AbstractShape{D}

Generic shape type for ImagePhantoms. The dimension D is likely 2 or 3, e.g., 2 for an Ellipse and 3 for an Ellipsoid.

Cone <: AbstractShape{3}

Cuboid <: AbstractShape{3}

Cylinder <: AbstractShape{3}

Dirac{D} <: AbstractShape{D}

Ellipse <: AbstractShape{2}

EllipsePhantomVersion

Parent type for different versions of ellipse phantoms:

• SheppLogan original CT version from Shepp&Logan paper
• SheppLoganEmis higher contrast version suitable for emission tomography
• SheppLoganBrainWeb integer index version based on brainweb
• SheppLoganToft higher contrast version from Toft, 1996
• SouthPark for fun

Ellipsoid <: AbstractShape{3}

Gauss2 <: AbstractShape{2}

Gauss3 <: AbstractShape{3}

Object(shape ; cx, cy, cz, wx=1, wy=wx, wz=wx, ϕ=0, θ=0, ψ=0, value=1)

3D object constructor from values (without tuples).

Object{S, D, V, C, A, Da}(center, width, angle, value) <: AbstractObject
    where {S <: AbstractShape, V <: Number, C,A <: RealU}

General container for 2D and 3D objects for defining image phantoms.

• center::NTuple{D,C} coordinates of "center" of this object
• width::NTuple{D,C} "width" along each axis (e.g., FWHM for Gauss, radii for Ellipse)
• angle::NTuple{D-1,A} angle of x' axis relative to x axis, in radians (or with units)
• value::V "intensity" value for this object

Example

julia> Object(Ellipse(), (0,0), (1,2), 0.0, 1//2)
Object2d{Ellipse, Rational{Int64}, Int64, Float64, 1, Float64} (S, D, V, ...)
 center::NTuple{2,Int64} (0, 0)
 width::NTuple{2,Int64} (1, 2)
 angle::Tuple{Float64} (0.0,)
 value::Rational{Int64} 1//2

Object(shape ; cx, cy, wx=1, wy=wx, ϕ=0, value=1)

2D object constructor from values (without tuples).

Object(shape, center=(0,…), width=(1,…), angle=(0,…), value=1)
Object(shape ; center, width=(1,…), angle=(0,…), value=1)

General outer Object constructor from tuples, as either positional arguments or named keyword arguments.

Object(ob::Object ; center, width, angle, value)

Make a copy of Object ob, optionally modifying some values.

Object2d{S,V,C} = Object{S,2,V,C} where {S <: AbstractShape, V,C <: Number}

For 2D objects

Object3d{S,V,C} = Object{S,3,V,C} where {S <: AbstractShape, V,C <: Number}

For 3D objects

Rect <: AbstractShape{2}

SheppLogan

Original version from : Larry A Shepp, Benjamin F Logan, "The Fourier reconstruction of a head section," IEEE Transactions on Nuclear Science, 21(3):21-42, June 1974. doi

Also in Kak and Slaney 1988 text, p. 255. doi

SheppLoganToft

Toft, Peter Aundal & Sørensen, John Aasted "The Radon transform-theory and implementation," Technical University of Denmark (DTU), 1996. Page 201. https://files.openpdfs.org/1ra51GP6gJO.pdf

(*)(ob::Object, x::Number)
(*)(x::Number, ob::Object)

Scale object value by x.

show(io::IO, ::MIME"text/plain", ob::Object)

Rxyz_inv(x::RealU, y::RealU, z::RealU, ϕ::RealU, θ::RealU, ψ::RealU)
Rxyz_inv(x::RealU, y::RealU, sinϕ, sinθ, sinψ, cosϕ, cosθ, cosψ)

Multiply Rz(-ϕ) * Ry(-θ) * Rx(-ψ) * [x, y, z] for 3D (inverse) rotation.

Rxyz_mul(x::RealU, y::RealU, z::RealU, ϕ::RealU, θ::RealU, ψ::RealU)
Rxyz_mul(x::RealU, y::RealU, sinϕ, sinθ, sinψ, cosϕ, cosθ, cosψ)

Multiply Rx(ψ) * Ry(θ) * Rz(ϕ) * [x, y, z] for 3D rotation, where ψ, θ, ϕ denote rotation about x, y, z axes with right-hand rule.

This is the reverse order of z-y′-x″ in https://en.wikipedia.org/wiki/Rotationformalismsinthreedimensions#Eulerangles(z-y%E2%80%B2-x%E2%80%B3intrinsic)%E2%86%92rotationmatrix.

(lower, upper) = _interval(a, b)

Determine the interval [lower, upper] corresponding to the set {x : b ≤ a x} where a is unitless but b and x have same units.

circle(x, y, r, v=1) (circle of radius `r` centered at `(x,y)`)
circle((x,y), r=1, v=1) ditto
circle(r, v=1) centered at origin

Construct circles as special cases of Ellipse.

cone(cx, cy, cz, wx, wy, wz, Φ, Θ, value::Number)
cone(center::NTuple{3,RealU}, width::NTuple{3,RealU}, angle::NTuple{3,RealU}, v)

Construct Object{Cone} from parameters; here width is the base radii for wx and wy and wz is the height.

coords(object::Object2d, x::RealU, y::RealU)

Put coordinates (x,y) in canonical axes associated with object.

coords(object::Object3d, x::RealU, y::RealU, y::RealU)

Put coordinates (x,y,z) in canonical axes associated with object.

cube(x, y, z, w, v=1) (cube of width `w` centered at `(x,y,z)`)
cube((x,y,z), w, v=1) ditto
cube(w, v=1) centered at origin

Construct cubes as special cases of Cuboid.

(ℓmin, ℓmax) = cube_bounds(p, e)

Bounds of ℓ corresponding to rect(x) for direction e from point p.

cuboid(cx, cy, cz, wx, wy, wz, Φ, Θ, value::Number)
cuboid(center::NTuple{3,RealU}, width::NTuple{3,RealU}, angle::NTuple{3,RealU}, v)

Construct Object{Cuboid} from parameters; here width is the full-width.

cylinder(cx, cy, cz, wx, wy, wz, Φ, Θ, value::Number)
cylinder(center::NTuple{3,RealU}, width::NTuple{3,RealU}, angle::NTuple{3,RealU}, v)

Construct Object{Cylinder} from parameters; here width is the radius in x,y and the height in z.

dirac2(...)

Construct Object2d{Dirac2} from parameters; here width is a scale factor: δ((x - c)/w) = w * δ(x - c).

dirac3(...)

Construct Object3d{Dirac3} from parameters; here width is a scale factor: δ((x - c)/w) = w * δ(x - c).

params = disk_phantom_params( ; ...)

Generate ndisk ellipse phantom parameter 6-tuples for a head-sized disk plus many disks within it, designed so that the disks have some minimum separation minsep to avoid overlap and to simplify patch-based model fitting.

Options

• fov::Real = 240 image field of view in mm
• rhead::Function = () -> 100 background radius for "head" [mm]
• muhead::Function = () -> 1000 "μ" (intensity) value for background head disk
• rdisk::Function = () -> 10 + 10 * rand() random disk radii [10,20]
• mudisk::Function = () -> 100 + 200 * rand() "μ" values for disks [100,300]
• ndisk::Function = () -> 10 # of random disks
• minsep::Real = 8 minimum disk separation in mm
• maxtry::Int = 500 give up on adding more disks if this is reached
• warn::Bool = false warn if maxtry reached?
• seed::Int = 0 if nonzero then use this seed

The function options can be replaced with rand() for other Distributions.

ellipse(cx, cy, rx=1, ry=rx, ϕ=0, value::Number=1)
ellipse(center::NTuple{2,RealU}, radii::NTuple{2,RealU}=(1,1), ϕ::RealU=0, v=1)

Construct Object{Ellipse} from parameters.

phantom = ellipse(Vector{Tuple{6 params}})

