From 802241ace3999becde9654353c1642a0633bab94 Mon Sep 17 00:00:00 2001 From: Omer S Date: Mon, 21 Oct 2019 20:09:04 +0300 Subject: [PATCH] Kappa and fista (#14) * added GD kappa effect & Image deblurring w/ FISTA * added GD kappa effect & Image deblurring w/ FISTA * Update README.md * Update README.md * Update README.md * Rename Image_Related/Image Deblurring/generic_grad.m to Image_Related/Image_Deblurring/generic_grad.m * Rename Image_Related/Image Deblurring/fista.m to Image_Related/Image_Deblurring/fista.m * Rename Image_Related/Image Deblurring/exact_quad.m to Image_Related/Image_Deblurring/exact_quad.m * Rename Image_Related/Image Deblurring/const_step.m to Image_Related/Image_Deblurring/const_step.m * Rename Image_Related/Image Deblurring/blur.m to Image_Related/Image_Deblurring/blur.m * Rename Image_Related/Image Deblurring/README.md to Image_Related/Image_Deblurring/README.md * Rename Image_Related/Image Deblurring/GD_vs_FISTA_wrapper.m to Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m * Rename --- .../Image_Deblurring/GD_vs_FISTA_wrapper.m | 98 ++++++++++++++++++ Image_Related/Image_Deblurring/README.md | 3 + Image_Related/Image_Deblurring/blur.m | 83 +++++++++++++++ Image_Related/Image_Deblurring/const_step.m | 10 ++ .../doc/GD_vs_FISTA_wrapper.pdf | Bin 0 -> 377020 bytes Image_Related/Image_Deblurring/exact_quad.m | 4 + Image_Related/Image_Deblurring/fista.m | 44 ++++++++ Image_Related/Image_Deblurring/generic_grad.m | 52 ++++++++++ .../Gradient_Descent_kappa_influence.m | 38 +++++++ .../README.md | 16 +++ .../const_step.m | 10 ++ .../generic_grad.m | 40 +++++++ .../sample_output.jpg | Bin 0 -> 17219 bytes README.md | 3 +- 14 files changed, 400 insertions(+), 1 deletion(-) create mode 100644 Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m create mode 100644 Image_Related/Image_Deblurring/README.md create mode 100644 Image_Related/Image_Deblurring/blur.m create mode 100644 Image_Related/Image_Deblurring/const_step.m create mode 100644 Image_Related/Image_Deblurring/doc/GD_vs_FISTA_wrapper.pdf create mode 100644 Image_Related/Image_Deblurring/exact_quad.m create mode 100644 Image_Related/Image_Deblurring/fista.m create mode 100644 Image_Related/Image_Deblurring/generic_grad.m create mode 100644 Optimization/Gradient_Descent_kappa_coef_effect/Gradient_Descent_kappa_influence.m create mode 100644 Optimization/Gradient_Descent_kappa_coef_effect/README.md create mode 100644 Optimization/Gradient_Descent_kappa_coef_effect/const_step.m create mode 100644 Optimization/Gradient_Descent_kappa_coef_effect/generic_grad.m create mode 100644 Optimization/Gradient_Descent_kappa_coef_effect/sample_output.jpg diff --git a/Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m b/Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m new file mode 100644 index 0000000..dd35369 --- /dev/null +++ b/Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m @@ -0,0 +1,98 @@ +% Basic Example +% sweden = [1,2,1,1,1;1,2,1,1,1;2,2,2,2,2;1,2,1,1,1;1,2,1,1,1]; +% colormap(parula) +% imagesc(sweden) + + +% Display the original and blurred images +% colormap ( gray ) +% imagesc ( reshape (x , 256 , 256 ) ) +% figure() +% colormap ( gray ) +% imagesc ( reshape (b , 256 , 256 ) ) +[A, b , x ] = blur ( 256 , 5 , 1 ) ; + +f = @(x) (norm(A*x-b))^2; + +% Cmputing