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tomasaschan/Cubic B-splines.ipynb

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Cubic B-splines.ipynb
{
"metadata": {
"language": "Julia",
"name": "Cubic B-splines"
},
"nbformat": 3,
"nbformat_minor": 0,
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"cell_type": "markdown",
"metadata": {},
"source": "# Cubic B-splines in Julia\n\nAfter having worked on this for a few days, pouring over every text on the subject that I've been able to find and implementing at least a dozen solutions that didn't work, I think it's time to be a little more formal and seek peer review of my thought process.\n\nThe implementation below is entirely based on [these lecture notes][1] by [Dr Y. Sue Liu][2] at the University of Edinburgh.\n\n[1]: http://www.geos.ed.ac.uk/~yliu23/docs/lect_spline.pdf\n[2]: http://www.geos.ed.ac.uk/~yliu23/\n\n### Introduction\n\nThe function we're interpolating is $f(x) = \\frac{1}{1+x^2}\\sin(x)$, on the interval $x\\in[-4,4]$, on a uniformly spaced grid with spacing 1, i.e. with grid points $x_k = -4,-3,\\ldots,3,4$ for $k = 0,\\ldots,N$, and corresponding data values $f_k=f(x_k)$."
},
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"input": "using Gadfly\n\nf(x) = sin(-x)./(1.+x.^2)\nxk = Float64[-4:4]\nfk = f(xk)\nplot(layer(f,-4,4),layer(x=xk,y=fk,Geom.point))",
"language": "python",
"metadata": {},
"outputs": [
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"source": "As described by Dr. Liu (eqn 3.8) the cubic B-splines are given by the following expression\n\n$$ B_0(x) =\\begin{cases}\n0 & x - x_0\\le -2 \\\\\n\\frac{1}{6} (2+(x-x_0))^3 & -2 \\le x-x_0 \\le -1 \\\\\n\\frac{2}{3} - \\frac{1}{2}(x-x_0)^2(2+(x-x_0)) & -1\\le x-x_0\\le 0 \\\\\n\\frac{2}{3} - \\frac{1}{2}(x-x_0)^2(2-(x-x_0)) & 0 \\le x-x_0\\le 1 \\\\\n\\frac{1}{6} (2-(x-x_0))^3 & 1 \\le x-x_0 \\le 2 \\\\\n0 & 2\\le x-x_0\n\\end{cases}$$\n\nwhere, in the context of this example, $x_0=-4$, while in Grid.jl we will have $x_0=1$. Liu also defines the translated spline\n\n$$B_k(x) = B_0(x-k+x_0)$$\n\ncorresponding to $B_0(x)$ shifted to the right by $k$ nodes. We now define the interpolating polynomial as a linear combination of translated base splines, i.e.\n\n$$S(x) = \\sum_{k=-1}^{N+1} a_k B_k(x)$$\n\nwhere the start and end of the sum is decided by what translations of the base spline that will be nonzero on our domain. (In principle, we could have summed from $-\\infty$ to $\\infty$, but these will be the only nonzero terms.)\n\n### Setting up the equation system\n\nRequiring that this polynomial is truly interpolating, we find\n\n$$S(x_k) = a_{k-1}B_{k-1}(x_k) + a_{k}B_{k}(x_k) + a_{k+1}B_{k+1}(x_k) + a_{k+2}B_{k+2}(x_k) = f_k$$\n\nUsing the translation relation and the definition of the base splines, one finds that\n\n$$ \\frac{1}{6} (a_{k-1}+4a_k+a_{k+1}) = f_k$$\n\nBy specifying the boundary condition $S''(x) = 0$ at both edges, one can also obtain four more equations:\n\n$$a_{-1} -2a_0 + a_1 = 0\\\\\na_0 = f_0\\\\\na_N = f_N\\\\\na_{N-1}-2a_N + a_{N+1} = 0$$\n\nand together they make up the following equation system\n\n$$\\begin{pmatrix}\n1 & -2 & 1 \\\\\n & 1 \\\\\n & 1 & 4 & 1 \\\\\n & & 1 & 4 & 1 \\\\\n & & & \\ddots & \\ddots & \\ddots \\\\\n & & & & 1 & 4 & 1 \\\\\n & & & & & 1 & 4 \\\\\n & & & & & & 1 \\\\\n & & & & & 1 & -2 & 1\n\\end{pmatrix} \n\\begin{pmatrix}a_{-1} \\\\ a_0 \\\\ \\ldots \\\\ \\\\\\\\\\\\\\\\\\\\ a_N \\\\ a_{N+1}\\end{pmatrix} \n= \n\\begin{pmatrix}0 \\\\ f_0 \\\\ 6*f_1 \\\\ 6*f_2 \\\\\\\\\\\\\\\\\\\\ f_N \\\\ f_{N+1}\\end{pmatrix} \n$$"
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"input": "# The system matrix consists of a 9x9 matrix for coefficients a0-a8 (located at -4,-3,...,-4) as given by eqn (3.21)\n# wrapped by a top and a bottom row for the boundary conditions, as given by eqn (3.19)\ndu = ones(Float64,10)\ndu[1:2] = [2 0]\ndu[end] = 0\ndl = copy(flipud(du))\nd = fill(convert(Float64,4),11)\nd[1:2] = d[end-1:end] = 1\nA = diagm(d) + diagm(du,1) + diagm(dl,-1)\nA[1,3] = A[end,end-2] = 1;\n\n# The right hand side of the system\nb = vcat(0., f(-4), 6.*f(Float64[-3:3]), f(4), 0.)\n\nA # Uncomment this row to see the full system matrix\n#A[2:end-1,2:end-1] # Uncomment this row to see the system matrix as in eqn (3.21) of Liu's notes",
