📄 mlsddemo.html
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<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Moving Least Squares</title> <meta name="generator" content="MATLAB 7.2"> <meta name="date" content="2006-09-13"> <meta name="m-file" content="MLSDdemo"><style>body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. On Gecko-based browsers, the shrink-to-fit doesn't work. */ p,h1,h2,div.content div { /* for MATLAB's browser */ width: 600px; /* for Mozilla, but the "width" tag overrides it anyway */ max-width: 600px; /* for IE */ width:expression(document.body.clientWidth > 620 ? "600px": "auto" );}pre.codeinput { background: #EEEEEE; padding: 10px;}@media print { pre.codeinput {word-wrap:break-word; width:100%;}} span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>Moving Least Squares</h1> <introduction><pre>The Moving Least Squares algorithm is a deformation technique thatallows to compute a map f:R2->R2 from the transformation of a set of Npivot points p in the new positions q. The map f is smooth (f in C2),preserves the identity (for q=p the map is the identity) and ensures thatthe points p are transformed in the points q (f(p)=q). In the work [1]the authors describes a simple optimization technique that allows to get,for each point of the R2 plane, the best linear transformation (affinity,similarity or rigid transformation) that moves the points according to aset of weights of the transformed pi-qi.The deformations are here implemented in matlab for points and imagesusing as pivots points and segments as explained in [1]. Each algorithm(the one with points as pivots and the one with segments as pivots)implements the three cases where the local transformation is 'affine','similar' or 'rigid'. Only for Matlab7.2.0 or newer there is the codethat allows to deform interactively points and images (really funny :).</pre><pre>[1] "Image Deformation Using Moving Least Squares", Scott Schaefer, Travis McPhail, Joe Warren</pre></introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">MLS for points using points as pivots</a></li> <li><a href="#2">MLS for images using points as pivots</a></li> <li><a href="#3">MLS for points using segments as pivots</a></li> <li><a href="#4">Interactive MLS deformation</a></li> <li><a href="#5">Other infos</a></li> </ul> </div> <h2>MLS for points using points as pivots<a name="1"></a></h2><pre>Here we try to use the MLS tools to deform a set of points. As firststep the points are required; then the pivots are required; finally thenew positions of the pivots (in the same order) are taken. Given this setof informations the mlsd object can be generated: an mlsd object is theprecomputation of a deformation on a set of points v given the pivots andthe deformation type. The mlsd allows to precompute all the values thatcan be computed when the destination positions of the pivots is unknown.This allows to get a real-time deformation tool that can work fast onimages too. Given the mlsd for a set of new pivot positions thetransformed points can be computed. The default transformation used isthe rigid one, the same work can be done using different transformationtypes.</pre><pre class="codeinput"><span class="comment">% Collecting the points:</span>f=figure; imshow(ones(100));v = getpoints;close(f);<span class="comment">% Requiring the pivots:</span>f=figure; axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>; plotshape(v,true,<span class="string">'g-'</span>);p = getpoints;close(f);<span class="comment">% Requiring the new pivots:</span>f=figure; axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>; plotshape(v,true,<span class="string">'g-'</span>); plotpointsLabels(p,<span class="string">'r.'</span>);q = getpoints;close(f);<span class="comment">% Precomputation of the mlsd:</span>mlsd = MLSD2DpointsPrecompute(p,v);<span class="comment">% Obtaining the transformed points:</span>fv = MLSD2DTransform(mlsd,q);<span class="comment">% Plotting:</span>figure;subplot(121); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(v,true,<span class="string">'g-'</span>); plotpoints(p,<span class="string">'r.'