Calibration, noise and coil compression

Calibration, noise and coil compression#

TL;DR

  • Sensitivity maps estimated from whitened and compressed data describe only data with the same channels, whitening, compression and voxel grid.

  • Prewhitening and coil compression are linear transforms of the channels; bartorch computes them and the estimate, and pulserver.recon holds, compares and routes the results.

  • Stored maps are reused only where every recorded field matches the unit; nothing is resampled, and a mismatch is reported.

A reconstruction from a receive-coil array needs the sensitivity of each channel, estimated from calibration data, and combines the channels with equal weight, which is the maximum-likelihood combination only where their noise is uncorrelated and of equal variance. Prewhitening makes the noise of the channels so, and coil compression reduces their number. Each of the three is expressed in a basis of the channels, and a quantity estimated in one basis does not describe data in another. This page states the model, the conditions under which pulserver.recon reuses sensitivity maps, and the interfaces that hold them. The transforms and the estimate are computed by bartorch; the reconstruction plugin passes the estimate to coil_maps().

Signal model#

The signal of each channel \(c\) of \(N_c\) is the image \(x\) weighted by the channel’s sensitivity \(S_c\) on the voxel grid, Fourier encoded, plus complex Gaussian noise \(\mathbf{n}\) of channel covariance \(\Psi = \mathrm{E}[\mathbf{n}\mathbf{n}^H]\): the encoding model bartorch’s encoding page states. A linear transform \(B\) of the channels, \(\tilde{\mathbf{y}} = B\mathbf{y}\), follows the same model with the sensitivities \(B\mathbf{S}(\mathbf{r})\), \(\mathbf{S} = (S_1, \dots, S_{N_c})\), and the noise covariance \(B\Psi B^H\). Two transforms are used:

  • prewhitening, \(B = W\) with \(W\Psi W^H = I\), after which the noise of the channels is uncorrelated with unit variance;

  • coil compression, \(B = A\), the \(n \times N_c\) matrix whose rows are the conjugated eigenvectors of the channel covariance of the calibration data that belong to its \(n\) largest eigenvalues, so that the \(n\) virtual channels are uncorrelated and ordered by decreasing power.

Data that is whitened and then compressed is transformed by \(AW\), and sensitivities estimated from it are \(AW\mathbf{S}\) on the voxel grid of the calibration k-space. A set of maps therefore describes only data with the same channels in the same order, the same whitening, the same compression and the same voxel grid at the same position and orientation.

Noise and prewhitening#

Readouts flagged IS_NOISE_MEASUREMENT sample \(\mathbf{n}\) with the RF off. From \(K\) samples the covariance is estimated as \(\hat\Psi = (K-1)^{-1}\sum_k \mathbf{n}_k\mathbf{n}_k^H\), and \(W = L^{-1}\) for its Cholesky factorisation \(\hat\Psi = LL^H\). Estimating \(\hat\Psi\) takes more noise samples than channels.

The variance of the noise in a sample is inversely proportional to the dwell time. A readout of dwell time \(\tau\) is multiplied by \(\sqrt{\tau/\tau_0}\,W\), \(\tau_0\) being the dwell time of the noise readouts, and by \(W\) alone where either dwell time is not stated. No other difference between the noise measurement and the readout is modelled.

Prewhiten takes \(W\) from the first of these:

  1. the noise readouts that precede the first readout of any other kind in the stream;

  2. the exam’s NOISE_COVARIANCE, stored by a noise series, when its coil labels are those of the header;

  3. none, in which case readouts pass unchanged and a warning is logged once, or MissingCalibration is raised for required=True.

Noise readouts after the first readout of another kind are consumed and not used. The gadget applies \(W\) on arrival, so the imaging and the calibration readouts of every unit are whitened alike.

Coil compression#

The basis \(A\) is that of the channel covariance \(\sum_k \mathbf{y}_k\mathbf{y}_k^H\) over the placed samples of the calibration k-space of the first unit the stream compresses, or of its imaging k-space where that unit holds no calibration k-space. One basis is estimated per stream, and every unit is projected onto it, so the virtual channels of successive units are the same.