Return vector of Object{Ellipse}, one for each element of input vector of tuples. Often the input comes from ellipse_parameters.

ellipse_parameters(case; fovs::NTuple{2}, u::NTuple{3}, disjoint::Bool)

Return vector of Tuples of ellipse parameters. By default the first four elements of each tuple are unitless "fractions of field of view", so elements 1,3 are scaled by xfov and elements 2,4 are scaled by yfov, where (xfov, yfov) = fovs. The optional 3-tuple u specifies scaling and/or units:

• elements 1-4 (center, radii) are scaled by u[1] (e.g., mm),
• elements 5 (angle) is scaled by u[2] (e.g., 1 or °),
• elements 6 (value) is scaled by u[3] (e.g., 1/cm) for an attenuation map.

If disjoint==true then the middle ellipse positions are adjusted to avoid overlap.

params = ellipse_parameters(SouthPark() ; fovs::NTuple{2,Number} = (100,100))

Ellipse parameters for "South Park" phantom.

ellipse_parameters_shepplogan( ; disjoint::Bool)

10 × 6 Matrix{Float64} of classic Shepp-Logan ellipse parameters. If disjoint==true then the middle ellipse positions are adjusted to avoid overlap.

ellipse_parameters_uscale(params, fovs, uc, ua, uv)

Return vector of Tuples after FOV and unit scaling.

ellipsoid(cx, cy, cz, rx=1, ry=1, rz=1, Φ=0, Θ=0, value::Number = 1)
ellipsoid(center::NTuple{3,RealU}, radii::NTuple{3,RealU}, angle::NTuple{3,RealU}, v)

Construct Object{Ellipsoid} from parameters.

oa = ellipsoid(Vector{Tuple{10 params}})

Return vector of Object{Ellipsoid}, one for each element of input vector of tuples. Often the input comes from ellipsoid_parameters.

ellipsoid_parameters(case; fovs::NTuple{3}, u::NTuple{3})

Return vector of Tuples of ellipsoid parameters. By default the parameters are for a 3D Shepp-Logan ellipsoid phantom. By default the first 6 elements of each tuple are unitless "fractions of field of view", so elements 1,4 are scaled by xfov and elements 2,5 are scaled by yfov, and elements 3,6 are scaled by zfov, where (xfov, yfov, zfov) = fovs. The optional 3-tuple u specifies scaling and/or units:

• elements 1-6 (center, radii) are scaled by u[1] (e.g., mm),
• elements 7-8 (angles) are scaled by u[2] (e.g., 1 or °),
• element 9 (value) is scaled by u[3] (e.g., 1/cm) for an attenuation map.

ellipsoid_parameters_shepplogan( ; disjoint::Bool)

12 × 10 Matrix{Float64} of 3D Shepp-Logan ellipsoid parameters (cx,cy,cz, rx,ry rz, Φ,Θ=0,Ψ=0, ρ). By default the first 6 columns are unitless "fractions of field of view".

ellipsoid_parameters_uscale(params, fovs, uc, ua, uv)

Return vector of Tuples after FOV and unit scaling.

focus_chart( ; radius=1, nspoke=30, value=1)

Generate nspoke Triangle phantom parameters for a focus chart. Returns Vector{Object2d{Triangle} for passing to phantom.

Options

• radius::RealU = 1 radius of phantom
• nspoke::Int = 60 # of spokes
• value::Number = 1 alternate between 0 and this value

s = fwhm2spread(w)

Convert FWHM w to equivalent Gaussian spread s for $\exp(-π (x/s)^2)$. exp(-π (fwhm/2/s)^2) = 1/2 means fwhm/2/s) = sqrt(log(2)/π) 2s/fwhm = sqrt(π/log(2)) 2s = fwhm * sqrt(π/log(2)) s = fwhm * sqrt(π / log(16))

gauss2(cx, cy, wx, wy=wx, ϕ=0, value::Number=1)
gauss2(center::NTuple{2,RealU}, width::NTuple{2,RealU}=(1,1), ϕ::RealU=0, v=1)
gauss2(w, v=1) (isotropic of width `w`)

Construct Object{Gauss2} from parameters; here width = FWHM (full-width at half-maximum).

