--- /dev/null
+// NOTE: Throughout, we use the symbol ⊙ for convolution.
+// Since all of our signals are symmetrical, discrete correlation and convolution
+// is the same operation, and so we won't make a difference in notation.
+
+
+#include <math.h>
+#include <assert.h>
+#include <Eigen/Dense>
+#include <Eigen/Cholesky>
+
+#include "deconvolution_sharpen_effect.h"
+#include "util.h"
+#include "opengl.h"
+
+using namespace Eigen;
+
+DeconvolutionSharpenEffect::DeconvolutionSharpenEffect()
+ : R(5),
+ circle_radius(2.0f),
+ gaussian_radius(0.0f),
+ correlation(0.95f),
+ noise(0.01f)
+{
+ register_int("matrix_size", &R);
+ register_float("circle_radius", &circle_radius);
+ register_float("gaussian_radius", &gaussian_radius);
+ register_float("correlation", &correlation);
+ register_float("noise", &noise);
+}
+
+std::string DeconvolutionSharpenEffect::output_fragment_shader()
+{
+ char buf[256];
+ sprintf(buf, "#define R %u\n", R);
+ return buf + read_file("deconvolution_sharpen_effect.frag");
+}
+
+namespace {
+
+// Integral of sqrt(r² - x²) dx over x=0..a.
+float circle_integral(float a, float r)
+{
+ assert(a >= 0.0f);
+ if (a <= 0.0f) {
+ return 0.0f;
+ }
+ if (a >= r) {
+ return 0.25f * M_PI * r * r;
+ }
+ return 0.5f * (a * sqrt(r*r - a*a) + r*r * asin(a / r));
+}
+
+// Yields the impulse response of a circular blur with radius r.
+// We basically look at each element as a square centered around (x,y),
+// and figure out how much of its area is covered by the circle.
+float circle_impulse_response(int x, int y, float r)
+{
+ if (r < 1e-3) {
+ // Degenerate case: radius = 0 yields the impulse response.
+ return (x == 0 && y == 0) ? 1.0f : 0.0f;
+ }
+
+ // Find the extents of this cell. Due to symmetry, we can cheat a bit
+ // and pretend we're always in the upper-right quadrant, except when
+ // we're right at an axis crossing (x = 0 or y = 0), in which case we
+ // simply use the evenness of the function; shrink the cell, make
+ // the calculation, and down below we'll normalize by the cell's area.
+ float min_x, max_x, min_y, max_y;
+ if (x == 0) {
+ min_x = 0.0f;
+ max_x = 0.5f;
+ } else {
+ min_x = abs(x) - 0.5f;
+ max_x = abs(x) + 0.5f;
+ }
+ if (y == 0) {
+ min_y = 0.0f;
+ max_y = 0.5f;
+ } else {
+ min_y = abs(y) - 0.5f;
+ max_y = abs(y) + 0.5f;
+ }
+ assert(min_x >= 0.0f && max_x >= 0.0f);
+ assert(min_y >= 0.0f && max_y >= 0.0f);
+
+ float cell_height = max_y - min_y;
+ float cell_width = max_x - min_x;
+
+ if (min_x * min_x + min_y * min_y > r * r) {
+ // Lower-left corner is outside the circle, so the entire cell is.
+ return 0.0f;
+ }
+ if (max_x * max_x + max_y * max_y < r * r) {
+ // Upper-right corner is inside the circle, so the entire cell is.
+ return 1.0f;
+ }
+
+ // OK, so now we know the cell is partially covered by the circle:
+ //
+ // \ .
+ // -------------
+ // |####|#\ |
+ // |####|##| |
+ // -------------
+ // A ###|
+ // ###|
+ //
+ // The edge of the circle is defined by x² + y² = r²,
+ // or x = sqrt(r² - y²) (since x is nonnegative).
+ // Find out where the curve crosses our given y values.
+ float mid_x1 = (max_y >= r) ? min_x : sqrt(r * r - max_y * max_y);
+ float mid_x2 = sqrt(r * r - min_y * min_y);
+ if (mid_x1 < min_x) {
+ mid_x1 = min_x;
+ }
+ if (mid_x2 > max_x) {
+ mid_x2 = max_x;
+ }
+ assert(mid_x1 >= min_x);
+ assert(mid_x2 >= mid_x1);
+ assert(max_x >= mid_x2);
+
+ // The area marked A in the figure above.
+ float covered_area = cell_height * (mid_x1 - min_x);
+
+ // The area marked B in the figure above. Note that the integral gives the entire
+ // shaded space down to zero, so we need to subtract the rectangle that does not
+ // belong to our cell.
