiff