clean up script for kernel DMD for hamlyn heart

examples/kernel_dmd_heart.m

-%% convert hamlyn heart video to image sequence
+%% Kernel DMD example for hamlyn heart dataset
 
-% workingDir = tempname;
-% mkdir(workingDir)
-% mkdir(workingDir,'images')
-% 
-% shuttleVideo = VideoReader('~/Downloads/hamlyn_heart.avi');
-% 
-% ii = 1;
-% while hasFrame(shuttleVideo)
-%    img = readFrame(shuttleVideo);
-%    filename = [sprintf('%03d',ii) '.jpg'];
-%    fullname = fullfile(workingDir,'images',filename);
-%    imwrite(img,fullname)    % Write out to a JPEG file (img1.jpg, img2.jpg, etc.)
-%    ii = ii+1;
-% end
-% 
-% imageNames = dir(fullfile(workingDir,'images','*.jpg'));
-% imageNames = {imageNames.name}';
-% 
-% 
-% for ii = 1:length(imageNames)
-%    img = imread(fullfile(workingDir,'images',imageNames{ii}));
-%    grayImages(:,:,ii) = rgb2gray(img);
-% end
-
-%% Try DMD on it
 clear variables
-load data/HamlynHeartImages.mat;
-
-disp('Data preparation...')
+load data/HamlynHeartImages.mat; % loads variable grayImages
 
+% choose images, initialize Kernel DMD object
 image_start  = 1;% frame number to start from
 num_images = 100;
-
-allImages = [];
-% for k = 1:size(grayImages,3)
-for k = image_start:image_start+num_images 
-    image_k = grayImages(:,:,k);
-%     allImages(:,k) = double(image_k(:));
-    allImages = [allImages, double(image_k(:))];
-end
-
-XallPrescaled = [allImages(:,2:end);
-                 allImages(:,1:end-1)];
-% XallPrescaled = allImages;
-
-% remove constant rows which make XallPrescaled lose rank
-[XallPrescaledProcessed,ProcessSettings] = removeconstantrows(XallPrescaled);
-XallPrescaledProcessed = XallPrescaledProcessed';
-
-
-%% do Kernel DMD via object implementation
 D = kernel_dmd(grayImages(:,:,image_start:image_start+num_images),'','gaussian','true');
-D.regression(1e6,49);
-rec = D.reconstruct();
-
-disp('DMD Regression...')
-%% kernel dmd
-disp('Scaling data...')
-[Xall,scaleFactor,scaleOffset] = ScaleData(XallPrescaledProcessed,'zscore'); % zscore may cause problems with body_length, which is constant
-% Xall = XallPrescaledProcessed;
-% scaleFactor = 1;
-% scaleOffset = 0;
-X = Xall(1:end-1,:);
-X2 = Xall(2:end,:);
-
-%% Kernel DMD
-disp('Doing Kernel DMD...')
-% kernel function
-% inputs: x1, x2 are vectors of the same size.
-% kernel = @(x1,x2) (1 + dot(x1,x2))^10; % polynomial kernel, doesnt work that well. often fails during reconstruction
-sigma2 = 1e6; 
-kernel = @(x1,x2) exp(-norm(x1-x2)^2/sigma2); % Gausssian kernel
-
-% for j = 1:size(X,1)
-%     disp(j)
-%     for k = 1:size(X2,1)
-%         Ahat(j,k) = kernel(X(k,:),X2(j,:)); % Y'Y2 is Ahat in Williams et al.
-%         Ghat(j,k) = kernel(X(k,:),X(j,:)); % Y*Y is the Gramian Ghat in Williams et al.
-%     end 
-% end
-
-tic;
-Ahat = GaussianKernel(X,X2,sqrt(sigma2));
-Ghat = GaussianKernel(X,X,sqrt(sigma2));
-
-%%
-
-% TODO: truncate to get low-order model.
-% use Ghat to compute Sigma (called 'S') and V
-% [Qfull,S2full] = eig(Ghat); % since it's true that Ghat*Q = Q*Sigma^2
-
-[Qfull, S2full] = eigs(Ghat,size(Ghat,1),'largestabs','Tolerance',1e-19); % use increased tolerance with GaussianKernel
-
-% S2full = flipud(fliplr(S2full)); % make largest singular values first instead of largest singular values last
-% Qfull = fliplr(Qfull); 
-Sfull = sqrtm(S2full);
-
-% truncate
-% r = size(Ghat,1)-1; % number of dimensions to keep
-r = floor(size(Ghat,1)/2);
-Q = Qfull(:,1:r);
-S = Sfull(1:r,1:r);
-
-figure(2);
-clf;
-semilogy(diag(Sfull),'o');
-hold on
-semilogy(diag(S),'.');
-
-% Sinv = diag(ones(size(S,1),1)./diag(S)); % better numerical stability?
-% Khat = (Sinv*Q')*Ahat*(Q*Sinv);
-Khat = (S\Q')*(Ahat)*(Q/S); % Khat, an approximation of the Koopman operator
-[Vhat, Lambda] = eig(Khat);
-
-if(rank(Vhat) < size(Vhat,1))
-    error('Vhat is not full rank (rank = %d). I didnt code up the case where Vhat isnt.', rank(Vhat));
-end
-
-Phi = Q*S*Vhat; % eigenfunctions of Khat 
-Xi = Phi\X; 
-toc;
-X2hat = Phi*Lambda*Xi;
-if(norm(imag(X2hat)) > 1e-3)
-    error('X2hat has non-negligible imaginary component.')
-end
-X2hat = real(X2hat); % this is quite an accurate reconstruction of X2.
 
