mimo

multiple-input-multiple-output

Syntax
Description
Example
Input Arguments
Output Arguments
Best Practice
More About
See Also
Code: mimo
Code: mimocalc

Syntax

[FFqmF,ADFFq] = mimo(seti,qROI,FmeasDelta)
[FFqmF,ADFFq] = mimo(seti,qROI,FmeasDelta,'adjOfDer')
[FFqMeas,~,FFqROI] = mimo(seti,qROI,'simo')
[FFqMeas,ADFFq,FFqROI] = mimo(seti,qROI,'simo')
[JA,JB] = mimo(seti,qROI,'jacobian')

Description

Adjoint of derivative ('adjOfDer')

[FFqmF, ADFFq] = mimo(seti,qROI,FmeasDelta)

and

[FFqmF, ADFFq] = mimo(seti,qROI,FmeasDelta,'adjOfDer')

computes the difference of the evalutated forward operator and the measured data, FFqmF, and
computes the adjoint of derivative, ADFFq.
You can leave 'adjOfDer' because it is used automatically if the last argument op is empty.

Single-input-multiple-output ('simo')

[FFqMeas,~,FFqROI] = mimo(seti,qROI,'simo')

computes the scattered field evaluated on receivers' positions for each transmitter, FFqMeas, and
the scattered field on ROI for each transmitter, FFqROI.
~ is the adjoint of derivative, ADFFq, but this is uninteresting because it is 0. (This is expected, because simo is used with input of FmeasDelta = 0 in mimocalc.)

Auxilary matrices JA and JB to compute the Jacobian matrix ('jacobian')

[JA,JB] = mimo(seti,qROI,'jacobian')

computes auxiliary matrices JA and JB, such that the Frechet derivative of $\mathcal{F}$ at $q$ evaluated on $h$ , i.e. $\mathcal{F}'(q)[h]$ , that is in finite-dimensional spaces is represented by the Jacobian matrix of $\mathcal{F}$ at $q$ evaluted on $h$ can easily computed by

$\texttt{DFF} = \mathcal{F}'(q)[h] = \texttt{JA*diag(h)*JB}$

Example

We give one example to compute the difference of the evalutated forward operator and the measured data, FFqmF, and the adjoint of derivative, ADFFq.

init;
seti.contrast = 'corner2D';
seti = setGeomSim(seti);
qROI = seti.qROIexact;
FmeasDelta = zeros(seti.measNb,seti.incNb);
[FFqmF,ADFFq] = mimo(seti,qROI,FmeasDelta,'adjOfDer');
figure(1); imagesc(real(FFqmF)); axis xy; colorbar;
figure(2); imagesc(real(seti.G(ADFFq))); axis xy; colorbar;

Input Arguments

seti : struct, details see below.
qROI : contrast in ROI (region of interest) (matrix as vector of size seti.nROI^seti.dim x 1)
FmeasDelta : $\mathcal{F}_\mathrm{meas}^\delta$ measurement data (on receivers) with noise level $\delta$ (matrix of size seti.measNb x seti.incNb)

struct seti

As in convenience functions adjOfDer.html, forward.html and derivative.html all necessary fields (and more) are set by

seti = setGeomSim(seti);

A list of all necessary (and not more) fields is below.

seti in setGeomSim.html:

seti.mCD : internal parameter, set seti.mCD = 0 (no coarse grid is used)
seti.tol : internal parameter, set 1E-6, is used in simo and solveLippmannSchwinger

seti in setGrid.html:

seti.dim : dimension of the problem (2 or 3)
seti.nCD : number of discretization points for each dimension of computational domain (CD)
seti.nROI : discretization points for each dimension of region of interest (ROI)
seti.gridROI : grid of region of interest (ROI) (matrix of size seti.dim x seti.nROI^seti.dim)
seti.ROImask : biggest square in 2D (cube in 3D) in ball with radius rCD/2 (logical array of size seti.nCD x seti.nCD)
seti.dV : pixel/voxel area/volume (number)

seti in setKernel.html:

seti.model : model (helmholtz2D or helmholtz3D)
seti.k : wave number
seti.kHat : Fourier coefficients of discretized integral operator (seti.nCD x seti.nCD)

seti in expSetup.html (subfiles pntsGeometry.html, setIncField.html, setMeasKer.html):

seti.measNb : number of receivers (number of measurements)
seti.incNb : number of transmitters (number of incident fields)
seti.dSInc : approximation of the infinitesimal element of closed contour with control points in case of transmitters
seti.dSMeas : approximation of the infinitesimal element of closed contour with control points in case of receivers
seti.incField : incident field, setIncField.html (matrix of size seti.nROI^seti.dim x seti.incNb)
seti.measKer : matrix to compute simulated measurement data from solution, setMeasKer.html (size seti.measNb x seti.nROI^seti.dim)

Output Arguments

More details of output arguments are in section More About. type: vector, matrix, etc...

