]> git.sesse.net Git - nageru/blobdiff - flow.cpp
Avoid infinities in a uniform.
[nageru] / flow.cpp
index dba6607b0eb3e967fcb2786bd31c5051ad72797a..11f4d4c87646ba31d5c9d2c485386f848c6613d1 100644 (file)
--- a/flow.cpp
+++ b/flow.cpp
@@ -15,6 +15,9 @@
 
 #include <assert.h>
 #include <stdio.h>
+#include <unistd.h>
+
+#include "util.h"
 
 #include <algorithm>
 #include <memory>
@@ -30,7 +33,7 @@ constexpr unsigned finest_level = 1;
 constexpr unsigned patch_size_pixels = 12;
 
 // Some global OpenGL objects.
-GLuint nearest_sampler, linear_sampler, mipmap_sampler;
+GLuint nearest_sampler, linear_sampler, smoothness_sampler;
 GLuint vertex_vbo;
 
 string read_file(const string &filename)
@@ -150,10 +153,10 @@ GLuint load_texture(const char *filename, unsigned width, unsigned height)
 
 GLuint link_program(GLuint vs_obj, GLuint fs_obj)
 {
-        GLuint program = glCreateProgram();
-        glAttachShader(program, vs_obj);
-        glAttachShader(program, fs_obj);
-        glLinkProgram(program);
+       GLuint program = glCreateProgram();
+       glAttachShader(program, vs_obj);
+       glAttachShader(program, fs_obj);
+       glLinkProgram(program);
        GLint success;
        glGetProgramiv(program, GL_LINK_STATUS, &success);
        if (success == GL_FALSE) {
@@ -191,9 +194,8 @@ GLuint fill_vertex_attribute(GLuint vao, GLuint glsl_program_num, const string &
        return vbo;
 }
 
-void bind_sampler(GLuint program, const char *uniform_name, GLuint texture_unit, GLuint tex, GLuint sampler)
+void bind_sampler(GLuint program, GLint location, GLuint texture_unit, GLuint tex, GLuint sampler)
 {
-       GLint location = glGetUniformLocation(program, uniform_name);
        if (location == -1) {
                return;
        }
@@ -221,6 +223,8 @@ private:
        GLuint sobel_fs_obj;
        GLuint sobel_program;
        GLuint sobel_vao;
+
+       GLuint uniform_tex, uniform_image_size;
 };
 
 Sobel::Sobel()
@@ -230,16 +234,14 @@ Sobel::Sobel()
        sobel_program = link_program(sobel_vs_obj, sobel_fs_obj);
 
        // Set up the VAO containing all the required position/texcoord data.
-        glCreateVertexArrays(1, &sobel_vao);
-        glBindVertexArray(sobel_vao);
+       glCreateVertexArrays(1, &sobel_vao);
+       glBindVertexArray(sobel_vao);
 
        GLint position_attrib = glGetAttribLocation(sobel_program, "position");
        glEnableVertexArrayAttrib(sobel_vao, position_attrib);
        glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
 
-       GLint texcoord_attrib = glGetAttribLocation(sobel_program, "texcoord");
-       glEnableVertexArrayAttrib(sobel_vao, texcoord_attrib);
-       glVertexAttribPointer(texcoord_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+       uniform_tex = glGetUniformLocation(sobel_program, "tex");
 }
 
 void Sobel::exec(GLint tex0_view, GLint grad0_tex, int level_width, int level_height)
@@ -247,9 +249,7 @@ void Sobel::exec(GLint tex0_view, GLint grad0_tex, int level_width, int level_he
        glUseProgram(sobel_program);
        glBindTextureUnit(0, tex0_view);
        glBindSampler(0, nearest_sampler);
-       glProgramUniform1i(sobel_program, glGetUniformLocation(sobel_program, "tex"), 0);
-       glProgramUniform1f(sobel_program, glGetUniformLocation(sobel_program, "inv_width"), 1.0f / level_width);
-       glProgramUniform1f(sobel_program, glGetUniformLocation(sobel_program, "inv_height"), 1.0f / level_height);
+       glProgramUniform1i(sobel_program, uniform_tex, 0);
 
        GLuint grad0_fbo;  // TODO: cleanup
        glCreateFramebuffers(1, &grad0_fbo);
@@ -257,7 +257,7 @@ void Sobel::exec(GLint tex0_view, GLint grad0_tex, int level_width, int level_he
 
        glViewport(0, 0, level_width, level_height);
        glBindFramebuffer(GL_FRAMEBUFFER, grad0_fbo);
-        glBindVertexArray(sobel_vao);
+       glBindVertexArray(sobel_vao);
        glUseProgram(sobel_program);
        glDisable(GL_BLEND);
        glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
@@ -274,6 +274,9 @@ private:
        GLuint motion_fs_obj;
        GLuint motion_search_program;
        GLuint motion_search_vao;
+
+       GLuint uniform_image_size, uniform_inv_image_size;
+       GLuint uniform_image0_tex, uniform_image1_tex, uniform_grad0_tex, uniform_flow_tex;
 };
 
 MotionSearch::MotionSearch()
@@ -283,32 +286,33 @@ MotionSearch::MotionSearch()
        motion_search_program = link_program(motion_vs_obj, motion_fs_obj);
 
