Merge branch 'master' into 'main'

Initial commit

See merge request sapme1/hexagonalsampling!1

ComputeMostHexagonalSampling.m

+function [packingDensity, L_optimum, isDegenerate, packingDensityCAIPI, L_CAIPI] = ...
+    ComputeMostHexagonalSampling(R_inplane, FOV_inplane, sliceSpacing, nSlicesSimultaneously, ...
+    nSlicesTotal, sliceThickness, sliceGap, doPlot, doDebug, gifName)
+% This function computes how to optimally sample the phase encoding plane for 3D/Multiband
+% EPI/Spiral to achieve the maximum packing density of circles in the image domain
+%
+% IN
+%   R_inplane               in-plane undersampling factor
+%
+%   FOV_inplane             in-plane FOV
+%                           [same unit as sliceSpacing, sliceThickness & sliceGap]
+%
+%   sliceSpacing            distance between simultaneously excited slices
+%                           (only used if all input arguments available)
+%                           [same unit as FOV_inplane, sliceThickness & sliceGap]
+%
+%   nSlicesSimultaneously   number of simultaneously excited slices
+%                           (only needed if sliceSpacing empty)
+%
+%   nSlicesTotal            total number of slices
+%                           (only needed if sliceSpacing empty)
+%
+%   sliceThickness          slice thickness
+%                           (only needed if sliceSpacing empty)
+%                           [same unit as FOV_inplane, sliceSpacing & sliceGap]
+%
+%   sliceGap                gap between slices
+%                           (only needed if sliceSpacing empty)
+%                           [same unit as FOV_inplane, sliceSpacing & sliceThickness]
+%
+%   doPlot                  Plot grid and packing density curve {false}
+%
+%   doDebug                 Plot all intermediate grids that are being tested {false}
+%
+%   gifName                 name of gif in case debug plot should be saved as gif
+%
+%
+% OUT
+%   packingDensity          circle packing density [%]
+%
+%   L_optimum               Optimum choice of L
+%                           (maximising packing density = most homogeneous sampling distribution)
+%
+%   isDegenerate            flag whether packing density metric is degenerate
+%
+%
+% Example: [packingDensity, L_optimum] = ComputeMostHexagonalSampling(1, 192, 19.2, 3);
+%          [packingDensity, L_optimum] = ComputeMostHexagonalSampling(1, 192, [], 3, 3, 4, 15.2)
+%
+% Author: Maria Engel
+% (c) Cardiff University Brain Research Imaging Centre (CUBRIC), Cardiff University, United Kingdom
+
+if nargin == 7
+    sliceSpacing = nSlicesTotal*(sliceThickness+sliceGap)/nSlicesSimultaneously;
+end
+
+if nargin < 8
+    doPlot = false;
+end
+
+if nargin < 9
+    doDebug = false;
+end
+
+if nargin < 10
+    gifName = [];
+end
+
+% in-plane spacing of lines in k-space
+deltak_inplane = 2*pi*R_inplane/FOV_inplane;
+kzSlabThickness = 2*pi/sliceSpacing;
+
+% As soon as L is so large that the distance to the chronologically next point is smaller than to
+% the next point after a down-blip, the grids are going to get only worse = less homogeneous, i.e.
+% no point in checking their packing density
+Lmax = ceil(sqrt(0.5+sqrt(0.25+(kzSlabThickness/deltak_inplane)^2)));
+% For L<2 there is an L>2 which produces a rotationally symmetric grid i.e. same packing density
+Lmin = 2;
+Lincrement = 0.01;%0.001;
+
+LArray = Lmin:Lincrement:Lmax;
+
+% Make sure you include cases shown in Narsude et al. MRM 2016, e.g. Fig 1d
+% They use DELTA rather than L and propose integer DELTA
+DELTA = 1:nSlicesSimultaneously;
+% Translate these DELTA's to L's and find which ones present unique patterns (i.e. not yet covered
+% by the previous L's)
+LNarsudeUnique = nSlicesSimultaneously./DELTA;
+LNarsudeUnique = LNarsudeUnique(mod(LNarsudeUnique,1)~=0);
+LNarsudeUnique = LNarsudeUnique(LNarsudeUnique>2);
+LArray = sort([LArray LNarsudeUnique]);
+
+howFar2Look = 10; % TODO: Find out what to put here!
+d_min = zeros(size(LArray));
+
+if doDebug
+    figure('Name','Sampling grid and alps');
