VoxelwiseConstantMapping.jl

Introduction

VoxelwiseConstantMapping is a simple Julia package that generalizes piecewise-constant functions to higher dimensions. In the numerical implementation of a piecewise-constant function, the main subtlety occurs at the transition point˜˜. For example, the Heaviside step function is defined as H(x) = 0 for x < 0 and H(x) = 1 for x > 0, but its value at the transition point x = 0 is ambiguous. One popular choice is to assign the average of the values on the two sides of the transition point (i.e., H(0) = 0.5 for the Heaviside step function), because such a choice reduces the discontinuity at the transition point.

VoxelwiseConstantMapping extends this concept to an arbitrary dimension. For example, consider a function f(x,y) defined on R². Suppose that we have pixel boundaries at x = 0 and y = 0. In other words,

f(x>0, y>0) = v₁
f(x<0, y>0) = v₂
f(x<0, y<0) = v₃
f(x>0, y<0) = v₄

Then, VoxelwiseConstantMapping defines the values of f at the transition lines as follows:

f(x=0, y>0) = (v₁+v₂)/2
f(x=0, y<0) = (v₃+v₄)/2
f(x>0, y=0) = (v₁+v₄)/2
f(x<0, y=0) = (v₂+v₃)/2
f(x=0, y=0) = (v₁+v₂+v₃+v₄)/4

Example

The main constructor to use is VoxelwiseConstant(...). The voxels are assumed having rectangular faces, and you can have as many voxels as you want. The voxel sizes can be nonuniform.

using VoxelwiseConstantMapping

vals = rand(3,4)  # 3 × 4 pixels (= 2D voxels)
xbounds = [-1, 0, 0.5, 0.6]  # 4 boundaries of 3 pixels in x-direction
ybounds = [-0.1, -0.05, 0, 4, 10]  # 5 boundaries of 4 pixels in y-direction
vc = VoxelwiseConstant(vals, (xbounds, ybounds))

v₀₀  = vc([0,0])  # returns value at (x,y) = (0,0)