4 The motion search is one of the two major components of DIS. It works more or less
5 like you'd expect; there's a bunch of overlapping patches (8x8 or 12x12 pixels) in
6 a grid, and for each patch, there's a search to try to find the most similar patch
9 Unlike in a typical video codec, the DIS patch search is based on gradient descent;
10 conceptually, you start with an initial guess (the value from the previous level,
11 or the zero flow for the very first level), subtract the reference (“template”)
12 patch from the candidate, look at the gradient to see in what direction there is
13 a lower difference, and then inch a bit toward that direction. (There is seemingly
14 nothing like AdaM, Momentum or similar, but the searched value is only in two
15 dimensions, so perhaps it doesn't matter as much then.)
17 DIS does a tweak to this concept. Since the procedure as outlined above requires
18 computing the gradient of the candidate patch, it uses the reference patch as
19 candidate (thus the “inverse” name), and thus uses _its_ gradient to understand
20 in which direction to move. (This is a bit dodgy, but not _that_ dodgy; after
21 all, the two patches are supposed to be quite similar, so their surroundings and
22 thus also gradients should also be quite similar.) It's not entirely clear whether
23 this is still a win on GPU, where calculations are much cheaper, especially
24 the way we parallelize the search, but we've kept it around for now.
26 The inverse search is explained and derived in the supplementary material of the
27 paper, section A. Do note that there's a typo; the text under equation 9 claims
28 that the matrix H is n x n (where presumably n is the patch size), while in reality,
31 Our GPU parallellization is fairly dumb right now; we do one patch per fragment
32 (ie., parallellize only over patches, not within each patch), which may not
33 be optimal. In particular, in the initial level, we only have 40 patches,
34 which is on the low side for a GPU, and the memory access patterns may also not
38 const uint patch_size = 12;
39 const uint num_iterations = 16;
42 in vec2 patch_bottom_left_texel; // Center of bottom-left texel of patch.
45 uniform sampler2D flow_tex, grad0_tex, image0_tex, image1_tex;
46 uniform vec2 image_size, inv_image_size;
50 // Lock patch_bottom_left_texel to an integer, so that we never get
51 // any bilinear artifacts for the gradient.
52 vec2 base = round(patch_bottom_left_texel * image_size)
55 // First, precompute the pseudo-Hessian for the template patch.
56 // This is the part where we really save by the inverse search
57 // (ie., we can compute it up-front instead of anew for each
62 // where S is the gradient at each point in the patch. Note that
63 // this is an outer product, so we get a (symmetric) 2x2 matrix,
66 vec2 grad_sum = vec2(0.0f); // Used for patch normalization.
67 float template_sum = 0.0f;
68 for (uint y = 0; y < patch_size; ++y) {
69 for (uint x = 0; x < patch_size; ++x) {
70 vec2 tc = base + uvec2(x, y) * inv_image_size;
71 vec2 grad = texture(grad0_tex, tc).xy;
72 H[0][0] += grad.x * grad.x;
73 H[1][1] += grad.y * grad.y;
74 H[0][1] += grad.x * grad.y;
76 template_sum += texture(image0_tex, tc).x;
82 // Make sure we don't get a singular matrix even if e.g. the picture is
83 // all black. (The paper doesn't mention this, but the reference code
84 // does it, and it seems like a reasonable hack to avoid NaNs. With such
85 // a H, we'll go out-of-bounds pretty soon, though.)
86 if (determinant(H) < 1e-6) {
91 mat2 H_inv = inverse(H);
93 // Fetch the initial guess for the flow. (We need the normalization step
94 // because densification works by accumulating; see the comments on the
96 vec3 prev_flow = texture(flow_tex, flow_tc).xyz;
98 if (prev_flow.z < 1e-3) {
99 initial_u = vec2(0.0, 0.0);
101 initial_u = prev_flow.xy / prev_flow.z;
104 // Note: The flow is in OpenGL coordinates [0..1], but the calculations
105 // generally come out in pixels since the gradient is in pixels,
106 // so we need to convert at the end.
109 for (uint i = 0; i < num_iterations; ++i) {
110 vec2 du = vec2(0.0, 0.0);
111 float warped_sum = 0.0f;
112 for (uint y = 0; y < patch_size; ++y) {
113 for (uint x = 0; x < patch_size; ++x) {
114 vec2 tc = base + uvec2(x, y) * inv_image_size;
115 vec2 grad = texture(grad0_tex, tc).xy;
116 float t = texture(image0_tex, tc).x;
117 float warped = texture(image1_tex, tc + u).x;
118 du += grad * (warped - t);
119 warped_sum += warped;
123 // Subtract the mean for patch normalization. We've done our
124 // sums without subtracting the means (because we didn't know them
127 // sum(S^T * ((x + µ1) - (y + µ2))) = sum(S^T * (x - y)) + (µ1 – µ2) sum(S^T)
129 // which gives trivially
131 // sum(S^T * (x - y)) = [what we calculated] - (µ1 - µ2) sum(S^T)
133 // so we can just subtract away the mean difference here.
134 du -= grad_sum * (warped_sum - template_sum) * (1.0 / (patch_size * patch_size));
136 u += (H_inv * du) * inv_image_size;
139 // Reject if we moved too far.
140 if (length((u - initial_u) * image_size) > patch_size) {