C.2 Jitter: CDF-PDF Convolution in n Dimensions

The following shows that the convolution of an n-dimensional spherical Gaussian probability density function (PDF) and a Gaussian cumulative distribution function (CDF) results in another Gaussian CDF.

Let f₁ (x) be a spherical Gaussian PDF in n-dimensions

and let F₂(x) be a Gaussian CDF of the form

This can be written as

where ŵ = w/|w| and σ₂ = 1/|w|.

The convolution of F₂ and f₁ is the n-dimensional integral

Separate x and α into components parallel and orthogonal to ŵ

where l and k are scalars and y and β are n-dimensional vectors orthogonal to ŵ.

Then

and

Thus F₂(x) * f₁(x) reduces to a one-dimensional convolution of a Gaussian CDF with standard deviation σ₂ = 1/|w| and a Gaussian PDF with standard deviation σ₁. It can be shown (see section C.3) that this is a Gaussian CDF with variance σ²₃ = σ²₁ + σ²₂.

Letting Z_a denote the Gaussian CDF function with standard deviation a,

Thus, the convolution of a Gaussian CDF and a Gaussian PDF can be computed by a simple scaling of the original CDF.