+// 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 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);
+}
+