Image Deformation Using Moving Least Squares（matlab实现）

function mlsd = MLSD2DlinesPrecompute(varargin) % MLSD2DLINESPRECOMPUTE Generates a 2D MLSD structure. % % function mlsd = MLSD2DlinesPrecompute(p,v,type) % % This function precomputes a set of values and generate an MLSD structure % that embeds infos about the deformation algorithm to be used and embeds % the parameters that must be used for the process. This function allows to % initialize a deformation process of one or more an images. For more % informations see [1]. % % [1] "Image Deformation Using Moving Least Squares", % Scott Schaefer, Travis McPhail, Joe Warren % % Parameters % ---------- % IN: % p = The starting position of the handles lines (as [a;b] vectors with % a and b column points that define the extrema of the segment). % v = The points to be moved (i.e. a grid). % type = The algorithm type ('affine','similar','rigid'). (def='rigid') % Parsing parameters: [p,v,type] = ParseParams(varargin{:}); % Removing coincidences: v = RemoveCoincidences(v,p); % Precomputing the weights: W = PrecomputeWeights(p,v); % Precomputing the centroids: [Pstar,denStar] = PrecomputeWCentroids(p,W); % Preparing the structure: mlsd.p = p; mlsd.v = v; mlsd.type = type; mlsd.W = W; mlsd.constr = 'lines'; % Selecting the generation function: switch type case 'affine' mlsd.data = PrecomputeAffine(p,v,W,Pstar); case 'similar' mlsd.data = PrecomputeSimilar(p,v,W,Pstar); case 'rigid' mlsd.data = PrecomputeRigid(p,v,W,Pstar); end % Other data: mlsd.data.denStar = denStar; % ------------------------ LOCAL FUNCTIONS ------------------------ % Removing coincidences: function v = RemoveCoincidences(v,p) % Iterating on points: for i=1:size(p,2) % Coincidences with a: ind = find(v(1,:)==p(1,i) & v(2,:)==p(2,i)); v(:,ind) = v(:,ind) + 1e-8; % Coincidences with b: ind = find(v(1,:)==p(3,i) & v(2,:)==p(4,i)); v(:,ind) = v(:,ind) + 1e-8; end % ----------------------------------------------------------------- % Precomputing the weights: function W = PrecomputeWeights(p,v) % Init W: W.d00 = zeros(size(p,2),size(v,2)); W.d01 = W.d00; W.d11 = W.d00; % Computing usefull quantities: a_b = p(1:2,:)-p(3:4,:); b_a = -a_b; for i=1:size(p,2) % Computing the a-v and b-v values: a_v = repmat(p(1:2,i),[1,size(v,2)])-v; b_v = repmat(p(3:4,i),[1,size(v,2)])-v; v_b = -b_v; % Computing the triplaned values: a_v_t = [-a_v(2,:);a_v(1,:)]; b_v_t = [-b_v(2,:);b_v(1,:)]; % Computing the big Delta and all the other combinations: Num1 = b_v(1,:).*b_a(1,i) + b_v(2,:).*b_a(2,i); Num2 = a_v(1,:).*a_b(1,i) + a_v(2,:).*a_b(2,i); Den1 = b_v_t(1,:).*b_a(1,i) + b_v_t(2,:).*b_a(2,i); Delta = a_v_t(1,:).*a_b(1,i) + a_v_t(2,:).*a_b(2,i); % Computing the angle theta: theta = atan2(Den1,Num1)-atan2(Delta,Num2); % Clearing: clear Num1 Num2 Den1 a_v_t b_v_t; % The beta vectors: beta00 = sum(a_v.^2,1); beta01 = sum(a_v.*v_b,1); beta11 = sum(v_b.^2,1); % Removing the Delta==0 case: ind = find(Delta==0); Delta(ind) = Delta(ind) + 1e-8; % Precomputing values: a_bNorm = norm(a_b(:,i)); a_bNorm_2Delta2 = a_bNorm./(2*Delta.^2); theta_Delta = theta./Delta; % Computing the weights: W.d00(i,:) = a_bNorm_2Delta2.*(beta01./beta00-beta11.*theta_Delta); W.d01(i,:) = a_bNorm_2Delta2.*(1-beta01.*theta_Delta); W.d11(i,:) = a_bNorm_2Delta2.*(beta01./beta11-beta00.*theta_Delta); % % Getting the case vector: % DeltaIsZero = Delta==0; % % % The non-zero case: % ind = find(~DeltaIsZero); % a_bNorm = norm(a_b(:,i)); % a_bNorm_2Delta2 = a_bNorm./(2*Delta(ind).^2); % theta_Delta = theta(ind)./Delta(ind); % W.d00(i,ind) = a_bNorm_2Delta2.*(beta01(ind)./beta00(ind)-beta11(ind).*theta_Delta); % W.d01(i,ind) = a_bNorm_2Delta2.*(1-beta01(ind).