        // Set up the VAO containing all the required position/texcoord data.
-        glCreateVertexArrays(1, &motion_search_vao);
-        glBindVertexArray(motion_search_vao);
+       glCreateVertexArrays(1, &motion_search_vao);
+       glBindVertexArray(motion_search_vao);
        glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
 
        GLint position_attrib = glGetAttribLocation(motion_search_program, "position");
        glEnableVertexArrayAttrib(motion_search_vao, position_attrib);
        glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
 
-       GLint texcoord_attrib = glGetAttribLocation(motion_search_program, "texcoord");
-       glEnableVertexArrayAttrib(motion_search_vao, texcoord_attrib);
-       glVertexAttribPointer(texcoord_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+       uniform_image_size = glGetUniformLocation(motion_search_program, "image_size");
+       uniform_inv_image_size = glGetUniformLocation(motion_search_program, "inv_image_size");
+       uniform_image0_tex = glGetUniformLocation(motion_search_program, "image0_tex");
+       uniform_image1_tex = glGetUniformLocation(motion_search_program, "image1_tex");
+       uniform_grad0_tex = glGetUniformLocation(motion_search_program, "grad0_tex");
+       uniform_flow_tex = glGetUniformLocation(motion_search_program, "flow_tex");
 }
 
 void MotionSearch::exec(GLuint tex0_view, GLuint tex1_view, GLuint grad0_tex, GLuint flow_tex, GLuint flow_out_tex, int level_width, int level_height, int width_patches, int height_patches)
 {
        glUseProgram(motion_search_program);
 
-       bind_sampler(motion_search_program, "image0_tex", 0, tex0_view, nearest_sampler);
-       bind_sampler(motion_search_program, "image1_tex", 1, tex1_view, linear_sampler);
-       bind_sampler(motion_search_program, "grad0_tex", 2, grad0_tex, nearest_sampler);
-       bind_sampler(motion_search_program, "flow_tex", 3, flow_tex, linear_sampler);
+       bind_sampler(motion_search_program, uniform_image0_tex, 0, tex0_view, nearest_sampler);
+       bind_sampler(motion_search_program, uniform_image1_tex, 1, tex1_view, linear_sampler);
+       bind_sampler(motion_search_program, uniform_grad0_tex, 2, grad0_tex, nearest_sampler);
+       bind_sampler(motion_search_program, uniform_flow_tex, 3, flow_tex, linear_sampler);
 
-       glProgramUniform1f(motion_search_program, glGetUniformLocation(motion_search_program, "image_width"), level_width);
-       glProgramUniform1f(motion_search_program, glGetUniformLocation(motion_search_program, "image_height"), level_height);
-       glProgramUniform1f(motion_search_program, glGetUniformLocation(motion_search_program, "inv_image_width"), 1.0f / level_width);
-       glProgramUniform1f(motion_search_program, glGetUniformLocation(motion_search_program, "inv_image_height"), 1.0f / level_height);
+       glProgramUniform2f(motion_search_program, uniform_image_size, level_width, level_height);
+       glProgramUniform2f(motion_search_program, uniform_inv_image_size, 1.0f / level_width, 1.0f / level_height);
 
        GLuint flow_fbo;  // TODO: cleanup
        glCreateFramebuffers(1, &flow_fbo);
@@ -316,7 +320,7 @@ void MotionSearch::exec(GLuint tex0_view, GLuint tex1_view, GLuint grad0_tex, GL
 
        glViewport(0, 0, width_patches, height_patches);
        glBindFramebuffer(GL_FRAMEBUFFER, flow_fbo);
-        glBindVertexArray(motion_search_vao);
+       glBindVertexArray(motion_search_vao);
        glUseProgram(motion_search_program);
        glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
 }
@@ -339,6 +343,9 @@ private:
        GLuint densify_fs_obj;
        GLuint densify_program;
        GLuint densify_vao;
+
+       GLuint uniform_width_patches, uniform_patch_size, uniform_patch_spacing;
+       GLuint uniform_image0_tex, uniform_image1_tex, uniform_flow_tex;
 };
 
 Densify::Densify()
@@ -348,31 +355,40 @@ Densify::Densify()
        densify_program = link_program(densify_vs_obj, densify_fs_obj);
 
        // Set up the VAO containing all the required position/texcoord data.
-        glCreateVertexArrays(1, &densify_vao);
-        glBindVertexArray(densify_vao);
+       glCreateVertexArrays(1, &densify_vao);
+       glBindVertexArray(densify_vao);
        glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
 
        GLint position_attrib = glGetAttribLocation(densify_program, "position");
        glEnableVertexArrayAttrib(densify_vao, position_attrib);
        glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_width_patches = glGetUniformLocation(densify_program, "width_patches");
+       uniform_patch_size = glGetUniformLocation(densify_program, "patch_size");
+       uniform_patch_spacing = glGetUniformLocation(densify_program, "patch_spacing");
+       uniform_image0_tex = glGetUniformLocation(densify_program, "image0_tex");
+       uniform_image1_tex = glGetUniformLocation(densify_program, "image1_tex");
+       uniform_flow_tex = glGetUniformLocation(densify_program, "flow_tex");
 }
 
 void Densify::exec(GLuint tex0_view, GLuint tex1_view, GLuint flow_tex, GLuint dense_flow_tex, int level_width, int level_height, int width_patches, int height_patches)
 {
        glUseProgram(densify_program);
 