If the signal of the channels lies in a subspace of dimension at most \(n\), compression discards none of it, and SENSE with compressed data and compressed maps reconstructs the image of the full set of channels to round-off. Otherwise the discarded signal is that of the \(N_c - n\) smallest eigenvalues. Maps estimated from uncompressed data are in another basis than compressed data, and are not transformed into this one.

Coil sensitivity maps#

Calibration k-space#

The calibration k-space of a unit is ReconData.ref. It comes from one of three arrangements.

Arrangement

Readouts

Maps are estimated by

Embedded calibration

Lines of the imaging space flagged IS_PARALLEL_CALIBRATION_AND_IMAGING

The imaging unit, which holds both data and ref

Calibration unit

Readouts flagged IS_PARALLEL_CALIBRATION in a unit with no imaging readouts, as in an encoding space of their own

That unit, whose data.data is None; the imaging units of the same slice take the maps from the stream

Calibration series

A series of its own

The reconstruction of that series, which stores the maps in the exam

The estimate receives the calibration k-space of a Cartesian encoding space zero-filled onto the whole grid of the space, the lines its readouts cover being at their grid positions, and returns maps on that grid. For a non-Cartesian space coil_maps() raises ValueError.

Resolution#

coil_maps() returns the maps of a unit from the first of these sources, the location being the unit’s slice counter:

  1. The unit’s own calibration k-space, from which the maps are estimated and stored for the slice, replacing any stored there.

  2. The maps stored in the stream for the slice.

  3. The exam’s COIL_SENSITIVITIES, which holds one set of maps.

  4. No source: MissingCalibration is raised, or None returned for required=False.

The slice is the location because a calibration unit and the imaging units that use its maps may be in different encoding spaces. A unit that needs maps before the unit that estimates them has closed finds none in the first three sources.

Reuse#

Stored maps are returned only to a unit for which every field of the CoilSensitivities is equal to that of the unit. The fields are compared in this order, and the first that differs is reported.

Field

Equal to

coils

The coil labels of the header in channel order

noise

The identifier of the prewhitening in context.noise, or None on both sides

basis

The identifier of the basis in context.coil_compression, or None on both sides

geometry.frame_of_reference

The frameOfReferenceUID of the header

geometry.table_position, geometry.position, geometry.read_dir, geometry.phase_dir, geometry.slice_dir

The same fields of the reference acquisition of the unit’s k-space, in the units the acquisition states

geometry.fov, geometry.matrix

The field of view in m and the matrix of the reconstruction space of the unit’s encoding space

geometry.grid

The shape of the voxel axes of the whole k-space of the unit’s encoding space

method and source, the estimate and the measurementID of the series that measured the maps, are recorded and appear in a refusal; they are not compared. The comparison establishes that the maps and the unit share the listed quantities. It does not test the accuracy of the maps.

Nothing is resampled, regridded or resliced to make maps fit. Maps on a grid or at a position other than the unit’s are not returned, and the message of MissingCalibration names each source and why it was not used:

no coil sensitivities for slice 0; sources consulted:
  this unit: it holds no calibration readouts
  this series, slice 0: no maps stored
  this exam: geometry.matrix differs: stored (96, 96), this unit (192, 192) (estimated by nlinv_maps from measurement 7)

Software abstraction#

Quantity

Interface

State

Whitening \(W\)

Prewhiten, a gadget

context.noise; the exam key NOISE_COVARIANCE

Compression basis \(A\)

CoilCompression, called from recon

context.coil_compression

Maps \(\mathbf{S}\)

coil_maps(), called from recon; CoilSensitivities

context.coil_maps by slice; the exam key COIL_SENSITIVITIES, read and written as context.coil_sensitivities

The state in context belongs to one stream. What a series leaves to the next is what it stores in context.exam under the exam keys, which Raw-data enrichment and routing describes. A calibration series publishes its whitening with Prewhiten.publish and its maps by assigning context.coil_sensitivities.

Whitening runs on arrival, in the gadgets of the plugin. Within recon, a unit is compressed before its maps are requested, because the maps record the basis in force when they are estimated. The whitening, the compression basis and the estimate are bartorch’s (tools.whiten, tools.cc and tools.ccapply, and the estimate the plugin names, such as apps.nlinv_maps); the interfaces above hold their results, compare them and route them between units and series. Reconstruction plugins shows the calls.

See also#