1 // NOTE: Throughout, we use the symbol ⊙ for convolution.
2 // Since all of our signals are symmetrical, discrete correlation and convolution
3 // is the same operation, and so we won't make a difference in notation.
6 #include <Eigen/Cholesky>
15 #include "deconvolution_sharpen_effect.h"
16 #include "effect_util.h"
19 using namespace Eigen;
21 DeconvolutionSharpenEffect::DeconvolutionSharpenEffect()
24 gaussian_radius(0.0f),
28 last_circle_radius(-1.0f),
29 last_gaussian_radius(-1.0f),
30 last_correlation(-1.0f),
33 register_int("matrix_size", &R);
34 register_float("circle_radius", &circle_radius);
35 register_float("gaussian_radius", &gaussian_radius);
36 register_float("correlation", &correlation);
37 register_float("noise", &noise);
40 std::string DeconvolutionSharpenEffect::output_fragment_shader()
43 sprintf(buf, "#define R %u\n", R);
46 assert(R <= 25); // Same limit as Refocus.
49 return buf + read_file("deconvolution_sharpen_effect.frag");
54 // Integral of sqrt(r² - x²) dx over x=0..a.
55 float circle_integral(float a, float r)
62 return 0.25f * M_PI * r * r;
64 return 0.5f * (a * sqrt(r*r - a*a) + r*r * asin(a / r));
67 // Yields the impulse response of a circular blur with radius r.
68 // We basically look at each element as a square centered around (x,y),
69 // and figure out how much of its area is covered by the circle.
70 float circle_impulse_response(int x, int y, float r)
73 // Degenerate case: radius = 0 yields the impulse response.
74 return (x == 0 && y == 0) ? 1.0f : 0.0f;
77 // Find the extents of this cell. Due to symmetry, we can cheat a bit
78 // and pretend we're always in the upper-right quadrant, except when
79 // we're right at an axis crossing (x = 0 or y = 0), in which case we
80 // simply use the evenness of the function; shrink the cell, make
81 // the calculation, and down below we'll normalize by the cell's area.
82 float min_x, max_x, min_y, max_y;
87 min_x = abs(x) - 0.5f;
88 max_x = abs(x) + 0.5f;
94 min_y = abs(y) - 0.5f;
95 max_y = abs(y) + 0.5f;
97 assert(min_x >= 0.0f && max_x >= 0.0f);
98 assert(min_y >= 0.0f && max_y >= 0.0f);
100 float cell_height = max_y - min_y;
101 float cell_width = max_x - min_x;
103 if (min_x * min_x + min_y * min_y > r * r) {
104 // Lower-left corner is outside the circle, so the entire cell is.
107 if (max_x * max_x + max_y * max_y < r * r) {
108 // Upper-right corner is inside the circle, so the entire cell is.
112 // OK, so now we know the cell is partially covered by the circle:
122 // The edge of the circle is defined by x² + y² = r²,
123 // or x = sqrt(r² - y²) (since x is nonnegative).
124 // Find out where the curve crosses our given y values.
125 float mid_x1 = (max_y >= r) ? min_x : sqrt(r * r - max_y * max_y);
126 float mid_x2 = sqrt(r * r - min_y * min_y);
127 if (mid_x1 < min_x) {
130 if (mid_x2 > max_x) {
133 assert(mid_x1 >= min_x);
134 assert(mid_x2 >= mid_x1);
135 assert(max_x >= mid_x2);
137 // The area marked A in the figure above.
138 float covered_area = cell_height * (mid_x1 - min_x);
140 // The area marked B in the figure above. Note that the integral gives the entire
141 // shaded space down to zero, so we need to subtract the rectangle that does not
142 // belong to our cell.
143 covered_area += circle_integral(mid_x2, r) - circle_integral(mid_x1, r);
144 covered_area -= min_y * (mid_x2 - mid_x1);
146 assert(covered_area <= cell_width * cell_height);
147 return covered_area / (cell_width * cell_height);
150 // Compute a ⊙ b. Note that we compute the “full” convolution,
151 // ie., our matrix will be big enough to hold every nonzero element of the result.
