Parameter estimation

Parameter estimation#

TL;DR

  • An estimator is fitted from the simulator of the acquisition it inverts, and returns one named map per unknown.

  • Dictionary matching and lookup tables search simulated signals; PERK regresses on a prior; a per-voxel fit of the model itself is bartorch’s apps.mobafit.

  • A grid grows with the product of the parameter ranges; a fit grows by one Jacobian column per parameter.

You give an estimator the simulator of your acquisition and the range of each unknown, and it returns parameter maps from a measured volume.

Capabilities#

  • Draws the training set from the simulator being inverted, and returns one named map per unknown.

    Show code and tests

    Code: fit(), map(). Tests: a mapping recovers a tissue it was trained over; the maps come back named and shaped like the volume (test_mapping.py).

  • Matches by normalized inner products, and returns M0 from the score and the stored atom norm.

    Show code and tests

    Code: DictionaryMatcher. Tests: dictionary matcher recovers complex scaled atoms (test_dictionary.py); a match answers with the density without touching an atom (test_uncertainty.py).

  • Works in a low-rank temporal subspace, fitted here or passed in, and reports the signal energy it retains.

    Show code and tests

    Code: Subspace. Tests: retained is the error it will cost (test_subspace.py); a subspace maps as well as the contrasts it replaces (test_mapping.py).

  • Clusters the dictionary and excludes whole groups before scoring their atoms.

    Show code and tests

    Code: DictionaryMatcher(groups=...). Tests: grouped matching finds what direct matching finds; most of the dictionary is ruled out (test_grouped.py).

  • Reads one unknown off a monotonic signal curve by interpolation.

    Show code and tests

    Code: LookupTable. Tests: interpolating beats the grid it was built on; a t1 map is read from the two blocks (test_lookup.py).

  • Trains a kernel ridge regression on random Fourier features.

    Show code and tests

    Code: PERK. Tests: streaming reaches what holding the dictionary reaches (test_perk.py); perk is comparable to dictionary matching under noise (test_mapping_comparison.py).

  • Takes separately measured maps, such as B1, in PERK.

    Show code and tests

    Code: fit(known=...). Tests: a known property reaches the simulator and is asked for again (test_mapping.py).

  • Returns an uncertainty map where the method defines one.

    Show code and tests

    Code: map(volume, uncertainty=True). Tests: perk states a spread without repeating itself; a method that answers with a grid point says it has none (test_uncertainty.py).

Division of responsibility#

  • Spatial regularization. Every estimator is voxel-wise; a spatial prior belongs to Model-based reconstruction.

  • Neural-network estimators. PERK’s estimate is differentiable with respect to the signal, so it can sit inside one.

  • Nonlinear least squares. A per-voxel Levenberg-Marquardt fit of the same simulator is bartorch’s apps.mobafit, over a ModelOperator; its answer depends on its starting point.

  • Declared equality constraints. Two fractions summing to one are written into the model, with one of them as the unknown.

Method#

Relative T2 error across the T2 range for five estimators, and median error against mapping time per voxel.

T2 and M0 from a 16-echo spin echo at 1 % noise. The lookup table reads the ratio of late to early echoes. On the same 24-point grid the dictionary is limited by the spacing and the table is not. PERK is least accurate at the short-T2 end of its prior. Times are measured at build time.#

Data requirements#

  • Dictionary matching: a grid. A subspace shortens each inner product to the rank; grouping reduces how many are taken. A prune set too small can exclude the group holding the match.

  • Lookup table: one unknown, and a combine that is monotonic in it, such as the MP2RAGE unified image.

  • PERK: a prior and the noise level of the scan. Mapping costs one feature projection per voxel, whatever the number of unknowns.

Nuisance parameters#

Normalization removes M0 and the receive phase from a match. M0 is recovered as the measurement norm times the score divided by the atom norm; PERK(normalize=True) regresses the reciprocal signal norm instead. A nuisance parameter that survives normalization, such as the noise floor of a magnitude image, has to be put on the grid.

Uncertainty#

PERK regresses the absolute error of its training estimates on its features, so its uncertainty includes the bias toward the prior; it holds at the signal amplitude it was trained at. It is read against crlb(), the bound for an unbiased estimator. Grid methods raise NotImplementedError when asked for one.

Examples#