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"text": "11x11 Array{Float64,2}:\n 1.0 2.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 1.0 4.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 1.0 4.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 1.0 4.0 1.0 0.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 1.0 4.0 1.0 0.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 1.0 4.0 1.0 0.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 1.0 4.0 1.0 0.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 4.0 1.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0\n 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 2.0 1.0"
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"cell_type": "markdown",
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"source": "We solve this system for the spline coefficients."
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"collapsed": false,
"input": "# Solving for the spline coefficients\nak = A\\b\nak'",
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"outputs": [
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"metadata": {},
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"prompt_number": 22,
"text": "1x11 Array{Float64,2}:\n 0.0872922 -0.0445178 0.00174343 \u2026 -0.00174343 0.0445178 -0.0872922"
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"cell_type": "markdown",
"metadata": {},
"source": "##Wait, what?\n\nOn page 7 in Liu's notes the values one should obtain by solving this system are tabulated. These aren't even close! For example, the second element from the left above, corresopnding to $a_0$, should be equal to $f(-4)$"
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"cell_type": "code",
"collapsed": false,
"input": "f(-4)",
"language": "python",
"metadata": {},
"outputs": [
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"prompt_number": 23,
"text": "-0.04451779384164284"
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"cell_type": "markdown",
"metadata": {},
"source": "OK, so given the same function with the same input, we get a different result compared to Liu. Yes, I tried working with degrees instead of radians too - that gave us this:"
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"cell_type": "code",
"collapsed": false,
"input": "f2(x) = sind(x) / (1+x^2)\nf2(-4)",
"language": "python",
"metadata": {},
"outputs": [
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"prompt_number": 27,
"text": "-0.004103321984948547"
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"prompt_number": 27
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"cell_type": "markdown",
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"source": "Still not the same as in the notes, not by a long shoSomething has gone awry already, but I have no idea what. Oh well, let's continue anyway. Just to see if we can get somewhere.\n\nWe start by defining the base spline function and the translated splines, and then express the interpolating polynomial through the `spline` function, which simply calculates the linear combination above."
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"input": "x0 = -4\nfunction base_spline_0(x) # Eqn (3.20)\n Dx = x-x0\n if Dx < -2 return 0 end\n if Dx < -1 return (2+Dx)^3/6 end\n if Dx < 0 return 2/3 - Dx^2*(2+Dx)/2 end\n if Dx < 1 return 2/3 + Dx^2*(2-Dx)/2 end\n if Dx < 2 return (2-Dx)^3/6 end\n return 0\nend\nbase_spline_k(x,k) = base_spline_0(x+x0-k)\n\nfunction spline(x)\n k = ifloor(x)\n sum([ak[k+2]*base_spline_k(x,k) for k = -1:9])\nend;",
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"metadata": {},
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"prompt_number": 25
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"cell_type": "markdown",
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"source": "Finally, we plot the resulting spline together with the original values:"
},
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"cell_type": "code",
"collapsed": false,
"input": "plot(layer(spline,-4,4),layer(f,-4,4))",
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"source": "Obviously, something went wrong. That's not an interpolation.\n\n###Things I've ruled out (I think)\n\n* The system matrix looks correct to me\n* The right-hand side does too. However, the function values don't seem to match up with the paper. "
}
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}
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