gradient elements beforehand +Y1 =A'*A; +Y = 2*Y1; +Z = 2*A'*b; +g = @(x) Y*x-Z; %2*A'*(A*x-b) + +x0 = zeros(size(x)); +% tic; +% % *** Part 1 *** +% [~ ,x_1,x_10,x_100,x_1000] = generic_grad_q3(f, g, exact_quad_q3(Y), x0,1000); +% disp('Generic Grad with Exact Quad complete. Time taken:') +% toc +% figure('Name','Image after 1 Iteration of GD with Exact Quad'); +% colormap ( gray ) +% imagesc ( reshape (x_1 , 256 , 256 ) ) +% +% figure('Name','Image after 10 Iterations of GD with Exact Quad'); +% colormap ( gray ) +% imagesc ( reshape (x_10 , 256 , 256 ) ) +% +% figure('Name','Image after 100 Iterations of GD with Exact Quad'); +% colormap ( gray ) +% imagesc ( reshape (x_100 , 256 , 256 ) ) +% +% figure('Name','Image after 1000 Iterations of GD with Exact Quad'); +% colormap ( gray ) +% imagesc ( reshape (x_1000 , 256 , 256 ) ) +% +% %*** Part 2 *** +L = 2*eigs(Y1,1); +% tic; +% [~ ,x_1,x_10,x_100,x_1000] = generic_grad_q3(f, g, const_step_q3(1/L), x0, 1000); +% disp('Generic Grad with const step complete. Time taken:') +% toc +% figure('Name','Image after 1 Iteration of GD with Const Step'); +% colormap ( gray ) +% imagesc ( reshape (x_1 , 256 , 256 ) ) +% +% figure('Name','Image after 10 Iterations of GD with Const Step'); +% colormap ( gray ) +% imagesc ( reshape (x_10 , 256 , 256 ) ) +% +% figure('Name','Image after 100 Iterations of GD with Const Step'); +% colormap ( gray ) +% imagesc ( reshape (x_100 , 256 , 256 ) ) +% +% figure('Name','Image after 1000 Iterations of GD with Const Step'); +% colormap ( gray ) +% imagesc ( reshape (x_1000 , 256 , 256 ) ) + +% *** Part 3 *** +% tic; +% [~ ,x_1,x_10,x_100,x_1000] = fista(f, g, L, x0, 1000); +% toc +% figure('Name','Image after 1 Iteration of FISTA'); +% colormap ( gray ) +% imagesc ( reshape (x_1 , 256 , 256 ) ) +% +% figure('Name','Image after 10 Iteration of FISTA'); +% colormap ( gray ) +% imagesc ( reshape (x_10 , 256 , 256 ) ) +% +% figure('Name','Image after 100 Iteration of FISTA'); +% colormap ( gray ) +% imagesc ( reshape (x_100 , 256 , 256 ) ) +% +% figure('Name','Image after 1000 Iteration of FISTA'); +% colormap ( gray ) +% imagesc ( reshape (x_1000 , 256 , 256 ) ) + +% Plot gs versus Time +eps = 10^(-5); +[x_const,fs_const,gs_const,ts_const] = generic_grad(f, g, const_step(1/L), x0, 1000); +[x_fista,fs_fista,gs_fista,ts_fista] = fista(f, g, L, x0, eps); +figure('Name', 'f vs Time Comparison'); +semilogy(ts_const, fs_const); +hold on; +semilogy(ts_fista, gs_fista); +title('Semilog Plot of f vs Time'); +xlabel('Time') ; +ylabel('f(x)') ; +legend({'const','fista'},'Location','northeast'); diff --git a/Image_Related/Image_Deblurring/README.md b/Image_Related/Image_Deblurring/README.md new file mode 100644 index 0000000..7873859 --- /dev/null +++ b/Image_Related/Image_Deblurring/README.md @@ -0,0 +1,3 @@ +[Results](/Image_Related/Image_Deblurring/doc/GD_vs_FISTA_wrapper.pdf) + +To reconstruct run [GD_vs_FISTA_wrapper.m](/Image_Related/Image_Deblurring/GD_vs_FISTA_wrapper.m) diff --git a/Image_Related/Image_Deblurring/blur.m b/Image_Related/Image_Deblurring/blur.m new file mode 100644 index 0000000..ab01967 --- /dev/null +++ b/Image_Related/Image_Deblurring/blur.m @@ -0,0 +1,83 @@ +function [A,b,x] = blur(N,band,sigma) +%BLUR Test problem: digital image