In 1D, the formula is g(x) = exp(-π ((x - cx) / sx)^2) where sx = fwhm2spread(w) = w * sqrt(π / log(16)), which, for cx=0, has 1D FT G(ν) = sx^2 exp(π (sx νx)^2).

gauss3(cx, cy, cz, wx, wy, wz, Φ=0, Θ=0, value::Number = 1)
gauss3(center::NTuple{3,RealU}, radii::NTuple{3,RealU}=(1,1,1), angle::NTuple{2,RealU}=(0,0), v=1)
gauss3(r, v=1) (isotropic of width `w`)

Construct Object{Gauss3} from parameters; here width = FWHM (full-width at half-maximum).

jinc(x)

Return jinc(x) = J1(π*x)/(2x), where J1 is a Bessel function of the first kind.

The argument x must be unitless.

mrismapbasis(mask ; kmax, kt, ki)

Construct Fourier basis for representing MRI sensitivity maps in terms of separable complex exponential signals, products of of the form basis = (k,N) -> exp.(2im * π * nfun(N) * kfun(k,N)). The default is nfun(N) = -(N÷2):(N÷2)-1 which is suitable for even N only. The default is kfun(k,N) = k / 2N, which is DCT-II like frequencies, leading to better boundary behavior than the DFT frequencies k/N.

Input

• mask::AbstractArray{Bool,D} binary support mask for region to reconstruct

Option

• kmax::Int = 5 default frequency index -kmax:kmax in all dimensions
• kfun::Function = (k,N) -> k / (2N) # DCT-II frequency
• deltas::NTuple{D,<:Number} = ones(D) pixel sizes

(For additional options kmaxs, kt, ki, T, see code.)

Output

• (; B, ν) where B is basis matrix of size count(mask) × nk where typically nk = (2*kmax+1)^D and ν is nk frequency tuples; each tuple has form ν = kfun.(Tuple(k), size(mask)) ./ deltas.

mri_smap_fit(smaps, embed ; mask, kwargs...)

Fit MRI sensitivity maps smaps using mri_smap_basis(mask ; kwargs...). Caller provides ImageGeoms.embed or equivalent.

Return named tuple (B, ν, coefs, nrmse, smaps):

• (B, ν) from mri_smap_basis
• coefs::Vector : [ncoil] each of length nk
• nrmse::Real : smaps vs smaps_fit
• smaps::Vector{Array{D}} : smaps_fit

mri_spectra(fx, fy, oa::Array{<:Object2d}, fit::NamedTuple)

Version of spectrum suitable for parallel MRI with sensitivity maps that were fit previously using mri_smap_fit. Returns a Vector of ncoil kspace data Vectors of dimension length(fx)

phantom(x::Vector, y::Vector, z::Vector, oa::Array{<:Object3d})

Return a 3D digital image of the phantom sampled at grid of (x,y,z) locations. Returned 3D image size is length(x) × length(y) × length(z).

phantom(x::Vector, y::Vector, oa::Array{<:Object2d}, oversample::Int; T)

Return a 2D digital image of the phantom sampled at grid of (x,y) locations, with over-sampling factor oversample and element type T.

phantom(x::Vector, y::Vector, oa::Array{<:Object2d})

Return a 2D digital image of the phantom sampled at grid of (x,y) locations. Returned 2D image size is length(x) × length(y).

phantom(itr, oa::Array{<:Object})

Return phantom values sampled at locations returned by generator (or iterator) itr. Returned array size matches size(itr).

phantom(oa::Array{<:Object2d})::Function

Return function of (x,y) that user can sample at any locations to make a 2D phantom image for an Array of 2D objects.

phantom(oa::Array{<:Object3d})::Function

Return function of (x,y,z) that user can sample at any locations to make a 3D phantom image for an Array of 3D objects.

