How to quantize surface normals

Question

I am trying to quantize surface normals into let's say 8 bins.

For example, when computing features like HOG to quantize 2D gradients [x,y] into 8 bins we just take the angle with the y plane i.e. arctan(y/x) which will give us an angle between 0-360.

My question is, given a 3D direction [x,y,z], a surface normal in this case, how can we histogram it in a similar way? Do we just project onto one plane and use that angle i.e. the dot product of [x,y,z] and [0,1,0] for example?

Thanks

EDIT

I also read a paper recently where they quantized surface normals by measuring angles between normal and precomputed vectors that which are arranged around a right circular cone shape. I have added a link to this paper in the question (section 3.3.2 last paragraph), is this an effective approach? And if so, how do we compute these vectors?

Answer 1

Quantizing a continuous topological space corresponds to partitioning it and assigning labels to each partition. The straightforward standard approach for this scenario (quantizing normals) is as follows.

1. Choose your favorite uniform polyhedron:
   • http://en.wikipedia.org/wiki/Tetrahedron (4 faces)
   • http://en.wikipedia.org/wiki/Cube (6 faces)
   • http://en.wikipedia.org/wiki/Octahedron (8 faces)
   • http://en.wikipedia.org/wiki/Dodecahedron (12 faces)
   • http://en.wikipedia.org/wiki/Icosahedron (20 faces)
   • In general: http://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol
2. Develop a mapping function from a normal on the unit sphere to the face of your chosen polyhedron that the normal intersects.
   • I would advise doing an argmax across polyhedron faces, taking the dot product of your normal and each polyhedron face normal. The one that gives the highest dot product is the face your normal should be binned into.
3. Use the face normal for each polyhedron face as the label for that face.

Prefer this approach to the approach suggested by others of mapping to spherical coordinates and then binning those. That approach suffers from too much sensitivity near the poles of the sphere.

Edit

In the paper you added to your question, the same idea is being used. There, however, the normals are restricted to a hemisphere - the only surfaces directly visible in an image have surface normals no more than 90 degrees away from the vector from the surface to the viewpoint.

The paper wants to quantize these surface normals into 8 values, represented by 8-bit integers with exactly one bit set to 1 and the rest set to 0. The 8 precomputed normals are computed as:

n^t_x = cos(a)*cos(t)

n^t_y = cos(a)*sin(t)

n^t_z = sin(a)

where a = pi/4 and t = 0, pi/4, 2*pi/4, 3*pi/4, ..., 7*pi/4.

Notice

[cos(a)*cos(t)]² + [cos(a)*sin(t)]² + [sin(a)]² = cos²(a)[cos²(t) + sin²(t)] + sin²(a) = cos²(a) + sin²(a) = 1