deblurring. +% +% function [A,b,x] = blur(N,band,sigma) +% +% The matrix A is an N*N-by-N*N symmetric, doubly block Toeplitz matrix that +% models blurring of an N-by-N image by a Gaussian point spread function. +% It is stored in sparse matrix format. +% +% In each Toeplitz block, only matrix elements within a distance band-1 +% from the diagonal are nonzero (i.e., band is the half-bandwidth). +% If band is not specified, band = 3 is used. +% +% The parameter sigma controls the width of the Gaussian point spread +% function and thus the amount of smoothing (the larger the sigma, the wider +% the function and the more ill posed the problem). If sigma is not +% specified, sigma = 0.7 is used. +% +% The vector x is a columnwise stacked version of a simple test image, while +% b holds a columnwise stacked version of the blurrred image; i.e, b = A*x. + +% Per Christian Hansen, IMM, 11/11/97. + +% Initialization. +if (nargin < 2), band = 3; end +band = min(band,N); +if (nargin < 3), sigma = 0.7; end + +% Construct the matrix as a Kronecker product. +z = [exp(-([0:band-1].^2)/(2*sigma^2)),zeros(1,N-band)]; +A = toeplitz(z); +A = sparse(A); +A = (1/(2*pi*sigma^2))*kron(A,A); + +% Generate x and b, if required. +if (nargout > 1) + + % Start with an image of all zeros. + x = zeros(N,N); + N2 = round(N/2); + N3= round(N/3); + N6 = round(N/6); + N12 = round(N/12); + + % Add a large ellipse. + T = zeros(N6,N3); + for i=1:N6, for j=1:N3 + if ( (i/N6)^2 + (j/N3)^2 < 1 ), T(i,j) = 1; end + end, end + T = [fliplr(T),T]; + T = [flipud(T);T]; + x(2+[1:2*N6],N3-1+[1:2*N3]) = T; + + % Add a smaller ellipse. + T = zeros(N6,N3); + for i=1:N6, for j=1:N3 + if ( (i/N6)^2 + (j/N3)^2 < 0.6 ), T(i,j) = 1; end + end, end + T = [fliplr(T),T]; + T = [flipud(T);T]; + x(N6+[1:2*N6],N3-1+[1:2*N3]) = x(N6+[1:2*N6],N3-1+[1:2*N3]) + 2*T; + % Correct for overlap. + f = find(x==3); + x(f) = 2*ones(size(f)); + + % Add a triangle. + T = triu(ones(N3,N3)); + [mT,nT] = size(T); + x(N3+N12+[1:nT],1+[1:mT]) = 3*T; + + % Add a cross. + T = zeros(2*N6+1,2*N6+1); + [mT,nT] = size(T); + T(N6+1,1:nT) = ones(1,nT); + T(1:mT,N6+1) = ones(mT,1); + x(N2+N12+[1:mT],N2+[1:nT]) = 4*T; + + % Make sure x is N-times-N, and stack the columns of x. + x = reshape(x(1:N,1:N),N^2,1); + + % Compute the blurred image. + b = A*x; +end diff --git a/Image_Related/Image_Deblurring/const_step.m b/Image_Related/Image_Deblurring/const_step.m new file mode 100644 index 0000000..8dd14f2 --- /dev/null +++ b/Image_Related/Image_Deblurring/const_step.m @@ -0,0 +1,10 @@ +function lsearch = const_step(s) + +% verify s is a real positive scalar +if s<=0 || ~isreal(s) || 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a/Image_Related/Image_Deblurring/fista.m b/Image_Related/Image_Deblurring/fista.m new file mode 100644 index 0000000..e8f141d --- /dev/null +++ b/Image_Related/Image_Deblurring/fista.m @@ -0,0 +1,44 @@ +function [x, fs, gs, ts] = fista(f, gf, L, x0, eps) + +% Input checks +if ~isa(f,'function_handle') || ~isa(gf,'function_handle') + error("f and gf must be functions") +end +if eps<=0 || ~isreal(eps) || ~isscalar(eps) + error("eps must be a real positive scalar") +end +if ~isreal(x0) + error("x0 must be real") +end + +% Start Timer +tic + +% Initialize Paramaters +x = x0; +y = x0; +grad = gf(x); +grad_norm = norm(grad); + +% Result containers +gs = grad_norm; +fs = f(x0); +ts = 0; +t_k = 1; + + while (grad_norm >= eps) + old_x = x; + x = y-((1/L )*grad); + old_t_k = t_k; + t_k = (1+sqrt(1+4*((old_t_k)^2)))/2; + y = x+((old_t_k-1)/t_k)*(x-old_x); + grad = gf(y); + grad_norm = norm(grad); + + % Recording interim results + fs = [fs, f(y)]; + gs = [gs, grad_norm]; + ts = [ts, toc]; + end +end + diff --git a/Image_Related/Image_Deblurring/generic_grad.m b/Image_Related/Image_Deblurring/generic_grad.m new file mode 100644 index 0000000..723489c --- /dev/null +++ b/Image_Related/Image_Deblurring/generic_grad.m @@ -0,0 +1,52 @@ +function [x ,x_1,x_10,x_100,x_1000] = generic_grad_q3(f , gf , lsearch , x0 , MAX_ITERATIONS) + +% Input checks +if ~isa(f,'function_handle') || ~isa(gf,'function_handle') || ~isa(lsearch,'function_handle') + error("f, gf and lsearch must be functions") +end +if ~isscalar(MAX_ITERATIONS) || ~(MAX_ITERATIONS==floor(MAX_ITERATIONS)) || ~isreal(MAX_ITERATIONS) + error("MAX_ITERATIONS must be a real scalar integer") +end +if ~isreal(x0) + error("x0 must be real") +end + +% Initialize Paramaters +x = x0; +grad = gf(x); +% Result containers + +iteration = 0; + +% Execute Generic Gradient Descent Algorithm +% while (grad_norm > eps && iteration < MAX_ITERATIONS) +while (iteration < MAX_ITERATIONS) + + t_k = lsearch(f,x,grad); + x = x - (t_k * grad); + grad = gf(x); + + % Recording interim results + iteration=iteration+1; + if(iteration == 1) + x_1=x; + disp(['Iteration ', num2str(iteration), ' complete']) + end + if(iteration == 10) + x_10=x; + disp(['Iteration ', num2str(iteration), ' complete']) + end + if(iteration == 100) + x_100=x; + end + if(iteration == 1000) + x_1000=x; + end + + if(mod(iteration,100) == 0) + disp(['Iteration ', num2str(iteration), ' complete']) + end + +end +end + diff --git a/Optimization/Gradient_Descent_kappa_coef_effect/Gradient_Descent_kappa_influence.m b/Optimization/Gradient_Descent_kappa_coef_effect/Gradient_Descent_kappa_influence.m new file mode 100644 index 0000000..ed7dc20 --- /dev/null +++ b/Optimization/Gradient_Descent_kappa_coef_effect/Gradient_Descent_kappa_influence.m @@ -0,0 +1,38 @@ +% Parameter Initialization +x0 = [1000;1000;1000]; + +F=[2,2,3; 2,5,5; 3,5,7]; +G=[10,8,9;8,13,11;9,11,15]; + +g=@(x) 0.5*x'*G*x; +f=@(x) 0.5*x'*F*x; + +grad_g=@(x) G*x; +grad_f=@(x) F*x; + +step_G = 1/eigs(G,1); +step_F = 1/eigs(F,1); + +K_G = cond(G) +K_F = cond(F) +k_ratio = K_F/K_G + +disp('Generic Grad for f complete. ') +[~, fs_f, iterations_f] = generic_grad(f, grad_f, const_step(step_F), x0, 100000, 10^-5); + +disp('Generic Grad for g complete.') +[~, fs_g, iterations_g]= generic_grad(g, grad_g, const_step(step_G), x0, 100000, 10^-5); + +% Plot fs versus Number of Iterations +figure('Name','fs vs Iters Comparison'); +semilogy(fs_f); +hold on; +semilogy(fs_g); + +title('Semilogy Plot of fs & gs vs Number of Iterations'); +xlabel('Iteration'); +ylabel('function value'); +legend({'fs\_f','fs\_g'},'Location','southwest'); + +iterations_ratio = iterations_f/iterations_g + diff --git a/Optimization/Gradient_Descent_kappa_coef_effect/README.md b/Optimization/Gradient_Descent_kappa_coef_effect/README.md new file mode 100644 index 0000000..c466b47 --- /dev/null +++ b/Optimization/Gradient_Descent_kappa_coef_effect/README.md @@ -0,0 +1,16 @@ +Sample Output: +```matlab +K_G = + 10.9129 +K_F = + 25.4916 +k_ratio = + 2.3359 +Generic Grad for f complete. +Total iterations taken: 250 +Generic Grad for g complete. +Total iterations taken: 105 +iterations_ratio = + 2.3810 +``` +![graph](/Optimization/Gradient_Descent_kappa_coef_effect/sample_output.jpg) \ No newline at end of file diff --git a/Optimization/Gradient_Descent_kappa_coef_effect/const_step.m b/Optimization/Gradient_Descent_kappa_coef_effect/const_step.m new file mode 100644 index 0000000..8dd14f2 --- /dev/null +++ b/Optimization/Gradient_Descent_kappa_coef_effect/const_step.m @@ -0,0 +1,10 @@ +function lsearch = const_step(s) + +% verify s is a real positive scalar +if s<=0 || ~isreal(s) || ~isscalar(s) + error("s must be a real positive scalar") +end + +lsearch =@(f,x_k,g_k) s; +end + diff --git a/Optimization/Gradient_Descent_kappa_coef_effect/generic_grad.m b/Optimization/Gradient_Descent_kappa_coef_effect/generic_grad.m new file mode 100644 index 0000000..f6b700f --- /dev/null +++ b/Optimization/Gradient_Descent_kappa_coef_effect/generic_grad.m @@ -0,0 +1,40 @@ +function [x, fs, iteration] = generic_grad_q5(f , gf , lsearch , x0 , MAX_ITERATIONS, eps) + +% Input checks +if ~isa(f,'function_handle') || ~isa(gf,'function_handle') || ~isa(lsearch,'function_handle') + error("f, gf and lsearch must be functions") +end +if ~isscalar(MAX_ITERATIONS) || ~(MAX_ITERATIONS==floor(MAX_ITERATIONS)) || ~isreal(MAX_ITERATIONS) + error("MAX_ITERATIONS must be a real scalar integer") +end +if ~isreal(x0) + error("x0 must be real") +end +if eps<=0 || ~isreal(eps) || ~isscalar(eps) + error("eps must be a real positive scalar") +end + +% Initialize Paramaters +x = x0; +grad = gf(x); +x_old = x; +t_k = lsearch(f,x,grad); +x = x_old - (t_k * grad); +iteration = 1; +fs = f(x); + +% Execute Generic Gradient Descent Algorithm +while (iteration < MAX_ITERATIONS && (abs((f(x_old)-f(x))) >= eps)) + + x_old = x; + grad = gf(x); + t_k = lsearch(f,x,grad); + x = x_old - (t_k * grad); + fs = [fs, f(x)]; + % Keeping track of currrent Iteration + iteration=iteration+1; + +end +disp(['Total iterations taken: ', num2str(iteration)]) +end + diff --git a/Optimization/Gradient_Descent_kappa_coef_effect/sample_output.jpg b/Optimization/Gradient_Descent_kappa_coef_effect/sample_output.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1f0171c03f4ba85a8bd4acc882c5c35e970d88f5 GIT binary patch literal 17219 zcmdVB1z1(x);7FoP(qY$1VkF?4w06WPU+mVq=cZLbR!{+G)T9AG$`FjH%NEw{Vkt3 zM?5|!{`0>7^?loGy7pRgu~=iyG464X05%Sr18&JkNJ{{4Z~y=Y{sUl8fIv=E<+12P z3Mm5{V`~!%Ne2Tva|GRkkj%C z@~{aovvaV0e+V2J8XEddbOH;hKgGNJRAZ7JR$-T5+WjawT8-lwDI;N;@w;pG#1BrYK-B`xzr zMO95*LsQGh*u>P#+``h)>6x>OtDC#u%UAvZuipelMn%WO#=VVCNYBX3dY_$>n^#&^ zUQt<9T~qt1rM0cSqqD1faA8*oa7!50Gz*DxnzI-?_{70u|?BL|RER8Wp?p zF0P@&zzsZVj(M8B@1p%C*}o^4@Bb&s{wCPp<(dT05#YehL%;@vfs^z5={`69aukt2 zIn7O5k30E5%w)JR`RNJ9)>UCdPfCXx^?JMKEkm%mC2@kFT=Fo ztqJ{I=Fwupw4AHsZBnY#A9vWd4ZA1Sq5BQLc!sFd}J>he&#Jm4t*5KvOm)oC-c;2hqnBv8t9ZnsM{PU%!rcy%dfyYPkKLT0cVIysSu z&AIeYnA3xAE-CjY6{D6RWp^y-TfHjH{LmIl?Bt}_t=fcJ5syVF&dhjusH$=ML5b|o z$67<~JRG$t-+Wt~d4V5uy!rO&G)8t}NV1U`6)!1V(Wf6)y?_Pxv`%){G#)b}zdRqp z^l8ff4Y$GDJCTB{?|rz<3*4T|W{VfE*K(bBgh-$yPxm7bJ_4fZgkBm}-UreHyZ2Z? 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