X-Git-Url: https://git.sesse.net/?a=blobdiff_plain;f=flow.cpp;h=11f4d4c87646ba31d5c9d2c485386f848c6613d1;hb=dac59bd006c3ee244634f1bcdfdd2b1237db99d1;hp=b46e0a10c74cff067b2a5c67675baece2adfe329;hpb=c240834b6e92f10da89c1cf6f4520c2b1b641312;p=nageru

diff --git a/flow.cpp b/flow.cpp
index b46e0a1..11f4d4c 100644
--- a/flow.cpp
+++ b/flow.cpp
@@ -33,7 +33,7 @@ constexpr unsigned finest_level = 1;
 constexpr unsigned patch_size_pixels = 12;
 
 // Some global OpenGL objects.
-GLuint nearest_sampler, linear_sampler;
+GLuint nearest_sampler, linear_sampler, smoothness_sampler;
 GLuint vertex_vbo;
 
 string read_file(const string &filename)
@@ -153,10 +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) {
@@ -224,7 +224,7 @@ private:
 	GLuint sobel_program;
 	GLuint sobel_vao;
 
-	GLuint uniform_tex, uniform_image_size, uniform_inv_image_size;
+	GLuint uniform_tex, uniform_image_size;
 };
 
 Sobel::Sobel()
@@ -234,15 +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));
 
 	uniform_tex = glGetUniformLocation(sobel_program, "tex");
-	uniform_inv_image_size = glGetUniformLocation(sobel_program, "inv_image_size");
 }
 
 void Sobel::exec(GLint tex0_view, GLint grad0_tex, int level_width, int level_height)
@@ -251,7 +250,6 @@ void Sobel::exec(GLint tex0_view, GLint grad0_tex, int level_width, int level_he
 	glBindTextureUnit(0, tex0_view);
 	glBindSampler(0, nearest_sampler);
 	glProgramUniform1i(sobel_program, uniform_tex, 0);
-	glProgramUniform2f(sobel_program, uniform_inv_image_size, 1.0f / level_width, 1.0f / level_height);
 
 	GLuint grad0_fbo;  // TODO: cleanup
 	glCreateFramebuffers(1, &grad0_fbo);
@@ -259,7 +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);
@@ -288,8 +286,8 @@ 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");
@@ -322,7 +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);
 }
@@ -357,8 +355,8 @@ 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");
@@ -388,6 +386,8 @@ void Densify::exec(GLuint tex0_view, GLuint tex1_view, GLuint flow_tex, GLuint d
 
 	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);
+	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);
@@ -399,11 +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) {
@@ -444,6 +834,16 @@ 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);
 
+	// 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,
 		0.0f, 0.0f,
@@ -457,13 +857,19 @@ 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);
@@ -516,9 +922,67 @@ int main(void)
 		// 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);