-       bind_sampler(densify_program, "image0_tex", 0, tex0_view, nearest_sampler);
-       bind_sampler(densify_program, "image1_tex", 1, tex1_view, linear_sampler);
-       bind_sampler(densify_program, "flow_tex", 2, flow_tex, nearest_sampler);
+       bind_sampler(densify_program, uniform_image0_tex, 0, tex0_view, nearest_sampler);
+       bind_sampler(densify_program, uniform_image1_tex, 1, tex1_view, linear_sampler);
+       bind_sampler(densify_program, uniform_flow_tex, 2, flow_tex, nearest_sampler);
 
-       glProgramUniform1i(densify_program, glGetUniformLocation(densify_program, "width_patches"), width_patches);
-       glProgramUniform2f(densify_program, glGetUniformLocation(densify_program, "patch_size"),
+       glProgramUniform1i(densify_program, uniform_width_patches, width_patches);
+       glProgramUniform2f(densify_program, uniform_patch_size,
                float(patch_size_pixels) / level_width,
                float(patch_size_pixels) / level_height);
 
        float patch_spacing_x = float(level_width - patch_size_pixels) / (width_patches - 1);
        float patch_spacing_y = float(level_height - patch_size_pixels) / (height_patches - 1);
-       glProgramUniform2f(densify_program, glGetUniformLocation(densify_program, "patch_spacing"),
+       if (width_patches == 1) patch_spacing_x = 0.0f;  // Avoid infinities.
+       if (height_patches == 1) patch_spacing_y = 0.0f;
+       glProgramUniform2f(densify_program, uniform_patch_spacing,
                patch_spacing_x / level_width,
                patch_spacing_y / level_height);
 
@@ -383,11 +399,401 @@ void Densify::exec(GLuint tex0_view, GLuint tex1_view, GLuint flow_tex, GLuint d
        glViewport(0, 0, level_width, level_height);
        glEnable(GL_BLEND);
        glBlendFunc(GL_ONE, GL_ONE);
-        glBindVertexArray(densify_vao);
+       glBindVertexArray(densify_vao);
        glBindFramebuffer(GL_FRAMEBUFFER, dense_flow_fbo);
        glDrawArraysInstanced(GL_TRIANGLE_STRIP, 0, 4, width_patches * height_patches);
 }
 
