Guide

Documentation may not be accurate as this is a beta stage package undergoing changes. Get started with tutorials which are more up to date.

Scalar & vector fields

Scalar & vector fields are represented as 2d/3d arrays of canonically scalars or vectors. Array values can alternatively be any type that Supports addition & multiplication.

Customizable grid

Main.EquivariantOperators.Grid — Type

Grid(cell, rmax::AbstractFloat)
Grid(
    cell::AbstractMatrix,
    sz::Union{AbstractVector,Tuple};
    origin = (sz .+ 1) ./ 2,
    )

Grid is specified by its discrete cell vectors (column-wise matrix), overall size and origin. For a uniform Cartesian 5x5 grid discretized at 0.1 with a centered origin, we get cell = [0.1 0; 0 0.1] & origin = [3, 3]. Grid cell can in general be noncartesian.

Particle mesh placement and interpolation

Base.get — Method

Base.get(field::AbstractArray, grid::Grid, rvec::AbstractVector)
Base.put!(
    field::AbstractArray,
    grid::Grid,
    rvec::AbstractVector,
    val::AbstractVector,
)

With grid info we can interpolate a scalar or vector field at any location. We can also place a scalar or vector point source anywhere with automatic normalization wrt discretization. Both work via a proximity weighted average of the closest grid points (in general up to 4 in 2d and 8 in 3d).

Finite difference equivariant operators

Main.EquivariantOperators.Op — Type

Op(radfunc, rmax, cell::Matrix; kw...)

constructs equivariant operator

function (m::Op)(x::AbstractArray, )

Main.EquivariantOperators.Del — Function

" Del(cell; pad = :same, border = :smooth)

constructs gradient operator

Main.EquivariantOperators.Laplacian — Function

Laplacian(cell; pad = :same, border = :smooth)

constructs Laplacian operator

Main.EquivariantOperators.Gaussian — Function

Gaussian(cell, σ, rmax; kw...)

constructs Gaussian diffusion operator

Lower level utilities

Convolutions

Feature rich convolution and cross correlation functions with options for padding, stride, boundary conditions, and custom products (tensor field convolutions).

Main.EquivariantOperators.cvconv — Function

cvconv(x, f; product = *, stride = 1, pad = 0)

"convolution" in computer vision for any dimension, same as Cross correlation. For convolution in signal processing , use dspconv instead.

x input array f filter array product product in convolution, eg *, dot pad amount of padding or padding option

any integer number of pixels on each boundary
:same: adds enough padding so output is same size as input
:outer: output size is size(x) .+ size(f) .- 1

border type of padding

0 value pixels
:replicate repeats edge values
:circular periodic BC
:smooth continuous derivatives at boundaries useful for differential operators
:reflect reflects interior across boundaries which are not repeated
:symmetric same as :reflect but with boundaries repeated

Convolutions in other Julia packages, fewer features but perhaps more optimized for speed in their specific use cases

ImageFiltering.imfilter. Its docs has excellent mathematical explaination of convolutions and correlation as well as padding/border options
DSP.conv DSP.xcor
Flux.conv

Main.EquivariantOperators.dspconv — Function

dspconv(x, f; product = *,pad = :outer,border=0)

Convolution in signal processing. For "convolution" in computer vision, use cvconv instead. By default output size is size(x) .+ size(f) .- 1. See cvconv for its keyword options which also apply here