phantom(ob::Object2d)::Function

Return function of (x,y) that user can sample at any locations to make a 2D phantom image for a single 2D object.

phantom(ob::Object3d)::Function

Return function of (x,y,z) that user can sample at any locations to make a 3D phantom image for a single 3D object.

image = phantom(x, y, z, oa::Array{<:Object3d}, oversample::Int; T)

Return a 3D digital image of the phantom sampled at grid of (x,y,z) locations, with over-sampling factor oversample and element type T.

phantom1(ob::Object2d{Ellipse}, (x,y))

Evaluate unit circle at (x,y), for unitless coordinates.

phantom1(ob::Object2d{Gauss2}, (x,y))

Evaluate unit gauss2 at (x,y), for unitless coordinates.

phantom1(ob::Object2d{Rect}, (x,y))

Evaluate unit square at (x,y), for unitless coordinates.

phantom1(ob::Object2d{Triangle}, (x,y))

Evaluate unit triangle at (x,y), for unitless coordinates.

phantom1(ob::Object3d{Cone}, (x,y,z))

Evaluate unit cone at (x,y,z), for unitless coordinates.

phantom1(ob::Object3d{Cuboid}, (x,y,z))

Evaluate unit cube at (x,y,z), for unitless coordinates.

phantom1(ob::Object3d{Cylinder}, (x,y,z))

Evaluate unit cylinder at (x,y,z), for unitless coordinates.

phantom1(ob::Object3d{Ellipsoid}, (x,y,z))

Evaluate unit sphere at (x,y,z), for unitless coordinates.

phantom1(ob::Object3d{Gauss3}, (x,y,z))

Evaluate Gauss3 (x,y,z), for unitless coordinates.

radon(u:Vector, v:Vector, ϕ:Vector, θ:Vector, oa::Array{<:Object3d})

Return parallel-beam projections sampled at grid of (u,v,ϕ,θ) locations. Returned array size is length(u) × length(v) × length(ϕ) × length(θ).

radon(r::Vector, ϕ::Vector, oa::Array{<:Object2d})

Return parallel-beam 2D sinogram sampled at grid of (r,ϕ) locations. Returned array size is length(r) × length(ϕ).

radon(u:Vector, v:Vector, ϕ:RealU, θ:RealU, oa::Array{<:Object3d})

Return parallel-beam projection view sampled at grid of (u,v) locations for a given (ϕ,θ) pair. Returned array size is length(u) × length(v).

radon(r::Vector, ϕ::RealU, oa::Array{<:Object2d})

Return parallel-beam projection sampled at grid of r locations for one ϕ value. Returned array size is length(r).

radon(itr, oa::Array{<:Object})

Return parallel-beam projections sampled at locations returned by generator (or iterator) itr. Returned array size matches size(itr).

radon(oa::Array{<:Object2d})::Function

Return function of (r,ϕ) that user can sample at any sinogram coordinates to make a phantom 2D sinogram for an array of 3D objects.

radon(oa::Array{<:Object3d})::Function

Return function of (u,v,ϕ,θ) that user can sample at any projection coordinates to make projection views for an array of 3D objects.

radon(ob::Object2d)::Function

Return function of (r,ϕ) that user can sample at any projection coordinates to make sinogram views of a 2D object.

The coordinate system used here is such that ϕ=0 corresponds to line integrals along the $y$ axis for an object $f(x,y)$. Then as ϕ increases, the line integrals rotate counter-clockwise.

radon(ob::Object3d)::Function

Return function of (u,v,ϕ,θ) that user can sample at any projection coordinates to make projection views of a 3D object.