+ covered_area += circle_integral(mid_x2, r) - circle_integral(mid_x1, r);
+ covered_area -= min_y * (mid_x2 - mid_x1);
+
+ assert(covered_area <= cell_width * cell_height);
+ return covered_area / (cell_width * cell_height);
+}
+
+// Compute a ⊙ b. Note that we compute the “full” convolution,
+// ie., our matrix will be big enough to hold every nonzero element of the result.
+MatrixXf convolve(const MatrixXf &a, const MatrixXf &b)
+{
+ MatrixXf result(a.rows() + b.rows() - 1, a.cols() + b.cols() - 1);
+ for (int yr = 0; yr < result.rows(); ++yr) {
+ for (int xr = 0; xr < result.cols(); ++xr) {
+ float sum = 0.0f;
+
+ // Given that x_b = x_r - x_a, find the values of x_a where
+ // x_a is in [0, a_cols> and x_b is in [0, b_cols>. (y is similar.)
+ //
+ // The second demand gives:
+ //
+ // 0 <= x_r - x_a < b_cols
+ // 0 >= x_a - x_r > -b_cols
+ // x_r >= x_a > x_r - b_cols
+ int ya_min = yr - b.rows() + 1;
+ int ya_max = yr;
+ int xa_min = xr - b.rows() + 1;
+ int xa_max = xr;
+
+ // Now fit to the first demand.
+ ya_min = std::max<int>(ya_min, 0);
+ ya_max = std::min<int>(ya_max, a.rows() - 1);
+ xa_min = std::max<int>(xa_min, 0);
+ xa_max = std::min<int>(xa_max, a.cols() - 1);
+
+ assert(ya_max >= ya_min);
+ assert(xa_max >= xa_min);
+
+ for (int ya = ya_min; ya <= ya_max; ++ya) {
+ for (int xa = xa_min; xa <= xa_max; ++xa) {
+ sum += a(ya, xa) * b(yr - ya, xr - xa);
+ }
+ }
+
+ result(yr, xr) = sum;
+ }
+ }
+ return result;
+}
+
+// Similar to convolve(), but instead of assuming every element outside
+// of b is zero, we make no such assumption and instead return only the
+// elements where we know the right answer. (This is the only difference
+// between the two.)
+// This is the same as conv2(a, b, 'valid') in Octave.
+//
+// a must be the larger matrix of the two.
+MatrixXf central_convolve(const MatrixXf &a, const MatrixXf &b)
+{
+ assert(a.rows() >= b.rows());
+ assert(a.cols() >= b.cols());
+ MatrixXf result(a.rows() - b.rows() + 1, a.cols() - b.cols() + 1);
+ for (int yr = b.rows() - 1; yr < result.rows() + b.rows() - 1; ++yr) {
+ for (int xr = b.cols() - 1; xr < result.cols() + b.cols() - 1; ++xr) {
+ float sum = 0.0f;
+
+ // Given that x_b = x_r - x_a, find the values of x_a where
+ // x_a is in [0, a_cols> and x_b is in [0, b_cols>. (y is similar.)
+ //
+ // The second demand gives:
+ //
+ // 0 <= x_r - x_a < b_cols
+ // 0 >= x_a - x_r > -b_cols
+ // x_r >= x_a > x_r - b_cols
+ int ya_min = yr - b.rows() + 1;
+ int ya_max = yr;
+ int xa_min = xr - b.rows() + 1;
+ int xa_max = xr;
+
+ // Now fit to the first demand.
+ ya_min = std::max<int>(ya_min, 0);
+ ya_max = std::min<int>(ya_max, a.rows() - 1);
+ xa_min = std::max<int>(xa_min, 0);
+ xa_max = std::min<int>(xa_max, a.cols() - 1);
+
+ assert(ya_max >= ya_min);
+ assert(xa_max >= xa_min);
+
+ for (int ya = ya_min; ya <= ya_max; ++ya) {
+ for (int xa = xa_min; xa <= xa_max; ++xa) {
+ sum += a(ya, xa) * b(yr - ya, xr - xa);
+ }
+ }
+
+ result(yr - b.rows() + 1, xr - b.cols() + 1) = sum;
+ }
+ }
+ return result;
+}
+
+void print_matrix(const MatrixXf &m)
+{
+ for (int y = 0; y < m.rows(); ++y) {
+ for (int x = 0; x < m.cols(); ++x) {
+ printf("%7.4f ", m(x, y));
+ }
+ printf("\n");
+ }
+}
+
+} // namespace
+
+void DeconvolutionSharpenEffect::set_gl_state(GLuint glsl_program_num, const std::string &prefix, unsigned *sampler_num)
+{
+ Effect::set_gl_state(glsl_program_num, prefix, sampler_num);
+
+ assert(R >= 1);
+ assert(R <= 25); // Same limit as Refocus.
