pcSENSE¶

template <typename T1> class pcSENSE

Phase-corrected multi-shot SENSE operator.

Extends the standard SENSE model with per-shot phase maps to correct for inter-shot phase inconsistencies in multi-shot non-Cartesian acquisitions. Uses a per-shot Gdft encoding operator to handle arbitrary trajectories.

The per-shot phase-corrected forward model is:

\[\mathbf{d}_{c,s} = G_s(P_s \circ S_c \circ \mathbf{x})\]

where \(P_s = e^{i\phi_s(\mathbf{r})}\) is the shot-specific phase map for shot \(s\), and \(G_s\) is the encoding operator for that shot's k-space trajectory.

Forward (per coil, per shot):

graph LR
    X["$$\mathbf{x}$$"] --> P["$$P_s \circ \mathbf{x}$$"]
    P --> S["$$S_c \circ$$"]
    S --> G["$$G_s$$"]
    G --> D["$$\mathbf{d}_{c,s}$$"]

Adjoint (combine all coils and shots):

graph RL
    D["$$\mathbf{d}_{c,s}$$"] --> GH["$$G_s^H$$"]
    GH --> SC["$$S_c^* \circ$$"]
    SC --> PC["$$P_s^* \circ$$"]
    PC --> SUM["$$\sum_{c,s}$$"]
    SUM --> XH["$$\hat{\mathbf{x}}$$"]

The adjoint sums over all coils and shots:

\[\hat{\mathbf{x}} = \sum_{s=1}^{N_s} \sum_{c=1}^{N_c} P_s^* \circ S_c^* \circ (G_s^H \mathbf{d}_{c,s})\]

Unlike SENSE, which assumes a single k-space trajectory for all data, pcSENSE handles multi-shot acquisitions (e.g., diffusion-weighted imaging) where each shot has a different \(B_0\)-induced phase error. This is essential for techniques such as navigated multi-shot DWI with non-Cartesian sampling.

// Per-shot k-space coordinates concatenated, readout times, maps
pcSENSE<float> P(kx, ky, kz, Nx, Ny, Nz, nc, tvec,
                  SENSEmap, fieldMap, shotPhaseMap);
Col<cx_float> kdata  = P * image;   // forward
Col<cx_float> recon  = P / kdata;   // adjoint

See : Liu, Moseley & Bammer, "Simultaneous phase correction and SENSE reconstruction for navigated multi-shot DWI with non-cartesian k-space sampling," MRM, 2005. https://doi.org/10.1002/mrm.20706

See : Holtrop & Sutton, "High spatial resolution diffusion weighted imaging on clinical 3T MRI scanners using multislab spiral acquisitions," J Med Imaging, 2016. https://doi.org/10.1117/1.JMI.3.2.023501

See : SENSE

See : pcSenseTimeSeg

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

Variables¶

Name	Description
Nd	Number of k-space samples per shot.
Ni	Number of image pixels.
Nc	Number of receiver coils.
Ns	Number of shots.
SMap	Coil sensitivity maps, size Ni x Nc (single source of truth, GPU-ready).
conjSMap	Conjugate sensitivity maps, size Ni x Nc.
PMap	Per-shot phase maps in radians, size Ni x Ns.
expiPMap	Complex exponential of phase maps (exp(i*PMap)), size Ni x Ns.
conjExpiPMap	Conjugate complex exponential of phase maps (exp(-i*PMap)), size Ni x Ns.
FMap	Off-resonance field map (rad/s), length Ni.
Kx	k-space x-coordinates matrix, size (samples/shot) x Ns.
Ky	k-space y-coordinates matrix, size (samples/shot) x Ns.
Kz	k-space z-coordinates matrix, size (samples/shot) x Ns.
Tvec	Readout time vector (s), size (samples/shot) x Ns.
Nx	Image size in x.
Ny	Image size in y.
Nz	Image size in z.
Ix	Image-space x-coordinates, length Ni.
Iy	Image-space y-coordinates, length Ni.
Iz	Image-space z-coordinates, length Ni.
i	Imaginary unit constant for phase computations.
type	Interpolation type: 1 = Hanning, 2 = exact min-max.
L	Number of time segments for field-map correction.
combinedMap_gpu	Array of per-shot encoding operator pointers.
conjCombinedMap_gpu	Conjugate combined maps for adjoint.
combinedMapCols	Precomputed per-(coil,shot) combined maps SMap.col(ii) % exp(+iPMap.col(jj)), stored as independent page-aligned forgeCol columns for zero-copy Metal weighting. Index col = jj + ii Ns (matches the k-space column layout and the CUDA combinedMap_gpu convention).
conjCombinedMapCols	Conjugate counterparts for the adjoint (exp(-i*PMap) weighting).

