{"worksheets": [{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["Mesh Deformation", "================", "", "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes.", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$", "$\\newcommand{\\norm}[1]{\\|#1\\|}$", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$", "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$", "$\\newcommand{\\CC}{\\mathbb{C}}$", "$\\newcommand{\\RR}{\\mathbb{R}}$", "$\\newcommand{\\EE}{\\mathbb{E}}$", "$\\newcommand{\\Zz}{\\mathcal{Z}}$", "$\\newcommand{\\Ww}{\\mathcal{W}}$", "$\\newcommand{\\Vv}{\\mathcal{V}}$", "$\\newcommand{\\Nn}{\\mathcal{N}}$", "$\\newcommand{\\NN}{\\mathcal{N}}$", "$\\newcommand{\\Hh}{\\mathcal{H}}$", "$\\newcommand{\\Bb}{\\mathcal{B}}$", "$\\newcommand{\\Ee}{\\mathcal{E}}$", "$\\newcommand{\\Cc}{\\mathcal{C}}$", "$\\newcommand{\\Gg}{\\mathcal{G}}$", "$\\newcommand{\\Ss}{\\mathcal{S}}$", "$\\newcommand{\\Pp}{\\mathcal{P}}$", "$\\newcommand{\\Ff}{\\mathcal{F}}$", "$\\newcommand{\\Xx}{\\mathcal{X}}$", "$\\newcommand{\\Mm}{\\mathcal{M}}$", "$\\newcommand{\\Ii}{\\mathcal{I}}$", "$\\newcommand{\\Dd}{\\mathcal{D}}$", "$\\newcommand{\\Ll}{\\mathcal{L}}$", "$\\newcommand{\\Tt}{\\mathcal{T}}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\al}{\\alpha}$", "$\\newcommand{\\la}{\\lambda}$", "$\\newcommand{\\ga}{\\gamma}$", "$\\newcommand{\\Ga}{\\Gamma}$", "$\\newcommand{\\La}{\\Lambda}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\Si}{\\Sigma}$", "$\\newcommand{\\be}{\\beta}$", "$\\newcommand{\\de}{\\delta}$", "$\\newcommand{\\De}{\\Delta}$", "$\\newcommand{\\phi}{\\varphi}$", "$\\newcommand{\\th}{\\theta}$", "$\\newcommand{\\om}{\\omega}$", "$\\newcommand{\\Om}{\\Omega}$"]}, {"cell_type": "markdown", "metadata": {}, "source": ["This tour explores deformation of 2D mesh using Laplacian interpolation.", "The dense deformation field is obtained from a sparse set of displaced", "anchor point by computing harmonic interpolation."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["from __future__ import division", "import nt_toolbox as nt", "from nt_solutions import meshdeform_5_deformation as solutions", "%matplotlib inline", "%load_ext autoreload", "%autoreload 2"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Mesh Creation", "-------------", "We create a simple mesh with fine scale details.", "", "", "We generate point on a square."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["p = 150", "[Y, X] = meshgrid(linspace(-1, 1, p), linspace(-1, 1, p))", "vertex0 = [X(: )'; Y(: )'; zeros(1, p^2)]", "n = size(vertex0, 2)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["We generate a triangulation of a square."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["I = reshape(1: p^2, p, p)", "a = I(1: p-1, 1: p-1); b = I(2: p, 1: p-1); c = I(1: p-1, 2: p)", "d = I(2: p, 1: p-1); e = I(2: p, 2: p); f = I(1: p-1, 2: p)", "faces = cat(1, [a(: ) b(: ) c(: )], [d(: ) e(: ) f(: )])'"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Width and height of the bumps."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["sigma = .03", "h = .35", "q = 8"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Elevate the surface using bumps."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["t = linspace(-1, 1, q + 2); t([1 length(t)]) = []", "vertex = vertex0", "for i in 1: q:", "for j in 1: q:", "        d = (X(: )'-t(i)).^2 + (Y(: )'-t(j)).^2", "        vertex(3, : ) = vertex(3, : ) + h * exp(-d/ (2*sigma^2))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display the surface."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["plot_mesh(vertex, faces)", "view(3)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute its geometric (cotan) Laplacian"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["W = sparse(n, n)", "for i in 1: 3:", "   i1 = mod(i-1, 3) + 1", "   i2 = mod(i  , 3) + 1", "   i3 = mod(i + 1, 3) + 1", "   pp = vertex(: , faces(i2, : )) - vertex(: , faces(i1, : ))", "   qq = vertex(: , faces(i3, : )) - vertex(: , faces(i1, : ))", "   % normalize the vectors   ", "   pp = pp ./ repmat(sqrt(sum(pp.^2, 1)), [3 1])", "   qq = qq ./ repmat(sqrt(sum(qq.^2, 1)), [3 1])", "   % compute angles", "   ang = acos(sum(pp.*qq, 1))", "   u = cot(ang)", "   u = clamp(u, 0.01, 100)", "   W = W + sparse(faces(i2, : ), faces(i3, : ), u, n, n)", "   W = W + sparse(faces(i3, : ), faces(i2, : ), u, n, n)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute the symmetric Laplacian matrix."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["d = full(sum(W, 1))", "D = spdiags(d(: ), 0, n, n)", "L = D - W"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Boundary Modification", "---------------------", "We modify the domain by modifying its boundary.", "", "", "Select boundary indexes."