+// Warp I_1 to I_w, and then compute the mean (I) and difference (I_t) of
+// I_0 and I_w. The prewarping is what enables us to solve the variational
+// flow for du,dv instead of u,v.
+//
+// See variational_refinement.txt for more information.
+class Prewarp {
+public:
+       Prewarp();
+       void exec(GLuint tex0_view, GLuint tex1_view, GLuint flow_tex, GLuint I_tex, GLuint I_t_tex, int level_width, int level_height);
+
+private:
+       GLuint prewarp_vs_obj;
+       GLuint prewarp_fs_obj;
+       GLuint prewarp_program;
+       GLuint prewarp_vao;
+
+       GLuint uniform_image0_tex, uniform_image1_tex, uniform_flow_tex;
+};
+
+Prewarp::Prewarp()
+{
+       prewarp_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       prewarp_fs_obj = compile_shader(read_file("prewarp.frag"), GL_FRAGMENT_SHADER);
+       prewarp_program = link_program(prewarp_vs_obj, prewarp_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &prewarp_vao);
+       glBindVertexArray(prewarp_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(prewarp_program, "position");
+       glEnableVertexArrayAttrib(prewarp_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_image0_tex = glGetUniformLocation(prewarp_program, "image0_tex");
+       uniform_image1_tex = glGetUniformLocation(prewarp_program, "image1_tex");
+       uniform_flow_tex = glGetUniformLocation(prewarp_program, "flow_tex");
+}
+
+void Prewarp::exec(GLuint tex0_view, GLuint tex1_view, GLuint flow_tex, GLuint I_tex, GLuint I_t_tex, int level_width, int level_height)
+{
+       glUseProgram(prewarp_program);
+
+       bind_sampler(prewarp_program, uniform_image0_tex, 0, tex0_view, nearest_sampler);
+       bind_sampler(prewarp_program, uniform_image1_tex, 1, tex1_view, linear_sampler);
+       bind_sampler(prewarp_program, uniform_flow_tex, 2, flow_tex, nearest_sampler);
+
+       GLuint prewarp_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &prewarp_fbo);
+       GLenum bufs[] = { GL_COLOR_ATTACHMENT0, GL_COLOR_ATTACHMENT1 };
+       glNamedFramebufferDrawBuffers(prewarp_fbo, 2, bufs);
+       glNamedFramebufferTexture(prewarp_fbo, GL_COLOR_ATTACHMENT0, I_tex, 0);
+       glNamedFramebufferTexture(prewarp_fbo, GL_COLOR_ATTACHMENT1, I_t_tex, 0);
+
+       glViewport(0, 0, level_width, level_height);
+       glDisable(GL_BLEND);
+       glBindVertexArray(prewarp_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, prewarp_fbo);
+       glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+}
+
+// From I, calculate the partial derivatives I_x and I_y. We use a four-tap
+// central difference filter, since apparently, that's tradition (I haven't
+// measured quality versus a more normal 0.5 (I[x+1] - I[x-1]).)
+// The coefficients come from
+//
+//   https://en.wikipedia.org/wiki/Finite_difference_coefficient
+//
+// Also computes β_0, since it depends only on I_x and I_y.
+class Derivatives {
+public:
+       Derivatives();
+       void exec(GLuint input_tex, GLuint I_x_y_tex, GLuint beta_0_tex, int level_width, int level_height);
+
+private:
+       GLuint derivatives_vs_obj;
+       GLuint derivatives_fs_obj;
+       GLuint derivatives_program;
+       GLuint derivatives_vao;
+
+       GLuint uniform_tex;
+};
+
+Derivatives::Derivatives()
+{
+       derivatives_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       derivatives_fs_obj = compile_shader(read_file("derivatives.frag"), GL_FRAGMENT_SHADER);
+       derivatives_program = link_program(derivatives_vs_obj, derivatives_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &derivatives_vao);
+       glBindVertexArray(derivatives_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(derivatives_program, "position");
+       glEnableVertexArrayAttrib(derivatives_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_tex = glGetUniformLocation(derivatives_program, "tex");
+}
+
+void Derivatives::exec(GLuint input_tex, GLuint I_x_y_tex, GLuint beta_0_tex, int level_width, int level_height)
+{
+       glUseProgram(derivatives_program);
+
+       bind_sampler(derivatives_program, uniform_tex, 0, input_tex, nearest_sampler);
+
+       GLuint derivatives_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &derivatives_fbo);
+       GLenum bufs[] = { GL_COLOR_ATTACHMENT0, GL_COLOR_ATTACHMENT1 };
+       glNamedFramebufferDrawBuffers(derivatives_fbo, 2, bufs);
+       glNamedFramebufferTexture(derivatives_fbo, GL_COLOR_ATTACHMENT0, I_x_y_tex, 0);
+       glNamedFramebufferTexture(derivatives_fbo, GL_COLOR_ATTACHMENT1, beta_0_tex, 0);
+
+       glViewport(0, 0, level_width, level_height);
+       glDisable(GL_BLEND);
+       glBindVertexArray(derivatives_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, derivatives_fbo);
+       glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+}
+
+// Calculate the smoothness constraints between neighboring pixels;
+// s_x(x,y) stores smoothness between pixel (x,y) and (x+1,y),
+// and s_y(x,y) stores between (x,y) and (x,y+1). We'll sample with
+// border color (0,0) later, so that there's zero diffusion out of
+// the border.
+//
+// See variational_refinement.txt for more information.
+class ComputeSmoothness {
+public:
+       ComputeSmoothness();
+       void exec(GLuint flow_tex, GLuint diff_flow_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, int level_width, int level_height);
+
+private:
+       GLuint smoothness_vs_obj;
+       GLuint smoothness_fs_obj;
+       GLuint smoothness_program;
+       GLuint smoothness_vao;
+
+       GLuint uniform_flow_tex, uniform_diff_flow_tex;
+};
+
+ComputeSmoothness::ComputeSmoothness()
+{
+       smoothness_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       smoothness_fs_obj = compile_shader(read_file("smoothness.frag"), GL_FRAGMENT_SHADER);
+       smoothness_program = link_program(smoothness_vs_obj, smoothness_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &smoothness_vao);
+       glBindVertexArray(smoothness_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(smoothness_program, "position");
+       glEnableVertexArrayAttrib(smoothness_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_flow_tex = glGetUniformLocation(smoothness_program, "flow_tex");
+       uniform_diff_flow_tex = glGetUniformLocation(smoothness_program, "diff_flow_tex");
+}
+
+void ComputeSmoothness::exec(GLuint flow_tex, GLuint diff_flow_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, int level_width, int level_height)
+{
+       glUseProgram(smoothness_program);
+
+       bind_sampler(smoothness_program, uniform_flow_tex, 0, flow_tex, nearest_sampler);
+       bind_sampler(smoothness_program, uniform_diff_flow_tex, 1, diff_flow_tex, nearest_sampler);
+
+       GLuint smoothness_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &smoothness_fbo);
+       GLenum bufs[] = { GL_COLOR_ATTACHMENT0, GL_COLOR_ATTACHMENT1 };
+       glNamedFramebufferDrawBuffers(smoothness_fbo, 2, bufs);
+       glNamedFramebufferTexture(smoothness_fbo, GL_COLOR_ATTACHMENT0, smoothness_x_tex, 0);
+       glNamedFramebufferTexture(smoothness_fbo, GL_COLOR_ATTACHMENT1, smoothness_y_tex, 0);
+
+       glViewport(0, 0, level_width, level_height);
+
+       // Make sure the smoothness on the right and upper borders is zero.
+       // We could have done this by making a (W-1)x(H-1) texture instead
+       // (we're sampling smoothness with all-zero border color), but we'd
+       // have to adjust the sampling coordinates, which is annoying.
+       glScissor(0, 0, level_width - 1, level_height - 1);
+       glEnable(GL_SCISSOR_TEST);
+
+       glDisable(GL_BLEND);
+       glBindVertexArray(smoothness_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, smoothness_fbo);
+       glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+
+       glDisable(GL_SCISSOR_TEST);
+}
+
+// Set up the equations set (two equations in two unknowns, per pixel).
+// We store five floats; the three non-redundant elements of the 2x2 matrix (A)
+// as 32-bit floats, and the two elements on the right-hand side (b) as 16-bit
+// floats. (Actually, we store the inverse of the diagonal elements, because
+// we only ever need to divide by them.) This fits into four u32 values;
+// R, G, B for the matrix (the last element is symmetric) and A for the two b values.
+// All the values of the energy term (E_I, E_G, E_S), except the smoothness
+// terms that depend on other pixels, are calculated in one pass.
+//