+    ah = subplot(1,2,1);
+end
+
+%% Initiate arrays
+packingDensityArray = zeros(numel(LArray),1);
+isBlippedCAIPI = zeros(numel(LArray),1);
+isNarsude = zeros(numel(LArray),1);
+
+%% Go through range of L's, compute the minimum distance between grid points and the resulting packing density
+for iL = 1:numel(LArray)
+    [kyArray, kzArray] = GetGridPoints(howFar2Look*ceil(LArray(iL)),deltak_inplane,sliceSpacing,LArray(iL));
+
+    d = sqrt(kyArray.^2 + kzArray.^2);
+    d_min(iL) = min(d(d>0));
+
+    packingDensityArray(iL) = pi/4*d_min(iL)^2/(2*pi*deltak_inplane/sliceSpacing);
+
+    isBlippedCAIPI(iL) = ceil(LArray(iL))==LArray(iL);% && LArray(iL)<=nSlicesSimultaneously; % This is being kind to blipped CAIPIRINHA!
+    isNarsude(iL) = ismember(LArray(iL), LNarsudeUnique);
+
+    if doDebug
+        [kyArray, kzArray] = GetGridPoints(3*Lmax, deltak_inplane, sliceSpacing, LArray(iL));
+        PlotGrid(kyArray, kzArray, sliceSpacing, ah);
+
+        subplot(1,2,2); hold on;
+        plot(LArray(iL),packingDensityArray(iL)*100,'.','Color','k');
+
+        if isBlippedCAIPI(iL) || isNarsude(iL)
+            plot(LArray(iL),packingDensityArray(iL)*100,'x','Color','r','MarkerSize',10,'LineWidth',3);
+        end
+        
+        xlabel('L'); ylabel('Packing density [%]');
+        xlim([Lmin Lmax]); ylim([min(packingDensityArray*100) 90.7]);
+        sgtitle(sprintf('L = %.2f',LArray(iL)));
+
+        %% Make GIF
+        if ~isempty(gifName)
+            frame = getframe(1);
+            im = frame2im(frame);
+            [imind,cm] = rgb2ind(im,256);
+
+            if iL == 1
+                imwrite(imind,cm,gifName,'gif', 'Loopcount',inf,"DelayTime",0.1);
+            else
+                imwrite(imind,cm,gifName,'gif','WriteMode','append',"DelayTime",0.1);
+            end
+        end
+    end
+end
+
+%% Find the L that maximizes the packing density
+[d_optimum, ind_optimum] = max(d_min);
+L_optimum = LArray(ind_optimum);
+
+isDegenerate = d_optimum > kzSlabThickness;
+
+packingDensity = ComputePackingDensity(d_optimum, deltak_inplane, sliceSpacing);
+
+%% Compute options that blipped-CAIPIRINHA would offer
+indCAIPI = find(isBlippedCAIPI + isNarsude);
+d_CAIPI = d_min(indCAIPI);
+L_CAIPI = LArray(indCAIPI);
+packingDensityCAIPI = ComputePackingDensity(d_CAIPI,deltak_inplane,sliceSpacing);
+
+
+%% Plot grid and resulting sampling density for all options
+if doPlot
+    % For plotting get grid points on a bit larger range
+    [kyArray, kzArray] = GetGridPoints(2*ceil(Lmax), deltak_inplane, sliceSpacing, L_optimum);
+    PlotGrid(kyArray, kzArray, sliceSpacing);
+
+    figure('Name','Packing density'); hold on;
+    plot(LArray, packingDensityArray*100);
+    plot(L_CAIPI, packingDensityCAIPI*100, 'x', 'MarkerSize', 10, 'LineWidth', 3);
+    xlabel('L');
+    ylabel('Packing density [%]');
+end
+
+end
+
+%% Compute coordinates of grid points for a given sampling scheme
+function [kyArray, kzArray] = GetGridPoints(mRange,deltak_inplane,sliceSpacing,L)
+% Get grid points in PE plane for given sampling pattern
+mArray = -mRange:mRange;
+kyArray = mArray.*deltak_inplane;
+kzArray = 2*pi/sliceSpacing*(mArray./L-floor(mArray./L));
+end
+
+%% Plot grid from given coordinates of grid points
+function ah = PlotGrid(ky,kz,sliceSpacing,ah)
+if nargin < 4 || isempty(ah)
+    figure('Name', 'Sampling grid');
+else
+    axes(ah);
+end
+scatter(ky, kz, 'MarkerEdgeColor', [0.5 0.5 0.5], 'MarkerFaceColor', [0.5 0.5 0.5], 'LineWidth', 5);
+
+hold on;
+
+yline(0,'r','LineWidth',3); 
+yline(2*pi/sliceSpacing,'r','LineWidth',3);
+
+axis equal;
+ylim(2*pi/sliceSpacing*[-1 2]);
+xlim([min(ky) max(ky)]);
+xlabel('k_y'); ylabel('k_z');
+hold off;
+end
+
+%% Compute packing density for a given smallest distance d in a grid
+function packingDensity = ComputePackingDensity(d, deltak_inplane, sliceSpacing)
+A_circle = pi/4*d.^2;
+A_unitcell = 2*pi*deltak_inplane/sliceSpacing;
+packingDensity = A_circle/A_unitcell;
+end
\ No newline at end of file