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["I = find(abs(X(: )) = =1 | abs(Y(: )) = =1)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute the deformation field (zeros outsize the handle, proportional to", "the normal otherwise)."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Delta0 = zeros(3, n)", "d = (vertex(1, I) + vertex(2, I)) / 2", "Delta0(3, I) = sign(d) .* abs(d).^3"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Modify the Laplacian to take into account the fixed handles."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["L1 = L", "L1(I, : ) = 0", "L1(I + (I-1)*n) = 1"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute the full deformation by solving for Laplacian=0 on each", "coordinate."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Delta = (L1 \\ Delta0')'"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute the deformed mesh."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["vertex1 = vertex + Delta"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["plot_mesh(vertex1, faces)", "view(-100, 15)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 1__", "", "Perform a more complicated deformation of the boundary.", "eform", "isplay it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo1()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 2__", "", "Move both the inside and the boundary.", "", "aplacian", "eform", "isplay it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo2()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 3__", "", "Apply the mesh deformation method to a real mesh, with both large scale", "and fine scale details."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo3()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Non-linear Deformation", "----------------------", "Linear methods give poor results for large deformation.", "", "", "It is possible to obtain better result by applying the linear deformation only", "to a low pass version of the mesh (coarse scale modifications). The", "remaining details are then added in the direction of the normal, in a", "local frame that is rotated to match the deformation of the coarse surface."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 4__", "", "Apply the deformation to the coarse mesh |vertex0| to obtain |vertex1|.", "*Important:* you need to compute and use the cotan Laplacian of the coarse", "mesh, not of the original mesh!", "ompute laplacian", "", "aplacian", "eform", "isplay it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo4()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute the residual vector contribution along the normal (which is vertical)."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["normal = compute_normal(vertex0, faces)", "d = repmat(sum(normal .* (vertex-vertex0)), [3 1])"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 5__", "", "Add the normal contribution |d.*normal| to |vertex1|, but", "after replacing the normal of |vertex0| by the normal of |vertex1|.", "isplay it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo5()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 6__", "", "Try on other surfaces. How can you compute |vertex0| for an arbitrary", "surface ?"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo6()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Bi-Laplacian Deformation", "------------------------", "To take into account the bending of the surface, one can use higher order", "derivative to interpolate the deformation field.", "", "", "The bi-laplacian |LL| corresponds to 4th order derivatives.", "It is the square of the Laplacian |LL=L*L|."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 7__", "", "Compute the bi-laplacian deformation of the coarse shape", "|vertex0| by using |LL| instead of |L|.", "What do you observe ?", "eform", "isplay it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo7()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 8__", "", "Compute the deformation obtained by moving from the Laplacian", "to the bi-laplacian, i.e. with |t*L+(1-t)*LL| for varying t."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo8()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 9__", "", "Apply the full model (Laplacian, bi-Laplacian and non-linear", "deformation) to the deformation of a complicated mesh."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo9()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}]}], "nbformat": 3, "metadata": {"name": ""}, "nbformat_minor": 0}