+// See variational_refinement.txt for more information.
+class SetupEquations {
+public:
+       SetupEquations();
+       void exec(GLuint I_x_y_tex, GLuint I_t_tex, GLuint diff_flow_tex, GLuint flow_tex, GLuint beta_0_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, GLuint equation_tex, int level_width, int level_height);
+
+private:
+       GLuint equations_vs_obj;
+       GLuint equations_fs_obj;
+       GLuint equations_program;
+       GLuint equations_vao;
+
+       GLuint uniform_I_x_y_tex, uniform_I_t_tex;
+       GLuint uniform_diff_flow_tex, uniform_flow_tex;
+       GLuint uniform_beta_0_tex;
+       GLuint uniform_smoothness_x_tex, uniform_smoothness_y_tex;
+
+};
+
+SetupEquations::SetupEquations()
+{
+       equations_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       equations_fs_obj = compile_shader(read_file("equations.frag"), GL_FRAGMENT_SHADER);
+       equations_program = link_program(equations_vs_obj, equations_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &equations_vao);
+       glBindVertexArray(equations_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(equations_program, "position");
+       glEnableVertexArrayAttrib(equations_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_I_x_y_tex = glGetUniformLocation(equations_program, "I_x_y_tex");
+       uniform_I_t_tex = glGetUniformLocation(equations_program, "I_t_tex");
+       uniform_diff_flow_tex = glGetUniformLocation(equations_program, "diff_flow_tex");
+       uniform_flow_tex = glGetUniformLocation(equations_program, "flow_tex");
+       uniform_beta_0_tex = glGetUniformLocation(equations_program, "beta_0_tex");
+       uniform_smoothness_x_tex = glGetUniformLocation(equations_program, "smoothness_x_tex");
+       uniform_smoothness_y_tex = glGetUniformLocation(equations_program, "smoothness_y_tex");
+}
+
+void SetupEquations::exec(GLuint I_x_y_tex, GLuint I_t_tex, GLuint diff_flow_tex, GLuint flow_tex, GLuint beta_0_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, GLuint equation_tex, int level_width, int level_height)
+{
+       glUseProgram(equations_program);
+
+       bind_sampler(equations_program, uniform_I_x_y_tex, 0, I_x_y_tex, nearest_sampler);
+       bind_sampler(equations_program, uniform_I_t_tex, 1, I_t_tex, nearest_sampler);
+       bind_sampler(equations_program, uniform_diff_flow_tex, 2, diff_flow_tex, nearest_sampler);
+       bind_sampler(equations_program, uniform_flow_tex, 3, flow_tex, nearest_sampler);
+       bind_sampler(equations_program, uniform_beta_0_tex, 4, beta_0_tex, nearest_sampler);
+       bind_sampler(equations_program, uniform_smoothness_x_tex, 5, smoothness_x_tex, smoothness_sampler);
+       bind_sampler(equations_program, uniform_smoothness_y_tex, 6, smoothness_y_tex, smoothness_sampler);
+
+       GLuint equations_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &equations_fbo);
+       glNamedFramebufferTexture(equations_fbo, GL_COLOR_ATTACHMENT0, equation_tex, 0);
+
+       glViewport(0, 0, level_width, level_height);
+       glDisable(GL_BLEND);
+       glBindVertexArray(equations_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, equations_fbo);
+       glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+}
+
+// Calculate the smoothness constraints between neighboring pixels;
+// s_x(x,y) stores smoothness between pixel (x,y) and (x+1,y),
+// and s_y(x,y) stores between (x,y) and (x,y+1). We'll sample with
+// border color (0,0) later, so that there's zero diffusion out of
+// the border.
+//
+// See variational_refinement.txt for more information.
+class SOR {
+public:
+       SOR();
+       void exec(GLuint diff_flow_tex, GLuint equation_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, int level_width, int level_height, int num_iterations);
+
+private:
+       GLuint sor_vs_obj;
+       GLuint sor_fs_obj;
+       GLuint sor_program;
+       GLuint sor_vao;
+
+       GLuint uniform_diff_flow_tex;
+       GLuint uniform_equation_tex;
+       GLuint uniform_smoothness_x_tex, uniform_smoothness_y_tex;
+};
+
+SOR::SOR()
+{
+       sor_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       sor_fs_obj = compile_shader(read_file("sor.frag"), GL_FRAGMENT_SHADER);
+       sor_program = link_program(sor_vs_obj, sor_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &sor_vao);
+       glBindVertexArray(sor_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(sor_program, "position");
+       glEnableVertexArrayAttrib(sor_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_diff_flow_tex = glGetUniformLocation(sor_program, "diff_flow_tex");
+       uniform_equation_tex = glGetUniformLocation(sor_program, "equation_tex");
+       uniform_smoothness_x_tex = glGetUniformLocation(sor_program, "smoothness_x_tex");
+       uniform_smoothness_y_tex = glGetUniformLocation(sor_program, "smoothness_y_tex");
+}
+
+void SOR::exec(GLuint diff_flow_tex, GLuint equation_tex, GLuint smoothness_x_tex, GLuint smoothness_y_tex, int level_width, int level_height, int num_iterations)
+{
+       glUseProgram(sor_program);
+
+       bind_sampler(sor_program, uniform_diff_flow_tex, 0, diff_flow_tex, nearest_sampler);
+       bind_sampler(sor_program, uniform_smoothness_x_tex, 1, smoothness_x_tex, smoothness_sampler);
+       bind_sampler(sor_program, uniform_smoothness_y_tex, 2, smoothness_y_tex, smoothness_sampler);
+       bind_sampler(sor_program, uniform_equation_tex, 3, equation_tex, nearest_sampler);
+
+       GLuint sor_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &sor_fbo);
+       glNamedFramebufferTexture(sor_fbo, GL_COLOR_ATTACHMENT0, diff_flow_tex, 0);  // NOTE: Bind to same as we render from!
+
+       glViewport(0, 0, level_width, level_height);
+       glDisable(GL_BLEND);
+       glBindVertexArray(sor_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, sor_fbo);
+
+       for (int i = 0; i < num_iterations; ++i) {
+               glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+               if (i != num_iterations - 1) {
+                       glTextureBarrier();
+               }
+       }
+}
+
+// Simply add the differential flow found by the variational refinement to the base flow.
+// The output is in diff_flow_tex; we don't need to make a new texture.
+class AddBaseFlow {
+public:
+       AddBaseFlow();
+       void exec(GLuint base_flow_tex, GLuint diff_flow_tex, int level_width, int level_height);
+
+private:
+       GLuint add_flow_vs_obj;
+       GLuint add_flow_fs_obj;
+       GLuint add_flow_program;
+       GLuint add_flow_vao;
+
+       GLuint uniform_base_flow_tex;
+};
+
+AddBaseFlow::AddBaseFlow()
+{
+       add_flow_vs_obj = compile_shader(read_file("vs.vert"), GL_VERTEX_SHADER);
+       add_flow_fs_obj = compile_shader(read_file("add_base_flow.frag"), GL_FRAGMENT_SHADER);
+       add_flow_program = link_program(add_flow_vs_obj, add_flow_fs_obj);
+
+       // Set up the VAO containing all the required position/texcoord data.
+       glCreateVertexArrays(1, &add_flow_vao);
+       glBindVertexArray(add_flow_vao);
+       glBindBuffer(GL_ARRAY_BUFFER, vertex_vbo);
+
+       GLint position_attrib = glGetAttribLocation(add_flow_program, "position");
+       glEnableVertexArrayAttrib(add_flow_vao, position_attrib);
+       glVertexAttribPointer(position_attrib, 2, GL_FLOAT, GL_FALSE, 0, BUFFER_OFFSET(0));
+
+       uniform_base_flow_tex = glGetUniformLocation(add_flow_program, "base_flow_tex");
+}
+
+void AddBaseFlow::exec(GLuint base_flow_tex, GLuint diff_flow_tex, int level_width, int level_height)
+{
+       glUseProgram(add_flow_program);
+
+       bind_sampler(add_flow_program, uniform_base_flow_tex, 0, base_flow_tex, nearest_sampler);
+
+       GLuint add_flow_fbo;  // TODO: cleanup
+       glCreateFramebuffers(1, &add_flow_fbo);
+       glNamedFramebufferTexture(add_flow_fbo, GL_COLOR_ATTACHMENT0, diff_flow_tex, 0);
+
+       glViewport(0, 0, level_width, level_height);
+       glEnable(GL_BLEND);
+       glBlendFunc(GL_ONE, GL_ONE);
+       glBindVertexArray(add_flow_vao);
+       glBindFramebuffer(GL_FRAMEBUFFER, add_flow_fbo);
+
+       glDrawArrays(GL_TRIANGLE_STRIP, 0, 4);
+}
+
 int main(void)
 {
        if (SDL_Init(SDL_INIT_EVERYTHING) == -1) {
@@ -428,11 +834,15 @@ int main(void)
        glSamplerParameteri(linear_sampler, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_EDGE);
        glSamplerParameteri(linear_sampler, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_EDGE);
 
