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
SMap	Number of k-space samples per shot.
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.

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	Array of per-shot encoding operator pointers.

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.

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

Number of k-space samples per shot.
Number of image pixels.
Number of receiver coils.
Number of shots.
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.

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)

Array of per-shot encoding operator pointers.
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¶

Nx¶

Ny¶

Nz¶

PMap¶

SMap¶

Tvec¶

conjExpiPMap¶

conjSMap¶

expiPMap¶

i¶

type¶

Operator Details¶

operator*¶

operator/¶

Function Details¶

pcSENSE¶

~pcSENSE¶

FMap ¶

Ix ¶

Iy ¶

Iz ¶

Kx ¶

Ky ¶

Kz ¶

L ¶

Nx ¶

Ny ¶

Nz ¶

PMap ¶

SMap ¶

Tvec ¶

conjExpiPMap ¶

conjSMap ¶

expiPMap ¶

i ¶

type ¶

pcSENSE ¶

~pcSENSE ¶