MakeGridAndAlpGif.m

+clear all;
+
+baseFolder = pwd;
+
+doSave = true;
+if doSave
+    gifName = fullfile(baseFolder,'GridAndAlpsGif.gif');
+else
+    gifName = [];
+end
+
+Nsim = 6;
+R = 1;
+FOVinplane = 220;
+
+sliceSpacing = FOVinplane/2/Nsim;
+
+[packingDensity, L_optimum, isDegenerate, packingDensityCAIPItmp, L_CAIPItmp] = ...
+    ComputeMostHexagonalSampling(R, FOVinplane, sliceSpacing, Nsim, [], [], [], [], true, gifName);
+
+[packingDensityCAIPI, indCAIPI] = max(packingDensityCAIPItmp);
+L_CAIPI = L_CAIPItmp(indCAIPI);

MakeOptDvsRPlot.m

+clear all;
+
+% # simultaneously excited slices to be investigated
+NSimArray  = 2:8;
+
+% Range of total undersampling factors to be investigated
+Rtot_start = 1;
+Rtot_end = 12;
+Rtot_increment = 0.001;
+
+FOVinplane = 220;
+
+doSave = true;
+saveFolder = pwd;
+if doSave
+    dfws = get(0,'DefaultFigureWindowStyle');
+    set(0,'DefaultFigureWindowStyle','normal');
+end
+
+fh = figure('Name','Packing density'); hold on;
+t = tiledlayout(numel(NSimArray),1,'TileSpacing','Compact');
+
+for iNSim = 1:numel(NSimArray)
+    Nsim = NSimArray(iNSim);
+
+    clear RArray packingDensity L_optimum packingDensityCAIPI L_CAIPI
+    RArray = (Rtot_start/Nsim):Rtot_increment:ceil(Rtot_end/Nsim);
+    sliceSpacing = FOVinplane/2/Nsim;
+    for iR = 1:numel(RArray)
+        [packingDensity(iR), L_optimum(iR), isDegenerate, packingDensityCAIPItmp, L_CAIPItmp] = ...
+            ComputeMostHexagonalSampling(RArray(iR), FOVinplane, sliceSpacing, Nsim);
+        [packingDensityCAIPI(iR), indCAIPI] = max(packingDensityCAIPItmp);
+        L_CAIPI(iR) = L_CAIPItmp(indCAIPI);
+        if isDegenerate
+            break
+        end
+    end
+
+    %%
+    if numel(RArray)>1
+        if doSave
+            name = [];
+        else
+            name = sprintf(', N_{sim} = %i',Nsim);
+        end
+        nexttile;  hold on;
+        plot(RArray(1:numel(packingDensity))*Nsim,packingDensity*100,...
+            'DisplayName', ['As hexagonal as possible' name],'LineWidth',2,'Color','k');
+        plot(RArray(1:numel(packingDensity))*Nsim,packingDensityCAIPI*100,...
+            'DisplayName',['Blipped CAIPIRINHA' name],'LineWidth',2)
+        yline(pi/2/sqrt(3)*100,'Color',[0.5 0.5 0.5],'LineWidth',2,'DisplayName', 'Maximum circle packing density, 90.7%');
+        xlim([Rtot_start Rtot_end]);
+        ylim([60 92]);
+        if iNSim==numel(NSimArray) || doSave==false
+            xlabel('R_{total}');
+            legend;
+        else
+            xlabel(''); xticks('');
+        end
+    end
+end
+
+if numel(RArray)>1
+    ylabel(t,'Packing density [%]')
+end
+
+if doSave
+    set(fh,'Position',[488.2 61.8 520.8 700])
+    hLegend = findobj(gcf, 'Type', 'Legend');
+    hLegend.Position = [0.476 0.834 0.421 0.062];
+    export_fig(fullfile(saveFolder, 'PackingDensityOverR'),'-png','-transparent','-a1','-r300',gcf);
+    set(0,'DefaultFigureWindowStyle',dfws);
+end
\ No newline at end of file

README.md

 # HexagonalSampling
-
-
+This repository contains code to find the optimum sampling for 3D/Multiband sampling in MRI using EPI or Spirals.
+ISMRM abstract Engel et al. 2024
 
 ## Getting started
+Reproduce results in abstract by running
 
-To make it easy for you to get started with GitLab, here's a list of recommended next steps.
-
-Already a pro? Just edit this README.md and make it your own. Want to make it easy? [Use the template at the bottom](#editing-this-readme)!
-
-## Add your files
-
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-
-```
-cd existing_repo
-git remote add origin https://git.cardiff.ac.uk/sapme1/hexagonalsampling.git
-git branch -M main
-git push -uf origin main
-```
-
-## Integrate with your tools
-
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-
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-
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-
-Use the built-in continuous integration in GitLab.
-
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-
-***
-
-# Editing this README
-
-When you're ready to make this README your own, just edit this file and use the handy template below (or feel free to structure it however you want - this is just a starting point!). Thank you to [makeareadme.com](https://www.makeareadme.com/) for this template.
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-## Suggestions for a good README
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-
-## Authors and acknowledgment
-Show your appreciation to those who have contributed to the project.
+MakeGridAndAlpGif.m (for Gif 1)
 
-## License
-For open source projects, say how it is licensed.
+and
 
-## Project status
-If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. You can also make an explicit request for maintainers.
+MakeOptDvsRPlot.m (for Figure 2, this requires https://se.mathworks.com/matlabcentral/fileexchange/23629-export_fig)
\ No newline at end of file