-       glCreateSamplers(1, &mipmap_sampler);
-       glSamplerParameteri(mipmap_sampler, GL_TEXTURE_MIN_FILTER, GL_LINEAR_MIPMAP_NEAREST);
-       glSamplerParameteri(mipmap_sampler, GL_TEXTURE_MAG_FILTER, GL_LINEAR);
-       glSamplerParameteri(mipmap_sampler, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_EDGE);
-       glSamplerParameteri(mipmap_sampler, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_EDGE);
+       // The smoothness is sampled so that once we get to a smoothness involving
+       // a value outside the border, the diffusivity between the two becomes zero.
+       glCreateSamplers(1, &smoothness_sampler);
+       glSamplerParameteri(smoothness_sampler, GL_TEXTURE_MIN_FILTER, GL_NEAREST);
+       glSamplerParameteri(smoothness_sampler, GL_TEXTURE_MAG_FILTER, GL_NEAREST);
+       glSamplerParameteri(smoothness_sampler, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_BORDER);
+       glSamplerParameteri(smoothness_sampler, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_BORDER);
+       float zero[] = { 0.0f, 0.0f, 0.0f, 0.0f };
+       glSamplerParameterfv(smoothness_sampler, GL_TEXTURE_BORDER_COLOR, zero);
 
        float vertices[] = {
                0.0f, 1.0f,
@@ -447,13 +857,23 @@ int main(void)
        // Initial flow is zero, 1x1.
        GLuint initial_flow_tex;
        glCreateTextures(GL_TEXTURE_2D, 1, &initial_flow_tex);
-       glTextureStorage2D(initial_flow_tex, 1, GL_RGB32F, 1, 1);
+       glTextureStorage2D(initial_flow_tex, 1, GL_RG16F, 1, 1);
 
