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Analytical Gradient and its intuition

zhengyuf opened this issue · 2 comments

zhengyuf commented
  • Analytical Gradient and its intuition

We find the canonical ray-surface intersection x_c in an interactive manner. Our goal is to determine how the change of geometry and deformation affects the location of x_c, in other words, the gradient of x_c w.r.t the geometry and deformation network.

We leverage the fact that the final x_c always satisfies two constraints: it’s on the surface (occ (x_c) = 0.5) and its deformed location is on the ray (P(x_d) = p). With any delta change of the geometry and deformation, the final location of x_c will change accordingly so that these two constraints still hold. Therefore, if we take the gradient of the constraints (rewrite as F(x_c) = 0) w.r.t the parameters of the geometry and deformation network, it will be equal to zero. Writing it into partial derivatives, we arrive at equation 10 in the main paper.
image

UestcJay commented
  • Analytical Gradient and its intuition

We find the canonical ray-surface intersection x_c in an interactive manner. Our goal is to determine how the change of geometry and deformation affects the location of x_c, in other words, the gradient of x_c w.r.t the geometry and deformation network.

We leverage the fact that the final x_c always satisfies two constraints: it’s on the surface (occ (x_c) = 0.5) and its deformed location is on the ray (P(x_d) = p). With any delta change of the geometry and deformation, the final location of x_c will change accordingly so that these two constraints still hold. Therefore, if we take the gradient of the constraints (rewrite as F(x_c) = 0) w.r.t the parameters of the geometry and deformation network, it will be equal to zero. Writing it into partial derivatives, we arrive at equation 10 in the main paper. image

Hi, Yufeng. Why the constant is deformed_x - points and not directly deformed_x the line code

zhengyuf commented

Hey,

If you don't optimize the camera, then they are the same. But if you do, subtracting points allow you to back-propagate to ray directions and camera locations.

Best,
Yufeng