Operators¶

Name	Description
operator*	Forward phase-corrected SENSE transform: image -> k-space. `d` : Input image vector of length Ni. Return : Stacked k-space vector of length NdNsNc.
operator/	Adjoint phase-corrected SENSE transform: k-space -> image. `d` : Input k-space vector of length NdNsNc. Return : Output image vector of length Ni.
operator*	Forward phase-corrected SENSE transform (forgeCol overload for Metal path).
operator/	Adjoint phase-corrected SENSE transform (forgeCol overload for Metal path).

Functions¶

Name	Description
pcSENSE	Default constructor.
~pcSENSE	Destructor — frees per-shot Gdft encoding operator array.
pcSENSE	Construct a phase-corrected SENSE operator.

Variable Details¶

FMap ¶

Col<T1> FMap

Off-resonance field map (rad/s), length Ni.

Ix ¶

Col<T1> Ix

Image-space x-coordinates, length Ni.

Iy ¶

Col<T1> Iy

Image-space y-coordinates, length Ni.

Iz ¶

Col<T1> Iz

Image-space z-coordinates, length Ni.

Kx ¶

Mat<T1> Kx

k-space x-coordinates matrix, size (samples/shot) x Ns.

Ky ¶

Mat<T1> Ky

k-space y-coordinates matrix, size (samples/shot) x Ns.

Kz ¶

Mat<T1> Kz

k-space z-coordinates matrix, size (samples/shot) x Ns.

L ¶

uword L

Number of time segments for field-map correction.

Nc ¶

uword Nc

Number of receiver coils.

Nd ¶

uword Nd

Number of k-space samples per shot.

Ni ¶

uword Ni

Number of image pixels.

Ns ¶

uword Ns

Number of shots.

Nx ¶

uword Nx

Image size in x.

Ny ¶

uword Ny

Image size in y.

Nz ¶

uword Nz

Image size in z.

PMap ¶

Mat<T1> PMap

Per-shot phase maps in radians, size Ni x Ns.

SMap ¶

forgeMat<forgeComplex<T1>> SMap

Coil sensitivity maps, size Ni x Nc (single source of truth, GPU-ready).

Tvec ¶

Mat<T1> Tvec

Readout time vector (s), size (samples/shot) x Ns.

combinedMapCols ¶

std::vector<forgeCol<forgeComplex<T1>>> combinedMapCols

Precomputed per-(coil,shot) combined maps SMap.col(ii) % exp(+i*PMap.col(jj)), stored as independent page-aligned forgeCol columns for zero-copy Metal weighting. Index col = jj + ii * Ns (matches the k-space column layout and the CUDA combinedMap_gpu convention).

combinedMap_gpu ¶

std::vector<forgeCol<forgeComplex<T1>>> combinedMap_gpu

Array of per-shot encoding operator pointers. Per-shot per-coil combined (SMap * expiPMap) maps, pre-uploaded to GPU. Index: jj + ii * Ns (shots vary faster than coils).

conjCombinedMapCols ¶

std::vector<forgeCol<forgeComplex<T1>>> conjCombinedMapCols

Conjugate counterparts for the adjoint (exp(-i*PMap) weighting).