        GLuint prev_level_flow_tex = initial_flow_tex;
 
        Sobel sobel;
        MotionSearch motion_search;
        Densify densify;
+       Prewarp prewarp;
+       Derivatives derivatives;
+       ComputeSmoothness compute_smoothness;
+       SetupEquations setup_equations;
+       SOR sor;
+       AddBaseFlow add_base_flow;
+
+       GLuint query;
+       glGenQueries(1, &query);
+       glBeginQuery(GL_TIME_ELAPSED, query);
 
        for (int level = coarsest_level; level >= int(finest_level); --level) {
                int level_width = WIDTH >> level;
@@ -487,26 +907,93 @@ int main(void)
                // Create an output flow texture.
                GLuint flow_out_tex;
                glCreateTextures(GL_TEXTURE_2D, 1, &flow_out_tex);
-               glTextureStorage2D(flow_out_tex, 1, GL_RG16F, width_patches, height_patches);
+               glTextureStorage2D(flow_out_tex, 1, GL_RGB16F, width_patches, height_patches);
 
                // And draw.
                motion_search.exec(tex0_view, tex1_view, grad0_tex, prev_level_flow_tex, flow_out_tex, level_width, level_height, width_patches, height_patches);
 
                // Densification.
 
-               // Set up an output texture.
+               // Set up an output texture (initially zero).
                GLuint dense_flow_tex;
                glCreateTextures(GL_TEXTURE_2D, 1, &dense_flow_tex);
-               //glTextureStorage2D(dense_flow_tex, 1, GL_RGB16F, level_width, level_height);
-               glTextureStorage2D(dense_flow_tex, 1, GL_RGBA32F, level_width, level_height);
+               glTextureStorage2D(dense_flow_tex, 1, GL_RGB16F, level_width, level_height);
 
                // And draw.
                densify.exec(tex0_view, tex1_view, flow_out_tex, dense_flow_tex, level_width, level_height, width_patches, height_patches);
 
-               // TODO: Variational refinement.
+               // Everything below here in the loop belongs to variational refinement.
+               // It is not done yet.
+
+               // Prewarping; create I and I_t.
+               GLuint I_tex, I_t_tex;
+               glCreateTextures(GL_TEXTURE_2D, 1, &I_tex);
+               glCreateTextures(GL_TEXTURE_2D, 1, &I_t_tex);
+               glTextureStorage2D(I_tex, 1, GL_R16F, level_width, level_height);
+               glTextureStorage2D(I_t_tex, 1, GL_R16F, level_width, level_height);
+               prewarp.exec(tex0_view, tex1_view, dense_flow_tex, I_tex, I_t_tex, level_width, level_height);
+
+               // Calculate I_x and I_y. We're only calculating first derivatives;
+               // the others will be taken on-the-fly in order to sample from fewer
+               // textures overall, since sampling from the L1 cache is cheap.
+               // (TODO: Verify that this is indeed faster than making separate
+               // double-derivative textures.)
+               GLuint I_x_y_tex, beta_0_tex;
+               glCreateTextures(GL_TEXTURE_2D, 1, &I_x_y_tex);
+               glCreateTextures(GL_TEXTURE_2D, 1, &beta_0_tex);
+               glTextureStorage2D(I_x_y_tex, 1, GL_RG16F, level_width, level_height);
+               glTextureStorage2D(beta_0_tex, 1, GL_R16F, level_width, level_height);
+               derivatives.exec(I_tex, I_x_y_tex, beta_0_tex, level_width, level_height);
+
+               // We need somewhere to store du and dv (the flow increment, relative
+               // to the non-refined base flow u0 and v0). It starts at zero.
+               GLuint du_dv_tex;
+               glCreateTextures(GL_TEXTURE_2D, 1, &du_dv_tex);
+               glTextureStorage2D(du_dv_tex, 1, GL_RG16F, level_width, level_height);
+
+               // And for smoothness.
+               GLuint smoothness_x_tex, smoothness_y_tex;
+               glCreateTextures(GL_TEXTURE_2D, 1, &smoothness_x_tex);
+               glCreateTextures(GL_TEXTURE_2D, 1, &smoothness_y_tex);
+               glTextureStorage2D(smoothness_x_tex, 1, GL_R16F, level_width, level_height);
+               glTextureStorage2D(smoothness_y_tex, 1, GL_R16F, level_width, level_height);
+
+               // And finally for the equation set. See SetupEquations for
+               // the storage format.
+               GLuint equation_tex;
+               glCreateTextures(GL_TEXTURE_2D, 1, &equation_tex);
+               glTextureStorage2D(equation_tex, 1, GL_RGBA32UI, level_width, level_height);
+
+               for (int outer_idx = 0; outer_idx < level + 1; ++outer_idx) {
+                       // Calculate the smoothness terms between the neighboring pixels,
+                       // both in x and y direction.
+                       compute_smoothness.exec(dense_flow_tex, du_dv_tex, smoothness_x_tex, smoothness_y_tex, level_width, level_height);
+
+                       // Set up the 2x2 equation system for each pixel.
+                       setup_equations.exec(I_x_y_tex, I_t_tex, du_dv_tex, dense_flow_tex, beta_0_tex, smoothness_x_tex, smoothness_y_tex, equation_tex, level_width, level_height);
+
+                       // Run a few SOR (or quasi-SOR, since we're not really Jacobi) iterations.
+                       // Note that these are to/from the same texture.
+                       sor.exec(du_dv_tex, equation_tex, smoothness_x_tex, smoothness_y_tex, level_width, level_height, 5);
+               }
+
+               // Add the differential flow found by the variational refinement to the base flow,
+               // giving the final flow estimate for this level.
+               // The output is in diff_flow_tex; we don't need to make a new texture.
+               add_base_flow.exec(dense_flow_tex, du_dv_tex, level_width, level_height);
 