152 MatrixXf convolve(const MatrixXf &a, const MatrixXf &b)
154 MatrixXf result(a.rows() + b.rows() - 1, a.cols() + b.cols() - 1);
155 for (int yr = 0; yr < result.rows(); ++yr) {
156 for (int xr = 0; xr < result.cols(); ++xr) {
159 // Given that x_b = x_r - x_a, find the values of x_a where
160 // x_a is in [0, a_cols> and x_b is in [0, b_cols>. (y is similar.)
162 // The second demand gives:
164 // 0 <= x_r - x_a < b_cols
165 // 0 >= x_a - x_r > -b_cols
166 // x_r >= x_a > x_r - b_cols
167 int ya_min = yr - b.rows() + 1;
169 int xa_min = xr - b.rows() + 1;
172 // Now fit to the first demand.
173 ya_min = std::max<int>(ya_min, 0);
174 ya_max = std::min<int>(ya_max, a.rows() - 1);
175 xa_min = std::max<int>(xa_min, 0);
176 xa_max = std::min<int>(xa_max, a.cols() - 1);
178 assert(ya_max >= ya_min);
179 assert(xa_max >= xa_min);
181 for (int ya = ya_min; ya <= ya_max; ++ya) {
182 for (int xa = xa_min; xa <= xa_max; ++xa) {
183 sum += a(ya, xa) * b(yr - ya, xr - xa);
187 result(yr, xr) = sum;
193 // Similar to convolve(), but instead of assuming every element outside
194 // of b is zero, we make no such assumption and instead return only the
195 // elements where we know the right answer. (This is the only difference
197 // This is the same as conv2(a, b, 'valid') in Octave.
199 // a must be the larger matrix of the two.
200 MatrixXf central_convolve(const MatrixXf &a, const MatrixXf &b)
202 assert(a.rows() >= b.rows());
203 assert(a.cols() >= b.cols());
204 MatrixXf result(a.rows() - b.rows() + 1, a.cols() - b.cols() + 1);
205 for (int yr = b.rows() - 1; yr < result.rows() + b.rows() - 1; ++yr) {
206 for (int xr = b.cols() - 1; xr < result.cols() + b.cols() - 1; ++xr) {
209 // Given that x_b = x_r - x_a, find the values of x_a where
210 // x_a is in [0, a_cols> and x_b is in [0, b_cols>. (y is similar.)
212 // The second demand gives:
214 // 0 <= x_r - x_a < b_cols
215 // 0 >= x_a - x_r > -b_cols
216 // x_r >= x_a > x_r - b_cols
217 int ya_min = yr - b.rows() + 1;
219 int xa_min = xr - b.rows() + 1;
222 // Now fit to the first demand.
223 ya_min = std::max<int>(ya_min, 0);
224 ya_max = std::min<int>(ya_max, a.rows() - 1);
225 xa_min = std::max<int>(xa_min, 0);
226 xa_max = std::min<int>(xa_max, a.cols() - 1);
228 assert(ya_max >= ya_min);
229 assert(xa_max >= xa_min);
231 for (int ya = ya_min; ya <= ya_max; ++ya) {
232 for (int xa = xa_min; xa <= xa_max; ++xa) {
233 sum += a(ya, xa) * b(yr - ya, xr - xa);
237 result(yr - b.rows() + 1, xr - b.cols() + 1) = sum;
245 void DeconvolutionSharpenEffect::update_deconvolution_kernel()
247 // Figure out the impulse response for the circular part of the blur.
248 MatrixXf circ_h(2 * R + 1, 2 * R + 1);
249 for (int y = -R; y <= R; ++y) {
250 for (int x = -R; x <= R; ++x) {
251 circ_h(y + R, x + R) = circle_impulse_response(x, y, circle_radius);
255 // Same, for the Gaussian part of the blur. We make this a lot larger
256 // since we're going to convolve with it soon, and it has infinite support
257 // (see comments for central_convolve()).