+
+ printf("circular blur radius: %5.3f\n", circle_radius);
+ printf("gaussian blur radius: %5.3f\n", gaussian_radius);
+ printf("correlation: %5.3f\n", correlation);
+ printf("noise factor: %5.3f\n", noise);
+ printf("\n");
+
+ // Figure out the impulse response for the circular part of the blur.
+ MatrixXf circ_h(2 * R + 1, 2 * R + 1);
+ for (int y = -R; y <= R; ++y) {
+ for (int x = -R; x <= R; ++x) {
+ circ_h(y + R, x + R) = circle_impulse_response(x, y, circle_radius);
+ }
+ }
+
+ // Same, for the Gaussian part of the blur. We make this a lot larger
+ // since we're going to convolve with it soon, and it has infinite support
+ // (see comments for central_convolve()).
+ MatrixXf gaussian_h(4 * R + 1, 4 * R + 1);
+ for (int y = -2 * R; y <= 2 * R; ++y) {
+ for (int x = -2 * R; x <= 2 * R; ++x) {
+ float val;
+ if (gaussian_radius < 1e-3) {
+ val = (x == 0 && y == 0) ? 1.0f : 0.0f;
+ } else {
+ float z = hypot(x, y) / gaussian_radius;
+ val = exp(-z * z);
+ }
+ gaussian_h(y + 2 * R, x + 2 * R) = val;
+ }
+ }
+
+ // h, the (assumed) impulse response that we're trying to invert.
+ MatrixXf h = central_convolve(gaussian_h, circ_h);
+ assert(h.rows() == 2 * R + 1);
+ assert(h.cols() == 2 * R + 1);
+
+ // Normalize the impulse response.
+ float sum = 0.0f;
+ for (int y = 0; y < 2 * R + 1; ++y) {
+ for (int x = 0; x < 2 * R + 1; ++x) {
+ sum += h(y, x);
+ }
+ }
+ for (int y = 0; y < 2 * R + 1; ++y) {
+ for (int x = 0; x < 2 * R + 1; ++x) {
+ h(y, x) /= sum;
+ }
+ }
+
+ // r_uu, the (estimated/assumed) autocorrelation of the input signal (u).
+ // The signal is modelled a standard autoregressive process with the
+ // given correlation coefficient.
+ //
+ // We have to take a bit of care with the size of this matrix.
+ // The pow() function naturally has an infinite support (except for the
+ // degenerate case of correlation=0), but we have to chop it off
+ // somewhere. Since we convolve it with a 4*R+1 large matrix below,
+ // we need to make it twice as big as that, so that we have enough
+ // data to make r_vv valid. (central_convolve() effectively enforces
+ // that we get at least the right size.)
+ MatrixXf r_uu(8 * R + 1, 8 * R + 1);
+ for (int y = -4 * R; y <= 4 * R; ++y) {
+ for (int x = -4 * R; x <= 4 * R; ++x) {
+ r_uu(x + 4 * R, y + 4 * R) = pow(correlation, hypot(x, y));
+ }
+ }
+
+ // Estimate r_vv, the autocorrelation of the output signal v.
+ // Since we know that v = h ⊙ u and both are symmetrical,
+ // convolution and correlation are the same, and
+ // r_vv = v ⊙ v = (h ⊙ u) ⊙ (h ⊙ u) = (h ⊙ h) ⊙ r_uu.
+ MatrixXf r_vv = central_convolve(r_uu, convolve(h, h));
+ assert(r_vv.rows() == 4 * R + 1);
+ assert(r_vv.cols() == 4 * R + 1);
+
+ // Similarly, r_uv = u ⊙ v = u ⊙ (h ⊙ u) = h ⊙ r_uu.
+ //MatrixXf r_uv = central_convolve(r_uu, h).block(2 * R, 2 * R, 2 * R + 1, 2 * R + 1);
+ MatrixXf r_uu_center = r_uu.block(2 * R, 2 * R, 4 * R + 1, 4 * R + 1);
+ MatrixXf r_uv = central_convolve(r_uu_center, h);
+ assert(r_uv.rows() == 2 * R + 1);
+ assert(r_uv.cols() == 2 * R + 1);
+
+ // Add the noise term (we assume the noise is uncorrelated,
+ // so it only affects the central element).
+ r_vv(2 * R, 2 * R) += noise;
+
+ // Now solve the Wiener-Hopf equations to find the deconvolution kernel g.
+ // Most texts show this only for the simpler 1D case:
+ //
+ // [ r_vv(0) r_vv(1) r_vv(2) ... ] [ g(0) ] [ r_uv(0) ]
+ // [ r_vv(-1) r_vv(0) ... ] [ g(1) ] = [ r_uv(1) ]
+ // [ r_vv(-2) ... ] [ g(2) ] [ r_uv(2) ]
+ // [ ... ] [ g(3) ] [ r_uv(3) ]
+ //
+ // (Since r_vv is symmetrical, we can drop the minus signs.)