-               prev_level_flow_tex = dense_flow_tex;
+               prev_level_flow_tex = du_dv_tex;
        }
+       glEndQuery(GL_TIME_ELAPSED);
+
+       GLint available;
+       do {
+               glGetQueryObjectiv(query, GL_QUERY_RESULT_AVAILABLE, &available);
+               usleep(1000);
+       } while (!available);
+       GLuint64 time_elapsed;
+       glGetQueryObjectui64v(query, GL_QUERY_RESULT, &time_elapsed);
+       fprintf(stderr, "GPU time used = %.1f ms\n", time_elapsed / 1e6);
 
        int level_width = WIDTH >> finest_level;
        int level_height = HEIGHT >> finest_level;
@@ -514,7 +1001,11 @@ int main(void)
        glGetTextureImage(prev_level_flow_tex, 0, GL_RGB, GL_FLOAT, level_width * level_height * 3 * sizeof(float), dense_flow.get());
 
        FILE *fp = fopen("flow.ppm", "wb");
+       FILE *flowfp = fopen("flow.flo", "wb");
        fprintf(fp, "P6\n%d %d\n255\n", level_width, level_height);
+       fprintf(flowfp, "FEIH");
+       fwrite(&level_width, 4, 1, flowfp);
+       fwrite(&level_height, 4, 1, flowfp);
        for (unsigned y = 0; y < unsigned(level_height); ++y) {
                int yy = level_height - y - 1;
                for (unsigned x = 0; x < unsigned(level_width); ++x) {
@@ -522,43 +1013,21 @@ int main(void)
                        float dv = dense_flow[(yy * level_width + x) * 3 + 1];
                        float w = dense_flow[(yy * level_width + x) * 3 + 2];
 
-                       du /= w;
-                       dv /= w;
-
-                       float angle = atan2(dv * level_width, du * level_height);
-                       float magnitude = min(hypot(du * level_width, dv * level_height) / 20.0, 1.0);
-                       
-                       // HSV to RGB (from Wikipedia). Saturation is 1.
-                       float c = magnitude;
-                       float h = (angle + M_PI) * 6.0 / (2.0 * M_PI);
-                       float X = c * (1.0 - fabs(fmod(h, 2.0) - 1.0));
-                       float r = 0.0f, g = 0.0f, b = 0.0f;
-                       if (h < 1.0f) {
-                               r = c; g = X;
-                       } else if (h < 2.0f) {
-                               r = X; g = c;
-                       } else if (h < 3.0f) {
-                               g = c; b = X;
-                       } else if (h < 4.0f) {
-                               g = X; b = c;
-                       } else if (h < 5.0f) {
-                               r = X; b = c;
-                       } else if (h < 6.0f) {
-                               r = c; b = X;
-                       } else {
-                               // h is NaN, so black is fine.
-                       }
-                       float m = magnitude - c;
-                       r += m; g += m; b += m;
-                       r = max(min(r, 1.0f), 0.0f);
-                       g = max(min(g, 1.0f), 0.0f);
-                       b = max(min(b, 1.0f), 0.0f);
-                       putc(lrintf(r * 255.0f), fp);
-                       putc(lrintf(g * 255.0f), fp);
-                       putc(lrintf(b * 255.0f), fp);
+                       du = (du / w) * level_width;
+                       dv = (-dv / w) * level_height;
+
+                       fwrite(&du, 4, 1, flowfp);
+                       fwrite(&dv, 4, 1, flowfp);
+
+                       uint8_t r, g, b;
+                       flow2rgb(du, dv, &r, &g, &b);
+                       putc(r, fp);
+                       putc(g, fp);
+                       putc(b, fp);
                }
        }
        fclose(fp);
+       fclose(flowfp);
 
        fprintf(stderr, "err = %d\n", glGetError());
 }