258 MatrixXf gaussian_h(4 * R + 1, 4 * R + 1);
259 for (int y = -2 * R; y <= 2 * R; ++y) {
260 for (int x = -2 * R; x <= 2 * R; ++x) {
262 if (gaussian_radius < 1e-3) {
263 val = (x == 0 && y == 0) ? 1.0f : 0.0f;
265 val = exp(-(x*x + y*y) / (2.0 * gaussian_radius * gaussian_radius));
267 gaussian_h(y + 2 * R, x + 2 * R) = val;
271 // h, the (assumed) impulse response that we're trying to invert.
272 MatrixXf h = central_convolve(gaussian_h, circ_h);
273 assert(h.rows() == 2 * R + 1);
274 assert(h.cols() == 2 * R + 1);
276 // Normalize the impulse response.
278 for (int y = 0; y < 2 * R + 1; ++y) {
279 for (int x = 0; x < 2 * R + 1; ++x) {
283 for (int y = 0; y < 2 * R + 1; ++y) {
284 for (int x = 0; x < 2 * R + 1; ++x) {
289 // r_uu, the (estimated/assumed) autocorrelation of the input signal (u).
290 // The signal is modelled a standard autoregressive process with the
291 // given correlation coefficient.
293 // We have to take a bit of care with the size of this matrix.
294 // The pow() function naturally has an infinite support (except for the
295 // degenerate case of correlation=0), but we have to chop it off
296 // somewhere. Since we convolve it with a 4*R+1 large matrix below,
297 // we need to make it twice as big as that, so that we have enough
298 // data to make r_vv valid. (central_convolve() effectively enforces
299 // that we get at least the right size.)
300 MatrixXf r_uu(8 * R + 1, 8 * R + 1);
301 for (int y = -4 * R; y <= 4 * R; ++y) {
302 for (int x = -4 * R; x <= 4 * R; ++x) {
303 r_uu(x + 4 * R, y + 4 * R) = pow(correlation, hypot(x, y));
307 // Estimate r_vv, the autocorrelation of the output signal v.
308 // Since we know that v = h ⊙ u and both are symmetrical,
309 // convolution and correlation are the same, and
310 // r_vv = v ⊙ v = (h ⊙ u) ⊙ (h ⊙ u) = (h ⊙ h) ⊙ r_uu.
311 MatrixXf r_vv = central_convolve(r_uu, convolve(h, h));
312 assert(r_vv.rows() == 4 * R + 1);
313 assert(r_vv.cols() == 4 * R + 1);
315 // Similarly, r_uv = u ⊙ v = u ⊙ (h ⊙ u) = h ⊙ r_uu.
316 MatrixXf r_uu_center = r_uu.block(2 * R, 2 * R, 4 * R + 1, 4 * R + 1);
317 MatrixXf r_uv = central_convolve(r_uu_center, h);
318 assert(r_uv.rows() == 2 * R + 1);
319 assert(r_uv.cols() == 2 * R + 1);
321 // Add the noise term (we assume the noise is uncorrelated,
322 // so it only affects the central element).
323 r_vv(2 * R, 2 * R) += noise;
325 // Now solve the Wiener-Hopf equations to find the deconvolution kernel g.
326 // Most texts show this only for the simpler 1D case:
328 // [ r_vv(0) r_vv(1) r_vv(2) ... ] [ g(0) ] [ r_uv(0) ]
329 // [ r_vv(-1) r_vv(0) ... ] [ g(1) ] = [ r_uv(1) ]
330 // [ r_vv(-2) ... ] [ g(2) ] [ r_uv(2) ]
331 // [ ... ] [ g(3) ] [ r_uv(3) ]
333 // (Since r_vv is symmetrical, we can drop the minus signs.)
335 // Generally, row i of the matrix contains (dropping _vv for brevity):
337 // [ r(0-i) r(1-i) r(2-i) ... ]
339 // However, we have the 2D case. We flatten the vectors out to
340 // 1D quantities; this means we must think of the row number
341 // as a pair instead of as a scalar. Row (i,j) then contains:
343 // [ 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) ... ]
345 // g and r_uv are flattened in the same fashion.
347 // Note that even though this matrix is block Toeplitz, it is _not_ Toeplitz,
348 // and thus can not be inverted through the standard Levinson-Durbin method.