*theta_Delta); % W.d11(i,ind) = a_bNorm_2Delta2.*(beta01(ind)./beta11(ind)-beta00(ind).*theta_Delta); % % % Clearing: % clear beta00 beta01 beta11 theta Delta theta_Delta; % % % The zero case: % ind = find(DeltaIsZero); % if ~isempty(ind) % % Computing the other elements: % a_bNorm5 = a_bNorm.^5; % a_vb_a = a_v(1,ind).*b_a(1,i) + a_v(2,ind).*b_a(2,i); % v_bb_a = v_b(1,ind).*b_a(1,i) + v_b(2,ind).*b_a(2,i); % W.d00(i,ind) = a_bNorm5./(3*v_bb_a.*a_vb_a.^3+1e-8); % W.d01(i,ind) = a_bNorm5./(6*v_bb_a.^2.*a_vb_a.^2+1e-8); % W.d11(i,ind) = a_bNorm5./(3*v_bb_a.^3.*a_vb_a+1e-8); % end % Clearing: clear a_vb_a v_bb_a ind DeltaIsZero a_v b_v v_b; end % ----------------------------------------------------------------- % Precomputing weighted centroids: function [Pstar,denStar] = PrecomputeWCentroids(p,W) % Computing the den star: denStar = sum(W.d00+2*W.d01+W.d11,1); % Initializing the Pstar array: Pstar = zeros(2,size(W.d00,2)); % Iterating on segments: for i=1:size(p,2) % Auxiliary: d00_d01 = W.d00(i,:) + W.d01(i,:); d01_d11 = W.d01(i,:) + W.d11(i,:); % Updating the centroids: Pstar = Pstar + repmat(p(1:2,i),[1,size(d00_d01,2)]).*[d00_d01;d00_d01] + ... repmat(p(3:4,i),[1,size(d00_d01,2)]).*[d01_d11;d01_d11]; end % Computing the centroids: Pstar = Pstar./[denStar;denStar]; % ----------------------------------------------------------------- % Precomputing the affine deformation: function data = PrecomputeAffine(p,v,W,Pstar) % Computing the ahat and bhat and the matrix elements: ahat = cell(1,size(p,2)); bhat = ahat; a = zeros(1,size(Pstar,2)); b = a; d = a; for i=1:size(p,2) % Computing ahat{i} and bhat{i}: ahat{i} = repmat(p(1:2,i),[1,size(Pstar,2)])-Pstar; bhat{i} = repmat(p(3:4,i),[1,size(Pstar,2)])-Pstar; % Computing the matrix elements: a = a + ahat{i}(1,:).^2.*W.d00(i,:) + ... 2*ahat{i}(1,:).*bhat{i}(1,:).*W.d01(i,:) + ... bhat{i}(1,:).^2.*W.d11(i,:); b = b + ahat{i}(1,:).*ahat{i}(2,:).*W.d00(i,:) + ... (ahat{i}(2,:).*bhat{i}(1,:)+ahat{i}(1,:).*bhat{i}(2,:)).*W.d01(i,:) + ... bhat{i}(1,:).*bhat{i}(2,:).*W.d11(i,:); d = d + ahat{i}(2,:).^2.*W.d00(i,:) + ... 2*ahat{i}(2,:).*bhat{i}(2,:).*W.d01(i,:) + ... bhat{i}(2,:).^2.*W.d11(i,:); end % Computing the inverse: det = a.*d - b.^2; det(find(det==0)) = 1e-8; Ia = d./det; Ib = -b./det; Id = a./det; % Computing the v-Pstar values: M1 = v - Pstar; % Computing the first product elements: F1 = [sum(M1.*[Ia;Ib],1);sum(M1.*[Ib;Id],1)]; % Computing the A values: A = cell(1,size(p,2)); for j=1:size(p,2) % All but W: a = sum(F1.*ahat{j},1); b = sum(F1.*bhat{j},1); % Multipling per W: A{j}.a = a.*W.d00(j,:) + b.*W.d01(j,:); A{j}.b = a.*W.d01(j,:) + b.*W.d11(j,:); end % The data structure: data.A = A; % ----------------------------------------------------------------- % Precomputing the Asimilar: function [A,R1] = PrecomputeA(Pstar,ahat,bhat,v,W) % Allocating: A = cell(1,numel(ahat)); % Fixed part: R1 = v - Pstar; R2 = [R1(2,:);-R1(1,:)]; % Iterating on segments: for i=1:numel(ahat) % The transformed versions: ahat_t = [ahat{i}(2,:);-ahat{i}(1,:)]; bhat_t = [bhat{i}(2,:);-bhat{i}(1,:)]; % Computing the first block: a11 = sum(ahat{i}.*R1,1); a12 = sum(ahat{i}.*R2,1); a21 = sum(ahat_t.*R1,1); a22 = sum(ahat_t.*R2,1); a31 = sum(bhat{i}.*R1,1); a32 = sum(bhat{i}.*R2,1); a41 = sum(bhat_t.*R1,1); a42 = sum(bhat_t.*R2,1); % Computing the values: A{i}.a11 = a11.*W.d00(i,:) + a31.*W.d01(i,:); A{i}.a12 = a12.*W.d00(i,:) + a32.*W.d01(i,:); A{i}.a21 = a21.*W.d00(i,:) + a41.*W.d01(i,:); A{i}.a22 = a22.*W.d00(i,:) + a42.*W.d01(i,:); A{i}.a31 = a11.*W.d01(i,:) + a31.*W.d11(i,:); A{i}.a32 = a12.*W.d01(i,:) + a32.*W.d11(i,:); A{i}.a41 = a21.*W.d01(i,:) + a41.*W.d11(i,:); A{i}.a42 = a22.*W.d01(i,:) + a42.*W.d