-disp('Finished computing kernel-DMD regression. Attempting simulation from IC.');
+% regression
+sigma2 = 1e6; % gaussian kernel variance. Try approximately length of state vector
+truncation_rank = 49; % number of modes to keep
+D.regression(sigma2,truncation_rank);
 
-%% simulate forward
-num_modes = size(Q,1); % i THINK this is the correct one
-tic;
-xhat(1,:) = X(1,:); 
-% xhat(1,1) = xhat(1,1) + 0.1;
-% xhat(1,:) = xhat(1,:) + 0.001*randn(1,size(xhat,2));
-for t = 2:2*size(X,1)
-    % calculate phi given x
-    prev_x = xhat(t-1,:);
-%     kernelmat = nan(1,num_modes);
-%     for M = 1:num_modes
-%         kernelmat(M) = kernel(prev_x,X(M,:));
-%     end
-    kernelmat = GaussianKernel(prev_x,X, sqrt(sigma2));
-    kernelmat = kernelmat(1:num_modes)';
-    phi_at_t = kernelmat*Q/S*Vhat;
-
-    % estimate 
-    xnext = phi_at_t*Lambda*Xi;
-    if(norm(imag(xnext)) > 1e-3)
-        error('xnext isnt real (has non-negligble imaginary component)')
-    end
-    xhat(t,:) = real(xnext);
-end
-toc;
-[xhatDmd,dmd_reconstruction] = D.predict(D.X(1,:),200);
-
-%% Check
-figure(1)
-clf
-subplot(2,1,1)
-plot(Xall(:,1));
-hold on
-plot([nan(1);X2hat(:,1)],'--'); 
-plot(xhat(:,1),'--');
-legend('true','1-step recon', 'n-step recon');
-title('Reconstruction element 1')
-subplot(2,1,2)
-plot(Xall(:,6));
-hold on
-plot([nan(1);X2hat(:,6)],'--'); 
-plot(xhat(:,6),'--');
-legend('true','1-step recon', 'n-step recon');
-title('Reconstruction element 6');
+% prediction
+initial_condition = D.X(1,:); % states are ROWS in this implementation
+num_timesteps = 200;
+[xhatDmd,dmd_reconstruction] = D.predict(initial_condition,num_timesteps);
 
-disp('Simulated from IC. Unscaling and casting data.');
 
-%% 
-% offsetx = repmat(scaleOffset,size(xhat,1),1);
-% reconstruction = xhat*diag(scaleFactor) + offsetx;
-for k = 1:size(xhat,1)
-    xhatRescaled(k,:) = xhat(k,:).*scaleFactor + scaleOffset;
-end
-xhatRescaled = xhatRescaled';
-xhatRescaled = removeconstantrows('reverse',xhatRescaled,ProcessSettings); % add removed rows back in.
 
-for k = 1:size(xhatRescaled,2) % now each state/image is a column, to comply with removeconstantrows()
-    reconstruction(:,:,k) = reshape(xhatRescaled(1:size(allImages,1),k),size(grayImages,1),size(grayImages,2));
-end
+%% plots and visualization
 
-disp('Done.')
-
-
-
-
-
-%% replay
-reconsructionHdl = implay(uint8(reconstruction));
-set(reconsructionHdl.Parent,'Name',sprintf('Reconstruction: %d modes, sigma2 = %.2e.',r,sigma2));
+% videos
 originalHdl = implay(grayImages(:,:,image_start:image_start+num_images));
 set(originalHdl.Parent,'Name','Original data');
 dmdReconstructionHdl = implay(dmd_reconstruction);
 set(dmdReconstructionHdl.Parent,'Name','DMD');
-%% 
-
-% figure(3);
-% clf
-% mode_number = 1;
-% % dont use uint8 and imshow, U0 is unitary so columns have very small
-% % values
-% singleMode = surf(flipud(reshape(Qfull(:,mode_number),size(grayImages,1),size(grayImages,2))),'LineStyle','none');
-% % singleMode = surf(flipud(reshape(U0(:,mode_number),size(grayImages,1),size(grayImages,2))),'LineStyle','none');
-% colormap(flipud(gray))
-% modeEig = Lambda(mode_number,mode_number);
-% title(sprintf('Mode %d, eig %.3f + %.3fi', mode_number, real(modeEig), imag(modeEig)));
-% % view from above
-%  axis([0 360 -0.001 288 -1 0])    
-%  view(0,90)
- 
- 
- function F = GaussianKernel(Xin,Yin,sigma)
-    X = Xin./sigma;
-    Y = Yin./sigma;
-    
-    % each term is |xi - yj|^2. Expanding:
-    % |x-y|^2 = x'x - 2*x'y + y'y. these three lines are the three terms.
-    X2 = sum(X.^2,2); %x'x as a column vector
-    Y2 = sum(Y.^2,2); %y'y as a column vector
-    XY = bsxfun(@minus, X2, (2*X)*Y.'); % add 2x'y terms
-    total = bsxfun(@plus, Y2.', XY); % add y'y terms
-    
-    F = exp(-total)'; 
 
-end
+% singular values
+figure(1);
+clf;
+semilogy(diag(D.Sfull),'o');
+hold on
+semilogy(diag(D.S),'.');
+legend('All singular values','Truncated singular values')
+xlabel('Mode number')
+ylabel('Singular Value')