sporco.fista.cbpdn

Classes for FISTA algorithm for the Convolutional BPDN problem

Classes

ConvBPDN(*args, **kwargs)	Base class for FISTA algorithm for the Convolutional BPDN (CBPDN) [26] problem.
ConvBPDNMask(*args, **kwargs)	FISTA algorithm for Convolutional BPDN with a spatial mask.

Class Descriptions

class sporco.fista.cbpdn.ConvBPDN(*args, **kwargs)

Bases: sporco.fista.fista.FISTADFT

Base class for FISTA algorithm for the Convolutional BPDN (CBPDN) [26] problem.

The generic problem form is

\[\mathrm{argmin}_\mathbf{x} \; f( \{ \mathbf{x}_m \} ) + \lambda g( \{ \mathbf{x}_m \} )\]

where \(f = (1/2) \left\| \sum_m \mathbf{d}_m * \mathbf{x}_m - \mathbf{s} \right\|_2^2\), and \(g(\cdot)\) is a penalty term or the indicator function of a constraint; with input image \(\mathbf{s}\), dictionary filters \(\mathbf{d}_m\), and coefficient maps \(\mathbf{x}_m\). It is solved via the FISTA formulation

Proximal step

\[\mathbf{x}_k = \mathrm{prox}_{t_k}(g) (\mathbf{y}_k - 1/L \nabla f(\mathbf{y}_k) ) \;\;.\]

Combination step

\[\mathbf{y}_{k+1} = \mathbf{x}_k + \left( \frac{t_k - 1}{t_{k+1}} \right) (\mathbf{x}_k - \mathbf{x}_{k-1}) \;\;,\]

with \(t_{k+1} = \frac{1 + \sqrt{1 + 4 t_k^2}}{2}\).

After termination of the solve method, attribute itstat is a list of tuples representing statistics of each iteration. The fields of the named tuple IterationStats are:

Iter : Iteration number

ObjFun : Objective function value

DFid : Value of data fidelity term \((1/2) \| \sum_m \mathbf{d}_m * \mathbf{x}_m - \mathbf{s} \|_2^2\)

RegL1 : Value of regularisation term \(\sum_m \| \mathbf{x}_m \|_1\)

Rsdl : Residual

L : Inverse of gradient step parameter

Time : Cumulative run time

This class supports an arbitrary number of spatial dimensions, dimN, with a default of 2. The input dictionary D is either dimN + 1 dimensional, in which case each spatial component (image in the default case) is assumed to consist of a single channel, or dimN + 2 dimensional, in which case the final dimension is assumed to contain the channels (e.g. colour channels in the case of images). The input signal set S is either dimN dimensional (no channels, only one signal), dimN + 1 dimensional (either multiple channels or multiple signals), or dimN + 2 dimensional (multiple channels and multiple signals). Determination of problem dimensions is handled by cnvrep.CSC_ConvRepIndexing.

Parameters

Darray_like

Dictionary array

Sarray_like

Signal array

lmbdafloat

Regularisation parameter

optConvBPDN.Options object

Algorithm options

dimK0, 1, or None, optional (default None)

Number of dimensions in input signal corresponding to multiple independent signals

dimNint, optional (default 2)

Number of spatial/temporal dimensions

class Options(opt=None)

Bases: sporco.fista.fista.FISTADFT.Options

ConvBPDN algorithm options

Options include all of those defined in fista.FISTADFT.Options, together with additional options:

NonNegCoef : Flag indicating whether to force solution to be non-negative.

NoBndryCross : Flag indicating whether all solution coefficients corresponding to filters crossing the image boundary should be forced to zero.

L1Weight : An array of weights for the \(\ell_1\) norm. The array shape must be such that the array is compatible for multiplication with the X/Y variables. If this option is defined, the regularization term is \(\lambda \sum_m \| \mathbf{w}_m \odot \mathbf{x}_m \|_1\) where \(\mathbf{w}_m\) denotes slices of the weighting array on the filter index axis.

Parameters

optdict or None, optional (default None)

ConvBPDN algorithm options

setdict(D=None)

Set dictionary array.

getcoef()

Get final coefficient array.

eval_grad()

Compute gradient in Fourier domain.

eval_Rf(Vf)

Evaluate smooth term in Vf.

eval_proxop(V)

Compute proximal operator of \(g\).

rsdl()

Compute fixed point residual in Fourier domain.

eval_objfn()

Compute components of objective function as well as total contribution to objective function.

obfn_dfd()

Compute data fidelity term \((1/2) \| \sum_m \mathbf{d}_m * \mathbf{x}_m - \mathbf{s} \|_2^2\). This function takes into account the unnormalised DFT scaling, i.e. given that the variables are the DFT of multi-dimensional arrays computed via rfftn, this returns the data fidelity term in the original (spatial) domain.

obfn_reg()

Compute regularisation term and contribution to objective function.

obfn_f(Xf=None)

Compute data fidelity term \((1/2) \| \sum_m \mathbf{d}_m * \mathbf{x}_m - \mathbf{s} \|_2^2\) This is used for backtracking. Since the backtracking is computed in the DFT, it is important to preserve the DFT scaling.

reconstruct(X=None)

Reconstruct representation.

class sporco.fista.cbpdn.ConvBPDNMask(*args, **kwargs)

Bases: sporco.fista.cbpdn.ConvBPDN

FISTA algorithm for Convolutional BPDN with a spatial mask.

Solve the optimisation problem

\[\mathrm{argmin}_\mathbf{x} \; (1/2) \left\| W \left(\sum_m \mathbf{d}_m * \mathbf{x}_m - \mathbf{s}\right) \right\|_2^2 + \lambda \sum_m \| \mathbf{x}_m \|_1 \;\;,\]

where \(W\) is a mask array.

See ConvBPDN for interface details.

Parameters

Darray_like

Dictionary matrix

Sarray_like

Signal vector or matrix

lmbdafloat

Regularisation parameter

Warray_like

Mask array. The array shape must be such that the array is compatible for multiplication with input array S (see cnvrep.mskWshape for more details).

optConvBPDNMask.Options object

Algorithm options

dimK0, 1, or None, optional (default None)

Number of dimensions in input signal corresponding to multiple independent signals

dimNint, optional (default 2)

Number of spatial dimensions

eval_grad()

Compute gradient in Fourier domain.

obfn_dfd()

Compute data fidelity term \((1/2) \| W (\sum_m \mathbf{d}_m * \mathbf{x}_{m} - \mathbf{s}) \|_2^2\)

obfn_f(Xf=None)

Compute data fidelity term \((1/2) \| W (\sum_m \mathbf{d}_m * \mathbf{x}_{m} - \mathbf{s}) \|_2^2\). This is used for backtracking. Since the backtracking is computed in the DFT, it is important to preserve the DFT scaling.