FFqMeas : scattered field evaluated on the receivers' positions for each transmitter (complex matrix of size seti.measNb x seti.incNb)
FFqmF : FFqMeas minus noisy data (complex matrix of size seti.measNb x seti.incNb)
ADFFq : adjoint of derivative (complex matrix written as vector of size seti.nROI^seti.dim x 1)
FFqROI : scattered field evaluated on ROI (region of interest) stored as vector for each transmitter (complex matrix of size seti.nROI^seti.dim x seti.incNb)
JA, JB : Auxilliary matrices (JA is a complex matrix of size seti.measNb x seti.nROI^seti.dim, JB is a complex matrix of size seti.nROI^seti.dim x seti.incNb)

Best Practice

If you are only interested in the scattered field evaluated on receivers' positions (i.e. measurements), FFqMeas, use

[FFqMeas,~,~] = mimo(seti,qROI,'simo')

instead of

[FFqMeas,~] = mimo(seti,qROI,0,'adjOfDer')

[FFqMeas,~] = mimo(seti,qROI,zeros(seti.measNb,seti.incNb),'adjOfDer')

The latter with 'adjOfDer' computes the adjoint of derivative additionally, which is more expensive than 'simo'.

More About

$\texttt{FF} = \mathcal{F}$ : contrast-to-measurement operator
$\mathcal{F}(q)$ is the matrix of TX-RX scattered fields (TX: transmitters, RX: receivers), i.e. multiple-input-multiple-output setting, i.e. the k-the column is the RX-field for TX(k).
$\texttt{FmeasDelta} = F_\mathrm{meas}^\delta$ , i.e. with noise level $\delta$ noisy simulated data (measurement) $F_\mathrm{meas}$ .
$f_\mathrm{dis} = (1/2)\ \|\mathcal{F}(q) - F_\mathrm{meas}^\delta\|_\mathrm{dis}^2$ : discrepancy
$\texttt{FFqMeas}$ is the scattered field evaluated on receivers' positions (measurements) (also called $\texttt{uScattRX}$ ). Note that $\texttt{FFqMeas} = \mathcal{F}(q)$ .
$\texttt{FFqmF} = \texttt{FFqMeas} - \texttt{FmeasDelta} = \mathcal{F}(q) - F_\mathrm{meas}^\delta$ .

$\texttt{FFqROI}$ is the scattered field evaluated on ROI stored for each transmitter for each transmitter (for one transmitter it is also called $\texttt{uScattROI}$ ).
Scattered field $\texttt{FFqROI} = u_\mathrm{ROI}^s = V(q \cdot u_\mathrm{ROI}^t)$ with volume potential .
Remind: FFqROI is on ROI, but FFqmF is on RX (receivers positions).

$\texttt{DFFqh} = \texttt{JA*diag(h)*JB} = \mathcal{F}'(q)[h]$ is the Frechet derivative with auxiliary matrices $\texttt{JA}$ and $\texttt{JB}$ .
$\texttt{ADFFq} = [\mathcal{F}'(q)]^\ast [\mathcal{F}(q) - F_\mathrm{meas}^\delta]$ is the adjoint of derivative.

[JA,JB] = mimo(seti, qROI,'jacobian') is used in case of primal-dual algorithm.
[FFqmF,ADFFq] = mimo(seti,qROI,seti.FmeasDelta) is used in case of soft-shrinkage (not available in public version).

Code: mimo

function varargout = mimo(seti, qROI, varargin)

if ~ischar(varargin{end})
    op = 'adjOfDer';
else
    op = varargin{end};
end

if strcmpi(op, 'adjOfDer') % old name: gradient
    [varargout{1}, varargout{2}] = mimocalc(seti, qROI, varargin{1}, op);
    % Note: varargin{1} = FmeasDelta.
    % Output documentation:
    % varargout{1} = FFqmF
    % varargout{2} = ADFFq

elseif strcmpi(op, 'simo')
    [varargout{1}, varargout{2}, varargout{3}] = mimocalc(seti, qROI, 0, op);
    % Output documentation:
    % varargout{1} = uScattRX = FFqMeas
    %     (and not FFqMeas - FmeasDelta, because FmeasDelta was set to 0).
    % varargout{2} = ADFFq = 0 (in this case)
    % varargout{3} = uScattROI = FFqROI

elseif strcmpi(op, 'jacobian')
    if strcmp(seti.model, 'helmholtz3D') ...
            || strcmp(seti.model, 'helmholtz2D') ...
            || strcmp(seti.model, 'helmholtzHMode2D')
        [~, ~, ~,varargout{1},varargout{2}] = mimocalc(seti, qROI, 0, op);
        % Output documentation:
        % varargout{1} = JA
        % varargout{2} = JB
    else
        disp('mimo.m: Error - Jacobian is not (yet ... ?) implemented');
        varargout{1} = [];
        varargout{2} = [];
    end
end

end

Code: mimocalc

function [FFqmF, ADFFq, FFqROI, JA, JB] = mimocalc(seti, qROI, FmeasDelta, op)
%Note: HMode is not available in public version.