The coordinate system used here is such that ϕ=0 corresponds to line integrals along the $y$ axis for an object $f(x,y,z)$. Then as ϕ increases, the line integrals rotate counter-clockwise.

radon_tri(r, sinϕ, cosϕ)

For a line integral at radial position r and angle ϕ, the locus of points along the line is {(r cos(ϕ), r sin(ϕ)) + τ (-sin(ϕ), cos(ϕ)) : τ ∈ ℝ}. This function treats the equilateral triangle with base [-1/2,1/2], pointing upwards, as the intersection of three half planes:

• H0 = {(x,y) : y ≥ 0}
• H1 = y ≤ √3 (1/2 - x)
• H2 = y ≤ √3 (1/2 + x).

Find the τ values where the line locus lies in each half planes, then take the length of the intersection of those three intervals.

For example, for H1 we have r sin(ϕ) + τ cos(ϕ) ≤ √3 (1/2 - (r cos(ϕ) - τ sin(ϕ))) or equivalently r (sin(ϕ) + √3 cos(ϕ)) - √3/2 ≤ τ (√3 sin(ϕ) - cos(ϕ)) which is a set the form b1 ≤ a1 τ, corresponding to some interval (l1,u1). Similarly for H0 and H2.

This approach might not be the most efficient, but it is simple.

See Peter Aundal Toft, "The Radon transform - theory and implementation", 1996 https://orbit.dtu.dk/en/publications/the-radon-transform-theory-and-implementation for a different approach to finding the Radon transform of a triangle.

radon_type(::Object)

Determine the element type of the Radon transform of an object (including units if applicable). Ensures that its precision is at least Float32.

rect(cx, cy, wx=1, wy=wx, ϕ=0, value::Number=1)
rect(center::NTuple{2,RealU}, width::NTuple{2,RealU}=(1,1), ϕ::RealU=0, v=1)

Construct Object{Rect} from parameters; here width is the full-width.

rotate(ob::Object2d, θ::RealU)

Rotate a 2D object.

rotate(ob::Object3d, (α,β))
rotate(ob::Object3d, α, β=0)

Rotate a 3D object.

scale(ob::Object, factor::RealU)
scale(ob::Object, factor::NTuple{D,RealU})

Scale the width(s) by factor.

image = shepp_logan(M, [N,] [case] ; options...)

Convenience method for generating M×N samples of Shepp-Logan phantoms.

In

• M::Int : horizontal size
• N::Int : vertical size, defaults to M
• case::EllipsePhantomVersion = SheppLogan()

Options

• oversample::Int = 3 (usually)
• yflip::Bool = true (reverse y samples for convenience.)
• T::Type{<:Number} default Float32 (except Int for BrainWeb version)
• kwargs... remaining options passed to ellipse_parameters for parameters.

Out

• image : M × N matrix

The default here is 3× over-sampling along both axes (9 samples per pixel), except for the SheppLoganBrainWeb phantom that consists of integer indices.

values = shepp_logan_values(::EllipsePhantomVersion)

Return 10 Shepp-Logan ellipse amplitudes for various versions.

spectrum(fx::Vector, fy::Vector, fz::Vector, oa::Array{<:Object3d})

Return 3D k-space array sampled at grid of (fx,fy,fz) locations. Returned 3D array size is length(fx) × length(fy) × length(fz).

spectrum(fx:Vector, fy:Vector, oa::Array{<:Object2d})

Return 2D k-space array sampled at grid of (fx,fy) locations. Returned 2D array size is length(fx) × length(fy).

spectrum(itr, oa::Array{<:Object})

Return spectrum of object(s) sampled at k-space locations returned by generator (or iterator) itr. Returned array size matches size(itr).

spectrum(oa::Array{<:Object2d})::Function

Return function kspace(fx,fy) that user can sample at any spatial frequency locations to evaluate the spectrum (2D Fourier transform) for an Array of 2D objects.

spectrum(oa::Array{<:Object3d})::Function

Return function kspace(fx,fy,fz) that user can sample at any spatial frequency locations to evaluate the spectrum (3D Fourier transform) for an Array of 3D objects.

spectrum(ob::Object2d, coefs::AbstractVector, f::)

Version of spectrum(ob) suitable for parallel MRI with sensitivity maps that were fit previously using mri_smap_fit for a single coil with fit coefficients coefs and frequencies f (array of tuples).

