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run_SlepianD.m
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run_SlepianD.m
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function run_SlepianD(exampleIndex,testEnergy)
%run_SlepianD compute and plot Slepian eigenfunctions on the sphere (8.27) and (8.29)
if nargin<1
exampleIndex=0; % default is to do Australia including Tasmania
end
if nargin<2
testEnergy=false;
end
%% Determine path for output frames
base=userpath; base(end)='/'; % ~/Documents/MATLAB/
frames_folder=[base 'slepian-frames/']; % ~/Documents/MATLAB/slepian-frames/
data_folder=[base 'slepian-data/']; % ~/Documents/MATLAB/slepian-data/
%% Parameters for various increments in degrees for different grids
intginc=0.1; % fine grid for integration (degrees)
medinc=0.5; % medium grid for smooth plotting (degrees)
plotinc=2.0; % coarse grid for grid plotting (degrees)
fprintf('\n@@ Fine Grid: %.2f (degrees)\n',intginc)
fprintf('@@ Medium Grid: %.2f (degrees)\n',medinc)
fprintf('@@ Coarse Grid: %.2f (degrees)\n',plotinc)
%% Close all figures
close all
%% Choose a region on the sphere
switch exampleIndex
case 0 % region is Australia including Tasmania
[tv_aus,pv_aus,mask_aus,tR,pR]=ausRegion(medinc,false,true);
basename='australia2';
case 1 % region is Australia excluding Tasmania
[tv_aus,pv_aus,mask_aus,tR,pR]=ausRegion(medinc,true,true);
basename='mainland';
case 2 % an open equatorial band
tr(1,:)=[60 100];
pr(1,:)=[0 300];
basename='band';
case 3 % anumber of non-intersecting regular subregions
% subregion 1
tr(1,:)=[30 60];
pr(1,:)=[-70 -10];
% subregion 2
tr(2,:)=[80 110];
pr(2,:)=[-70 -10];
% subregion 3
tr(3,:)=[20 70];
pr(3,:)=[0 40];
basename='reg3';
case 4 % rectangle bounding Australia
[tv_aus,pv_aus,mask_aus,tR,pR]=ausRegion(medinc,false,true);
tr(1,:)=[tv_aus(1) tv_aus(end)]*180/pi;
pr(1,:)=[pv_aus(1) pv_aus(end)]*180/pi;
basename='bnd-aust';
end
%% Populate the N_tot x N_tot Hermitian D matrix (8.27)
for L_max=15:5:20
fprintf('\n@@ L_max: %d\n',L_max)
N_tot=(L_max+1)^2;
switch exampleIndex
case {0,1,4} % irregular non-simply-connected (uses mask)
D=SlepianDH(L_max,tv_aus,pv_aus,mask_aus,true);
case {2,3}
D=zeros(N_tot,N_tot); % allocate and initialize to zero
for r=1:size(tr,1) % loop over subregions
tv_intg=(tr(r,1):intginc:tr(r,2))*pi/180;
pv_intg=(pr(r,1):intginc:pr(r,2))*pi/180;
D=D+SlepianDH(L_max,tv_intg,pv_intg); % accumulate
fprintf('@@ Completed D for Region: %d\n',r)
end
end
fprintf('@@ Size of D matrix: %dx%d\n', size(D))
%% Save the D matrix
dataName=sprintf('%s_%04d',[data_folder basename],L_max);
save([dataName '.mat'],'L_max','D');
%% Get spectral eigenstructure
[V,lamD]=eig(D); % eigen-structure
% stem(flip(diag(lamD))) % plot the eigenvalues
maxSlepian=1.0;
for eigindex=0:3;%N_tot-1 % eigenvalue index in descending energy order
if eigindex>N_tot-1
break;
end
lambda=min(max(lamD(end-eigindex,end-eigindex),0.0),1.0); % pin to [0,1] (superfluous)
fprintf('@@ Eigenvalue %04d: %8.6f\n',eigindex,lambda)
%% eigindex'th eigenvector w with eigenvalue lambda in spectral domain
w=V(:,end-eigindex); % eigenvector - SHT of Slepian function
w=w/norm(w); % normalize eigenvector (actually this is superfluous)
if testEnergy % test spatial energy of slepian function is unity
spectralEnergy=norm(w)^2; % will be 1 but just check
fprintf('@@ Spectral Energy: %8.6f\n',spectralEnergy)
% eigindex'th Slepian eigenfunction in spatial domain (0'th is dominant)
tv_intg=(0:intginc:180)*pi/180;
pv_intg=(0:intginc:360)*pi/180;
% spatial Slepian eigenfunction [slepian,theta,phi]
[slepian,~,~]=ishtRectGrid(w,tv_intg,pv_intg); % ISHT; theta,phi unused
spatialEnergy=trapSphereMaskedR(abs(slepian).^2,tv_intg,pv_intg);
fprintf('@@ Spatial Energy: %8.6f\n',spatialEnergy)
end
%% Plot eigenfunction in spatial domain
close % prepare fresh figure
bump=0.3; ref=1.0; ptype=1;
tv_plot=(0:medinc:180)*pi/180;
pv_plot=(0:medinc:360)*pi/180;
[slepian_plot,theta_plot,phi_plot]=ishtRectGrid(w,tv_plot,pv_plot,true);
if eigindex==0
maxSlepian=max(abs(slepian_plot(:))); % for plot normalization
fprintf('@@ Max Slepian: %8.6f\n',maxSlepian)
end
s=spatialPlot(slepian_plot/maxSlepian,theta_plot,phi_plot, bump,ref,ptype);
s.EdgeColor='none'; % no lines
s.FaceAlpha=0.8;
colormap('cool') % or copper
set(gcf,'MenuBar','none');
set(gcf,'ToolBar','none');
hold on
switch exampleIndex
case {0,1,4} % fill Australia area on zero/reference surface
aus_coast=ref*[sin(tR).*cos(pR); sin(tR).*sin(pR); cos(tR)]/ ...
