Gdft

template <typename T1> class Gdft

Non-uniform field-corrected discrete Fourier transform (DFT) operator.

Computes the MRI signal model \(d = G x\) where \(x\) is the image and \(d\) is the k-space data. Supports optional off-resonance correction via a field map FM and a per-sample time vector t.

The forward model is:

\[d_j = \sum_k x_k \, e^{-i(2\pi \mathbf{k}_j \cdot \mathbf{r}_k + \omega_k t_j)}\]

where \(\omega_k\) = FM[k] is the field map in rad/s, \(\mathbf{k}_j\) are k-space coordinates (cycles/FOV), \(\mathbf{r}_k\) are image-space coordinates, and \(t_j\) are readout times (s).

Use G * x for the forward transform and G / d for the adjoint.

See : GdftR2 for extension with R2* gradient maps.

See : Gnufft for the fast gridding-based NUFFT alternative.

See : Fessler & Sutton, "Nonuniform Fast Fourier Transforms Using Min-Max Interpolation," IEEE TSP, 2003. https://doi.org/10.1109/TSP.2002.807005

Col<float> kx(4096), ky(4096), kz(4096, fill::zeros);
Col<float> ix(4096), iy(4096), iz(4096, fill::zeros);
Col<float> FM(4096, fill::zeros);  // field map (rad/s)
Col<float> t(4096, fill::zeros);   // readout times (s)
Gdft<float> G(4096, 4096, kx, ky, kz, ix, iy, iz, FM, t);
Col<cx_float> kdata = G * image;   // forward
Col<cx_float> recon = G / kdata;   // adjoint

T1 : Floating-point precision type (float or double).

Variables

Name	Description
kx	Number of k-space samples (output length of forward transform).
ky	k-space y-coordinates (length n1).
kz	k-space z-coordinates (length n1).
ix	Image-space x-coordinates (length n2).
iy	Image-space y-coordinates (length n2).
iz	Image-space z-coordinates (length n2).
FM	Off-resonance field map in rad/s (length n2).
t	Per-sample readout time vector in seconds (length n1).
metalCtx	Opaque Metal DFT context handle (MetalDFTContext*); nullptr when T1 != float.

Operators

Name	Description
operator*	Forward DFT: image -> k-space. Computes \(d_j = \sum_k x_k \, e^{-i(2\pi \mathbf{k}_j \cdot \mathbf{r}_k + \omega_k t_j)}\). d : Input image vector of length n2. Return : Output k-space vector of length n1.
operator/	Adjoint DFT: k-space -> image. Computes the conjugate transpose of the forward operator (\(G^H\)). d : Input k-space vector of length n1. Return : Output image vector of length n2.
operator*	Forward DFT (forgeCol overload for Metal path).
operator/	Adjoint DFT (forgeCol overload for Metal path).

Functions

Name	Description
Gdft	Default constructor.
Gdft	Construct a Gdft operator with the given trajectory and field map.
~Gdft	Destructor — releases the Metal DFT context if allocated.

Variable Details

FM

forgeCol<T1> FM

Off-resonance field map in rad/s (length n2).

ix

forgeCol<T1> ix

Image-space x-coordinates (length n2).

iy

forgeCol<T1> iy

Image-space y-coordinates (length n2).

iz

forgeCol<T1> iz

Image-space z-coordinates (length n2).

kx

forgeCol<T1> kx

Number of k-space samples (output length of forward transform).
Number of image pixels (input length of forward transform).
k-space x-coordinates (length n1).

ky

forgeCol<T1> ky

k-space y-coordinates (length n1).

kz

forgeCol<T1> kz

k-space z-coordinates (length n1).

metalCtx

void* metalCtx

Opaque Metal DFT context handle (MetalDFTContext*); nullptr when T1 != float.

t

forgeCol<T1> t

Per-sample readout time vector in seconds (length n1).

Operator Details

operator*

Col<CxT1> operator*(const Col<CxT1>& d) const

Forward DFT: image -> k-space.

Computes \(d_j = \sum_k x_k \, e^{-i(2\pi \mathbf{k}_j \cdot \mathbf{r}_k + \omega_k t_j)}\).

d : Input image vector of length n2.

Return : Output k-space vector of length n1.

forgeCol<forgeComplex<T1>> operator*(const forgeCol<forgeComplex<T1>>& d) const

Forward DFT (forgeCol overload for Metal path).

d : Input image vector of length n2.

Return : Output k-space vector of length n1.

operator/

Col<CxT1> operator/(const Col<CxT1>& d) const

Adjoint DFT: k-space -> image.

Computes the conjugate transpose of the forward operator (\(G^H\)).

d : Input k-space vector of length n1.

Return : Output image vector of length n2.

forgeCol<forgeComplex<T1>> operator/(const forgeCol<forgeComplex<T1>>& d) const

Adjoint DFT (forgeCol overload for Metal path).

d : Input k-space vector of length n1.

Return : Output image vector of length n2.

Function Details

Gdft

Gdft()

Default constructor. Produces an uninitialized operator.

Gdft(uword a, uword b, const Col<T1>& k1, const Col<T1>& k2, const Col<T1>& k3, const Col<T1>& i1, const Col<T1>& i2, const Col<T1>& i3, const Col<T1>& f1, const Col<T1>& t1)

Construct a Gdft operator with the given trajectory and field map.

a : Number of k-space samples (n1, output size of forward transform).

b : Number of image pixels (n2, input size of forward transform).

k1 : k-space x-coordinates, length a (units: cycles/FOV, range [-N/2, N/2)).

k2 : k-space y-coordinates, length a.

k3 : k-space z-coordinates, length a (set to zeros for 2-D).

i1 : Image-space x-coordinates, length b (units: fraction of FOV, range [0, 1)).

i2 : Image-space y-coordinates, length b.

i3 : Image-space z-coordinates, length b (set to zeros for 2-D).

f1 : Off-resonance field map, length b (units: rad/s).

t1 : Per-sample readout time vector, length a (units: s).

~Gdft

~Gdft()

Destructor — releases the Metal DFT context if allocated.