spectrum(ob::Object2d, fit::NamedTuple, coil::Int)

Version of spectrum(ob) suitable for parallel MRI with sensitivity maps that were fit previously using mri_smap_fit.

spectrum(ob::Object2d)::Function

Return function of (fx,fy) that user can sample at any spatial frequency locations to evaluate the spectrum (2D Fourier transform) of a single 2D object. Units of spatial frequencies should be the reciprocal of the units defining the object.

spectrum(ob::Object3d)::Function

Return function of (fx,fy,fz) that user can sample at any spatial frequency locations to evaluate the spectrum (3D Fourier transform) of a single 3D object. Units of spatial frequencies should be the reciprocal of the units defining the object.

spectrum1(::Object2d{Dirac2}, (kx,ky))

Spectrum of Dirac2 at (kx,ky), for unitless spatial frequency coordinates.

spectrum1(::Object2d{Ellipse}, (kx,ky))

Spectrum of unit circle at (kx,ky), for unitless spatial frequency coordinates.

spectrum1(::Object2d{Gauss2}, (kx,ky))

Spectrum of unit gauss2 at (kx,ky), for unitless spatial frequency coordinates.

spectrum1(::Object2d{Rect}, (kx,ky))

Spectrum of unit square at (kx,ky), for unitless spatial frequency coordinates.

spectrum1(ob::Object2d{Triangle}, (kx,ky))

Spectrum of unit triangle at (kx,ky), for unitless spatial frequency coordinates.

spectrum1(::Object3d{Cuboid}, (kx,ky,kz))

Spectrum of unit cube at (kx,ky,kz), for unitless spatial frequency coordinates.

spectrum(::Object3d{Cylinder}, (kx,ky,kz))

Spectrum of unit cylinder at (kx,ky,kz), for unitless spatial frequency coordinates.

spectrum1(::Object3d{Dirac3}, (kx,ky,kz))

Spectrum of Dirac3 at (kx,ky,kz), for unitless spatial frequency coordinates.

spectrum1(::Object3d{Ellipsoid}, (kx,ky,kz))

Spectrum of unit sphere at (kx,ky,kz), for unitless spatial frequency coordinates.

spectrum1(::Object3d{Gauss3}, (kx,ky,kz))

Spectrum of unit sphere at (kx,ky,kz), for unitless spatial frequency coordinates.

sphere(x, y, z, r,v=1) (sphere of radius `r` centered at `(x,y,z)`)
sphere((x,y,z), r, v=1) ditto
sphere(r, v=1) centered at origin

Construct spheres as special cases of Ellipsoid.

sphere_transform(f::Real)

Fourier transform of unit-radius sphere. The argument f is the radial coordinate in k-space and is unitless. See p253 of Bracewell 1978, The Fourier transform and its applications, or https://doi.org/10.1002/mrm.21292.

Formula: 4/3 π for f ≈ 0, otherwise (sin(2πf) - 2πf cos(2πf)) / (2 * π^2 * f^3).

square(x, y, w,v=1) (square of width `w` centered at `(x,y)`)
square((x,y), w=1, v=1) ditto
square(w, v=1) centered at origin

Construct squares as special cases of Rect.

translate(ob::Object, shift::NTuple{D,RealU})
translate(ob::Object{S,2}, xshift, yshift)
translate(ob::Object{S,3}, xshift, yshift, zshift)

Translate the center coordinates of an object by shift

trapezoid(t::Real, t1, t2, t3, t4)

Unit-height trapezoid with breakpoints t1, t2, t3, t4.

triangle(cx, cy, wx=1, wy=wx, ϕ=0, value::Number=1, param::Real=0.5)
triangle(center::NTuple{2,RealU}, width::NTuple{2,RealU}=(1,1), ϕ::RealU=0, v=1, param=0.5)

Construct Object{Triangle} from parameters. In the typical case where param=0.5 and width[1] == width[2], this is an equilateral triangle with base width[1] centered along the x axis.