conjCombinedMap_gpu ¶

std::vector<forgeCol<forgeComplex<T1>>> conjCombinedMap_gpu

Conjugate combined maps for adjoint.

conjExpiPMap ¶

forgeMat<forgeComplex<T1>> conjExpiPMap

Conjugate complex exponential of phase maps (exp(-i*PMap)), size Ni x Ns.

conjSMap ¶

forgeMat<forgeComplex<T1>> conjSMap

Conjugate sensitivity maps, size Ni x Nc.

expiPMap ¶

forgeMat<forgeComplex<T1>> expiPMap

Complex exponential of phase maps (exp(i*PMap)), size Ni x Ns.

i ¶

CxT1 i

Imaginary unit constant for phase computations.

type ¶

uword type

Interpolation type: 1 = Hanning, 2 = exact min-max.

Operator Details¶

operator*¶

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

Forward phase-corrected SENSE transform: image -> k-space.

d : Input image vector of length Ni.

Return : Stacked k-space vector of length NdNsNc.

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

Forward phase-corrected SENSE transform (forgeCol overload for Metal path).

d : Input image vector of length Ni.

Return : Stacked k-space vector of length NdNsNc.

operator/¶

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

Adjoint phase-corrected SENSE transform: k-space -> image.

d : Input k-space vector of length NdNsNc.

Return : Output image vector of length Ni.

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

Adjoint phase-corrected SENSE transform (forgeCol overload for Metal path).

d : Input k-space vector of length NdNsNc.

Return : Output image vector of length Ni.

Function Details¶

pcSENSE ¶

pcSENSE()

Default constructor.

pcSENSE(Col<T1> kx, Col<T1> ky, Col<T1> kz, uword nx, uword ny, uword nz, uword nc, Col<T1> t, Col<CxT1> SENSEmap, Col<T1> FieldMap, Col<T1> ShotPhaseMap)

Construct a phase-corrected SENSE operator.

kx : k-space x-coordinates (all shots concatenated), length Nd*Ns.

ky : k-space y-coordinates, length Nd*Ns.

kz : k-space z-coordinates, length Nd*Ns.

nx : Image size in x.

ny : Image size in y.

nz : Image size in z.

nc : Number of receiver coils.

t : Per-sample readout time vector (s), length Nd*Ns.

SENSEmap : Sensitivity maps as flat vector, length Ni*Nc.

FieldMap : Off-resonance field map (rad/s), length Ni.

ShotPhaseMap : Per-shot phase maps in radians, length Ni*Ns.

~pcSENSE ¶

~pcSENSE()

Destructor — frees per-shot Gdft encoding operator array.

pcSENSE¶

Variables¶

Operators¶

Functions¶

Variable Details¶

FMap¶

Ix¶

Iy¶

Iz¶

Kx¶

Ky¶

Kz¶

L¶

Nc¶

Nd¶

Ni¶

Ns¶

Nx¶

Ny¶

Nz¶

PMap¶

SMap¶

Tvec¶

combinedMapCols¶

combinedMap_gpu¶

conjCombinedMapCols¶

conjCombinedMap_gpu¶

conjExpiPMap¶

conjSMap¶

expiPMap¶

i¶

type¶

Operator Details¶

operator*¶

operator/¶

Function Details¶

pcSENSE¶

~pcSENSE¶

FMap ¶

Ix ¶

Iy ¶

Iz ¶

Kx ¶

Ky ¶

Kz ¶

L ¶

Nc ¶

Nd ¶

Ni ¶

Ns ¶

Nx ¶

Ny ¶

Nz ¶

PMap ¶

SMap ¶

Tvec ¶

combinedMapCols ¶

combinedMap_gpu ¶

conjCombinedMapCols ¶

conjCombinedMap_gpu ¶

conjExpiPMap ¶

conjSMap ¶

expiPMap ¶

i ¶

type ¶

pcSENSE ¶

~pcSENSE ¶