nnROI = length(seti.gridROI(1,:));
ADFFq = 0;
FFqROI = zeros(nnROI, seti.incNb);
FFqmF = zeros(seti.measNb, seti.incNb);

if strcmpi(op, 'adjOfDer')
    ADFFq = zeros(size(qROI));
end
if strcmpi(op, 'jacobian')
    if strcmp(seti.model, 'helmholtz3D') || strcmp(seti.model, 'helmholtz2D')
        JA = zeros(seti.measNb, nnROI); JB = zeros(nnROI, seti.incNb);
    elseif strcmp(seti.model, 'helmholtzHMode2D')
        JA = zeros(seti.measNb, nnROI,2); JB = zeros(nnROI, seti.incNb,2);
    else
        disp('mimo.m: Error - Jacobian is not (yet ... ?) implemented');
    end
end

dSInc = seti.dSInc.*ones(seti.incNb,1);

[~, S, VGStar, VStar, TStar, ~, QStarU, ~, ~] = intOpsFuncs(qROI,seti);

for jj = 1:seti.incNb
    SL = dSInc(jj)*seti.incField(:,jj);
    % SL = uIncROI (i.e. single layer potential/Herglotz wave function)
    % SL \phi = uIncROI
    [uMeas, uScattROI] = S(SL); % uMeas is uScattRX (i.e. u^s on measPnts)

    h = uMeas;

    if strcmpi(op, 'adjOfDer')
        h = h - FmeasDelta(:,jj); % result: h = FFqmF = FF(q) - FmeasDelta
    end;
    FFqmF(:,jj) = h; % In case of adjOfDer (default): FFqmF = FF(q) - FmeasDelta
                     % otherwise so just FFq:         FFqmF = FF(q)
    h( isnan(h) ) = 0; % workaround
    uTotROI = uScattROI + SL; % uTotROI = uScattROI + uIncROI
    if strcmpi(op, 'adjOfDer')
        if strcmp(seti.model, 'helmholtz3D') || strcmp(seti.model, 'helmholtz2D')
            f = VGStar(h);
            v = conj(uTotROI).*(f + VStar(TStar(QStarU(f)))); %VStar NOT VqStar
            ADFFq = ADFFq + v;
            %figure(101); x = ADFFq; imagesc(real(extendROItoCD(x,seti.ROImask))); axis xy; colorbar; caxis([-3,3]); error('stop');
        elseif strcmp(seti.model, 'helmholtzHMode2D')
            ADFFq = ADFFhelmHMode2D(h,seti,uTotROI);
        else
            disp('mimo.m: Error - model is not (yet ... ?) implemented');
        end
    end
    if strcmpi(op, 'simo')
        FFqROI(:,jj) = reshape(uScattROI,[nnROI 1]);
        ADFFq = 0;
    end
    if strcmpi(op, 'jacobian')
        if strcmp(seti.model, 'helmholtz3D') || strcmp(seti.model, 'helmholtz2D')
            JB(:,jj) = uTotROI;
        elseif strcmp(seti.model, 'helmholtzHMode2D')
            [uTotROI1, uTotROI2] = gradientHMode(uTotROI,seti);
            JB(:,jj,1) = uTotROI1;
            JB(:,jj,2) = uTotROI2;
        else
            disp('mimo.m: Error - Jacobian is not (yet ... ?) implemented');
        end
    end
end
if strcmpi(op, 'jacobian')
    dSMeas = seti.dSMeas.*ones(seti.measNb,1);
    if strcmp(seti.model, 'helmholtz3D') || strcmp(seti.model, 'helmholtz2D')
        for jj = 1:seti.measNb
            f = VGStar( ((1:seti.measNb)==jj)' * seti.dV/dSMeas(jj) );
            JA(jj,:) = conj(f + VStar(TStar(QStarU(f)))); %VStar NOT VqStar
        end;
    elseif strcmp(seti.model, 'helmholtzHMode2D')
        for jj = 1:seti.measNb
            [f1,f2] = helmholtzHMode2DZStar( ((1:seti.measNb)==jj)' * seti.dV/dSMeas(jj), seti); % Z*(g_j)
            qf1 = conj(seti.qROIexact).*f1; qf2 = conj(seti.qROIexact).*f2;

%             qf1Ext = 1.0i*pi/seti.rCD * fft2(extendROItoCD(qf1,seti.ROImask)) * seti.gradMat;
%             qf2Ext = 1.0i*pi/seti.rCD * seti.gradMat * fft2(extendROItoCD(qf2,seti.ROImask));
%             qf1 = restrictCDtoROI(ifft2(qf1Ext+qf2Ext),seti.ROImask);

            qf1 = divHMode(seti,qf1,qf2);

            qf1 = TStar(qf1);

            [qf1, qf2] = helmholtzHMode2DNablaVStar(qf1,seti);

            JA(jj,:,1) = conj(f1 + qf1); JA(jj,:,2) = conj(f2 + qf2);

        end;
    else
        disp('mimo.m: Error - Jacobian is not (yet ... ?) implemented');
    end
end

end