</span>);subplot(122); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(fv,true,<span class="string">'k-'</span>); plotpoints(q,<span class="string">'b.'</span>);<span class="comment">% Other transformations:</span>fv_rigid = fv;<span class="comment">% Transforming the same points using a similarity:</span>mlsd = MLSD2DpointsPrecompute(p,v,<span class="string">'similar'</span>);fv_similar = MLSD2DTransform(mlsd,q);<span class="comment">% Transforming the same points using an affinity:</span>mlsd = MLSD2DpointsPrecompute(p,v,<span class="string">'affine'</span>);fv_affine = MLSD2DTransform(mlsd,q);<span class="comment">% Plotting:</span>figure;subplot(141); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(v,true,<span class="string">'g-'</span>); plotpoints(p,<span class="string">'r.'</span>);title(<span class="string">'Original'</span>);subplot(142); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(fv_rigid,true,<span class="string">'k-'</span>); plotpoints(q,<span class="string">'b.'</span>);title(<span class="string">'Rigid'</span>);subplot(143); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(fv_similar,true,<span class="string">'k-'</span>); plotpoints(q,<span class="string">'b.'</span>);title(<span class="string">'Similar'</span>);subplot(144); axis <span class="string">equal</span> <span class="string">ij</span>; hold <span class="string">on</span>;plotshape(fv_affine,true,<span class="string">'k-'</span>); plotpoints(q,<span class="string">'b.'</span>);title(<span class="string">'Affine'</span>);</pre><img vspace="5" hspace="5" src="MLSDdemo_01.png"> <img vspace="5" hspace="5" src="MLSDdemo_02.png"> <h2>MLS for images using points as pivots<a name="2"></a></h2><pre>The same work done for points can be used for images using a dense setof points over a grid that cover the image and using the interpolation toget all the transformations. The function MLSD2DWarp do exactly thiswork. The same mlsd must be computed to obtain the warping, only the setof points is different because a grid is required, so a step must bechosen.</pre><pre class="codeinput"><span class="comment">% The step size:</span>step = 15;<span class="comment">% Reading an image:</span>img = imread(<span class="string">'image.jpg'</span>);<span class="comment">% Requiring the pivots:</span>f=figure; imshow(img);p = getpoints;close(f);<span class="comment">% Requiring the new pivots:</span>f=figure; imshow(img); hold <span class="string">on</span>; plotpointsLabels(p,<span class="string">'r.'</span>);q = getpoints;close(f);<span class="comment">% Generating the grid:</span>[X,Y] = meshgrid(1:step:size(img,2),1:step:size(img,1));gv = [X(:)';Y(:)'];<span class="comment">% Generating the mlsd:</span>mlsd = MLSD2DpointsPrecompute(p,gv);<span class="comment">% The warping can now be computed:</span>imgo = MLSD2DWarp(img,mlsd,q,X,Y);<span class="comment">% Plotting the result:</span>figure; imshow(imgo); hold <span class="string">on</span>; plotpoints(q,<span class="string">'r.'</span>);</pre><img vspace="5" hspace="5" src="MLSDdemo_03.png"> <h2>MLS for points using segments as pivots<a name="3"></a></h2><pre>The function MLSD2DlinesPrecompute allows to get mlsd object that can beused to compute teh MLS deformation of points or images where theconstraints (pivots) are segments. Here an exemple is shown only for theimage.</pre><pre class="codeinput"><span class="comment">% Requiring the pivots:</span>f=figure; imshow(img);lp = getpoints;close(f);<span class="comment">% Rearranging the pivots:</span><span class="keyword">if</span> mod(size(lp,2),2)==1 lp = lp(:,1:end-1);<span class="keyword">end</span>p = [lp(:,1:2:end);lp(:,2:2:end)];<span class="comment">% Requiring the new pivots:</span>f=figure; imshow(img); hold <span class="string">on</span>;plotpointsLabels(lp,<span class="string">'r.'</span>);plot([p(1,:);p(3,:)],[p(2,:);p(4,:)],<span class="string">'r-'</span>);lq = getpoints;close(f);<span class="comment">% Rearranging the pivots:</span><span class="keyword">if</span> mod(size(lq,2),2)==1 lq = lq(:,1:end-1);<span class="keyword">end</span>⌨️ 快捷键说明
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