349 // There exists a block Levinson-Durbin method, which we may or may not
350 // want to use later. (Eigen's solvers are fast enough that for big matrices,
351 // the convolution operation and not the matrix solving is the bottleneck.)
353 // One thing we definitely want to use, though, is the symmetry properties.
354 // Since we know that g(i, j) = g(|i|, |j|), we can reduce the amount of
355 // unknowns to about 1/4th of the total size. The method is quite simple,
356 // as can be seen from the following toy equation system:
358 // A x0 + B x1 + C x2 = y0
359 // D x0 + E x1 + F x2 = y1
360 // G x0 + H x1 + I x2 = y2
362 // If we now know that e.g. x0=x1 and y0=y1, we can rewrite this to
364 // (A+B+D+E) x0 + (C+F) x2 = 2 y0
365 // (G+H) x0 + I x2 = y2
367 // This both increases accuracy and provides us with a very nice speed
369 MatrixXf M(MatrixXf::Zero((R + 1) * (R + 1), (R + 1) * (R + 1)));
370 MatrixXf r_uv_flattened(MatrixXf::Zero((R + 1) * (R + 1), 1));
371 for (int outer_i = 0; outer_i < 2 * R + 1; ++outer_i) {
372 int folded_outer_i = abs(outer_i - R);
373 for (int outer_j = 0; outer_j < 2 * R + 1; ++outer_j) {
374 int folded_outer_j = abs(outer_j - R);
375 int row = folded_outer_i * (R + 1) + folded_outer_j;
376 for (int inner_i = 0; inner_i < 2 * R + 1; ++inner_i) {
377 int folded_inner_i = abs(inner_i - R);
378 for (int inner_j = 0; inner_j < 2 * R + 1; ++inner_j) {
379 int folded_inner_j = abs(inner_j - R);
380 int col = folded_inner_i * (R + 1) + folded_inner_j;
381 M(row, col) += r_vv((inner_i - R) - (outer_i - R) + 2 * R,
382 (inner_j - R) - (outer_j - R) + 2 * R);
385 r_uv_flattened(row) += r_uv(outer_i, outer_j);
389 LLT<MatrixXf> llt(M);
390 MatrixXf g_flattened = llt.solve(r_uv_flattened);
391 assert(g_flattened.rows() == (R + 1) * (R + 1)),
392 assert(g_flattened.cols() == 1);
394 // Normalize and de-flatten the deconvolution matrix.
395 g = MatrixXf(R + 1, R + 1);
397 for (int i = 0; i < g_flattened.rows(); ++i) {
400 if (y == 0 && x == 0) {
401 sum += g_flattened(i);
402 } else if (y == 0 || x == 0) {
403 sum += 2.0f * g_flattened(i);
405 sum += 4.0f * g_flattened(i);
408 for (int i = 0; i < g_flattened.rows(); ++i) {
411 g(y, x) = g_flattened(i) / sum;
414 last_circle_radius = circle_radius;
415 last_gaussian_radius = gaussian_radius;
416 last_correlation = correlation;
420 void DeconvolutionSharpenEffect::set_gl_state(GLuint glsl_program_num, const std::string &prefix, unsigned *sampler_num)
422 Effect::set_gl_state(glsl_program_num, prefix, sampler_num);
426 if (fabs(circle_radius - last_circle_radius) > 1e-3 ||
427 fabs(gaussian_radius - last_gaussian_radius) > 1e-3 ||
428 fabs(correlation - last_correlation) > 1e-3 ||
429 fabs(noise - last_noise) > 1e-3) {
430 update_deconvolution_kernel();
432 // Now encode it as uniforms, and pass it on to the shader.
433 float samples[4 * (R + 1) * (R + 1)];
434 for (int y = 0; y <= R; ++y) {
435 for (int x = 0; x <= R; ++x) {
436 int i = y * (R + 1) + x;
437 samples[i * 4 + 0] = x / float(width);
438 samples[i * 4 + 1] = y / float(height);
439 samples[i * 4 + 2] = g(y, x);
440 samples[i * 4 + 3] = 0.0f;
444 set_uniform_vec4_array(glsl_program_num, prefix, "samples", samples, (R + 1) * (R + 1));