+ //
+ // Generally, row i of the matrix contains (dropping _vv for brevity):
+ //
+ // [ r(0-i) r(1-i) r(2-i) ... ]
+ //
+ // However, we have the 2D case. We flatten the vectors out to
+ // 1D quantities; this means we must think of the row number
+ // as a pair instead of as a scalar. Row (i,j) then contains:
+ //
+ // [ r(0-i,0-j) r(1-i,0-j) r(2-i,0-j) ... r(0-i,1-j) r_(1-i,1-j) r(2-i,1-j) ... ]
+ //
+ // g and r_uv are flattened in the same fashion.
+ //
+ // Note that even though this matrix is block Toeplitz, it is _not_ Toeplitz,
+ // and thus can not be inverted through the standard Levinson-Durbin method.
+ // There exists a block Levinson-Durbin method, which we may or may not
+ // want to use later. (Eigen's solvers are fast enough that for big matrices,
+ // the convolution operation and not the matrix solving is the bottleneck.)
+ //
+ // One thing we definitely want to use, though, is the symmetry properties.
+ // Since we know that g(i, j) = g(|i|, |j|), we can reduce the amount of
+ // unknowns to about 1/4th of the total size. The method is quite simple,
+ // as can be seen from the following toy equation system:
+ //
+ // A x0 + B x1 + C x2 = y0
+ // D x0 + E x1 + F x2 = y1
+ // G x0 + H x1 + I x2 = y2
+ //
+ // If we now know that e.g. x0=x1 and y0=y1, we can rewrite this to
+ //
+ // (A+B+D+E) x0 + (C+F) x2 = 2 y0
+ // (G+H) x0 + I x2 = y2
+ //
+ // This both increases accuracy and provides us with a very nice speed
+ // boost. We could have gone even further and went for 8-way symmetry
+ // like the shader does, but this is good enough right now.
+ MatrixXf M(MatrixXf::Zero((R + 1) * (R + 1), (R + 1) * (R + 1)));
+ MatrixXf r_uv_flattened(MatrixXf::Zero((R + 1) * (R + 1), 1));
+ for (int outer_i = 0; outer_i < 2 * R + 1; ++outer_i) {
+ int folded_outer_i = abs(outer_i - R);
+ for (int outer_j = 0; outer_j < 2 * R + 1; ++outer_j) {
+ int folded_outer_j = abs(outer_j - R);
+ int row = folded_outer_i * (R + 1) + folded_outer_j;
+ for (int inner_i = 0; inner_i < 2 * R + 1; ++inner_i) {
+ int folded_inner_i = abs(inner_i - R);
+ for (int inner_j = 0; inner_j < 2 * R + 1; ++inner_j) {
+ int folded_inner_j = abs(inner_j - R);
+ int col = folded_inner_i * (R + 1) + folded_inner_j;
+ M(row, col) += r_vv((inner_i - R) - (outer_i - R) + 2 * R,
+ (inner_j - R) - (outer_j - R) + 2 * R);
+ }
+ }
+ r_uv_flattened(row) += r_uv(outer_i, outer_j);
+ }
+ }
+
+ LLT<MatrixXf> llt(M);
+ MatrixXf g_flattened = llt.solve(r_uv_flattened);
+ assert(g_flattened.rows() == (R + 1) * (R + 1)),
+ assert(g_flattened.cols() == 1);
+
+ // Normalize and de-flatten the deconvolution matrix.
+ MatrixXf g(R + 1, R + 1);
+ sum = 0.0f;
+ for (int i = 0; i < g_flattened.rows(); ++i) {
+ int y = i / (R + 1);
+ int x = i % (R + 1);
+ if (y == 0 && x == 0) {
+ sum += g_flattened(i);
+ } else if (y == 0 || x == 0) {
+ sum += 2.0f * g_flattened(i);
+ } else {
+ sum += 4.0f * g_flattened(i);
+ }
+ }
+ for (int i = 0; i < g_flattened.rows(); ++i) {
+ int y = i / (R + 1);
+ int x = i % (R + 1);
+ g(y, x) = g_flattened(i) / sum;
+ }
+
+ // Now encode it as uniforms, and pass it on to the shader.
+ // (Actually the shader only uses about half of the elements.)
+ float samples[4 * (R + 1) * (R + 1)];
+ for (int y = 0; y <= R; ++y) {
+ for (int x = 0; x <= R; ++x) {
+ int i = y * (R + 1) + x;
+ samples[i * 4 + 0] = x / float(width);
+ samples[i * 4 + 1] = y / float(height);
+ samples[i * 4 + 2] = g(y, x);
+ samples[i * 4 + 3] = 0.0f;
+ }
+ }
+
+ set_uniform_vec4_array(glsl_program_num, prefix, "samples", samples, R * R);
+}
projects / movit / blobdiff