GdftR2

template <typename T1> class GdftR2

Non-uniform DFT operator extended with R2* (transverse relaxation) gradient maps.

Extends Gdft with spatial gradients of the field map, enabling a first-order Taylor expansion of the off-resonance phase term about each voxel center. The standard DFT signal model assumes a spatially constant field within each voxel; when \(R_2^*\) decay or \(\omega\) varies significantly across the FOV, intravoxel dephasing causes signal loss that the zeroth-order model cannot capture.

GdftR2 corrects this by including gradient terms \((G_x, G_y, G_z)\):

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

where \(\nabla\omega_k = (G_x, G_y, G_z)_k\) is the spatial gradient of the field map at voxel \(k\) (computed by calcGradientMaps) and \(\Delta\mathbf{r}\) is the displacement within the voxel.

When to use GdftR2 vs Gdft: prefer GdftR2 when \(R_2^*\) decay or the off-resonance field varies significantly across the FOV (e.g., near air-tissue interfaces). For slowly varying field maps, the simpler Gdft is sufficient and faster.

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

See : Gdft for the base DFT without gradients.

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)
int Nx = 64, Ny = 64, Nz = 1;
GdftR2<float> G(4096, 4096, kx, ky, kz, ix, iy, iz, FM, t, Nx, Ny, Nz);
// Gradient maps are computed internally from FM via calcGradientMaps
Col<cx_float> kdata = G * image;   // forward with R2* gradients
Col<cx_float> recon = G / kdata;   // adjoint with R2* gradients

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).
Gx	R2* gradient map in the x-direction (length n2).
Gy	R2* gradient map in the y-direction (length n2).
Gz	R2* gradient map in the z-direction (length n2).
t	Per-sample readout time vector in seconds (length n1).
numX	Image x-dimension in pixels.
metalCtx	Opaque Metal DFT context handle (MetalDFTContext*); nullptr when T1 != float.

Operators

Name	Description
operator*	Forward DFT with R2 gradients: image -> k-space. d : Input image vector of length n2. Return* : Output k-space vector of length n1.
operator/	Adjoint DFT with R2 gradients: k-space -> image. d : Input k-space vector of length n1. Return* : Output image vector of length n2.
operator*	Forward DFT with R2* gradients (forgeCol overload for Metal path).
operator/	Adjoint DFT with R2* gradients (forgeCol overload for Metal path).

Functions

Name	Description
GdftR2	Default constructor.
GdftR2	Construct a GdftR2 operator with trajectory, field map, and image dimensions.
~GdftR2	Destructor — releases the Metal DFT context if allocated.
calcGradientMaps	Compute R2* gradient maps from the field map using finite differences.
Cd	Forward finite difference operator along dimension dim. d : Input vector. dim : Spatial dimension (0 = x, 1 = y, 2 = z). Return : Forward difference of d along dim.

Variable Details

FM

forgeCol<T1> FM

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

Gx

Col<T1> Gx

R2* gradient map in the x-direction (length n2).

Gy

Col<T1> Gy

R2* gradient map in the y-direction (length n2).

Gz

Col<T1> Gz

R2* gradient map in the z-direction (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.

numX

int numX, numY, numZ

Image x-dimension in pixels.
Image y-dimension in pixels.
Image z-dimension in pixels.

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 with R2* gradients: image -> k-space.

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 with R2* gradients (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 with R2* gradients: k-space -> image.

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 with R2* gradients (forgeCol overload for Metal path).

d : Input k-space vector of length n1.

Return : Output image vector of length n2.

Function Details

Cd

Col<T1> Cd(const Col<T1>& d, uword dim) const

Forward finite difference operator along dimension `dim.`

d : Input vector.

dim : Spatial dimension (0 = x, 1 = y, 2 = z).

Return : Forward difference of d along dim.

GdftR2

GdftR2()

Default constructor. Produces an uninitialized operator.

GdftR2(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, const int numX, const int numY, const int numZ)

Construct a GdftR2 operator with trajectory, field map, and image dimensions.

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).

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

k3 : k-space z-coordinates, length a.

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

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

i3 : Image-space z-coordinates, length b.

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

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

numX : Image x-dimension (pixels).

numY : Image y-dimension (pixels).

numZ : Image z-dimension (pixels; use 1 for 2-D).

calcGradientMaps

void calcGradientMaps(Col<T1>& Gx, Col<T1>& Gy, Col<T1>& Gz)

Compute R2* gradient maps from the field map using finite differences.

Populates Gx, Gy, and Gz with the spatial gradients of the field map.

Gx : Output: x-gradient of the field map (length n2).

Gy : Output: y-gradient of the field map (length n2).

Gz : Output: z-gradient of the field map (length n2).

~GdftR2

~GdftR2()

Destructor — releases the Metal DFT context if allocated.