(ref+bump);
idx = any([~isnan(tR);~isnan(tR)],1);
fill3(aus_coast(1,idx),aus_coast(2,idx),aus_coast(3,idx), ...
'w','EdgeColor','None');
% draw Australia coastline on Slepian surface
rR=interpn(theta_plot,phi_plot,slepian_plot,tR,pR); % very slick
F=spatialMap(rR/maxSlepian,ptype);
rad=abs(ref + bump*F)/(ref+bump)*1.02;
aus_coast=[rad.*sin(tR).*cos(pR); rad.*sin(tR).*sin(pR); rad.*cos(tR)];
plot3(aus_coast(1,:),aus_coast(2,:),aus_coast(3,:),'w','LineWidth',2);
view(-100,-30) % set viewpoint
% view(mode(tR)*180/pi,0) % set viewpoint
set(gca,'CameraViewAngle',9) % zoom into scene
case {2,3} % Compute energy in sub-regions
lambda_est=0;
for r=1:size(tr,1) % loop over subregions (fine grid)
tv_intg=(tr(r,1):intginc:tr(r,2))*pi/180;
pv_intg=(pr(r,1):intginc:pr(r,2))*pi/180;
reg=ishtRectGrid(w,tv_intg,pv_intg);
lambda_est=lambda_est+trapSphereMaskedR(abs(reg).^2,tv_intg,pv_intg);
end
fprintf('@@ Region Energy: %8.6f (eigenvalue %8.6f)\n',lambda_est,lambda)
% Plot sub-regions
for r=1:size(tr,1) % loop over subregions (coarse grid)
ttcr=(tr(r,1):plotinc:tr(r,2))*pi/180;
ppcr=(pr(r,1):plotinc:pr(r,2))*pi/180;
[r_reg,theta_reg,phi_reg]=ishtRectGrid(w,ttcr,ppcr);
s=spatialPlot(1.01*r_reg/maxSlepian,theta_reg,phi_reg);
s.EdgeColor='y'; % no lines
end
end
%% Annotate plot
llabel=sprintf('$L_{\\mathrm{max}}=%d$\n$\\lambda_{%d}=%8.6f$', L_max,eigindex,lambda);
delete(findall(gcf,'Tag','myLabel'));
a=annotation('textbox',[0.695,0.895,0.28,0.1],'String',llabel);
a.Interpreter='latex';
a.FontSize=18;
a.LineStyle='none';
a.Tag='myLabel';
hold off
%% Output to png file (directory needs to exist)
set(gcf,'InvertHardCopy','off');
set(gcf,'color','w'); % Set the figure frame color to white
set(gcf,'PaperUnits','inches','PaperPosition',[0 0 6 6]) %150dpi
outname=sprintf('%s_%04d_%04d',[frames_folder basename],L_max,eigindex);
saveas(gcf,outname,'png')
end
%% Render eigenfunction movie on osx; needs avconv - get via brew install libav
if ~system('which avconv >/dev/null')
renderMovWithAvconv=sprintf(...
'avconv -framerate 2 -y -v quiet -f image2 -i %s_%04d_%%04d.png %s-%04d.mov',...
[frames_folder basename],L_max,[frames_folder basename],L_max);
system(renderMovWithAvconv); % make compressed mov
cleanup=sprintf('rm %s_%04d*.png',[frames_folder basename],L_max);
system(cleanup); % delete frames
end
end