Zernike Coefficients Displaced Gaussian-.25.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Cubature"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Z (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function R(n::Int64, m::Int64, ρ::Float64)\n",
    "    if (n - m) % 2 != 0\n",
    "        0\n",
    "    else\n",
    "        function summand(k)\n",
    "            n = big(n)\n",
    "            k = big(k)\n",
    "            (-1)^k * factorial(n-k)/(factorial(k)*factorial(Int((n+m)/2) - k)*factorial(Int((n-m)/2) - k))*(ρ)^(n-2*k)\n",
    "        end\n",
    "        mapreduce(summand, +, Array(range(0, stop=Int((n-m)/2), step=1)))\n",
    "    end\n",
    "end\n",
    "\n",
    "function Z(n, m, ρ, θ)\n",
    "    if m < 0\n",
    "        R(n, abs(m), ρ) * sin(abs(m) * θ)\n",
    "    else\n",
    "        R(n, m, ρ) * cos(m * θ)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "integrand (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "σ1 = .1\n",
    "σ2 = .1\n",
    "displacement = .25\n",
    "function integrand(n, m)\n",
    "    function rtn(coor)\n",
    "        ρ = coor[1]\n",
    "        θ = coor[2]\n",
    "        x = ρ * cos(θ) + displacement\n",
    "        y = ρ * sin(θ)\n",
    "        Z(n, m, ρ, θ) * exp(-x^2/σ1^2 - y^2/σ2^2) * ρ\n",
    "    end\n",
    "    rtn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cond_eval (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function neumann(m)\n",
    "    if m == 0\n",
    "        2\n",
    "    else \n",
    "        1\n",
    "    end\n",
    "end\n",
    "function cond_eval(n, m)\n",
    "    if -n ≤ m ≤ n\n",
    "        (2*n+2)/(π*neumann(m)) * hcubature(integrand(n, m), [0., 0.], [1., 2*π], maxevals=1000)[1]\n",
    "    else\n",
    "        0\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "maxn=100\n",
    "coeffs1 = [sum([c[1]^2 for c in [cond_eval(n, m) for n in range(0, maxn, step=1)]]) for m in range(0, maxn, step=1)];\n",
    "coeffs2 = [sum([c[1]^2 for c in [cond_eval(n, m) for n in range(0, maxn, step=1)]]) for m in range(-maxn, 0, step=1)];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr()\n",
    "maxm = 10\n",
    "\n",
    "heatmap(hcat([[c[1]^2 for c in [cond_eval(n, m) for n in range(0, 4*maxm, step=1)]] for m in range(0, maxm, step=1)]...))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     "execution_count": 11,
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   "source": [
    "heatmap(hcat([[c[1]^2 for c in [cond_eval(n, m) for n in range(0, 4*maxn, step=1)]] for m in range(-maxn, 0, step=1)]...))"
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  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 92,
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   ],
   "source": [
    "coeffs1[1] = 0"
   ]
  },
  {
   "cell_type": "code",
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       "  2023.57,155.698 2162.65,155.698 \n",
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       "<path clip-path=\"url(#clip870)\" d=\"M 0 0 M2199.68 175.385 Q2197.87 180.015 2196.16 181.427 Q2194.45 182.839 2191.58 182.839 L2188.17 182.839 L2188.17 179.274 L2190.67 179.274 Q2192.43 179.274 2193.4 178.44 Q2194.38 177.607 2195.56 174.505 L2196.32 172.561 L2185.83 147.052 L2190.35 147.052 L2198.45 167.329 L2206.55 147.052 L2211.07 147.052 L2199.68 175.385 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip870)\" d=\"M 0 0 M2216.95 169.042 L2224.58 169.042 L2224.58 142.677 L2216.27 144.343 L2216.27 140.084 L2224.54 138.418 L2229.21 138.418 L2229.21 169.042 L2236.85 169.042 L2236.85 172.978 L2216.95 172.978 L2216.95 169.042 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /></svg>\n"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot((coeffs1 + reverse(coeffs2))[1:end])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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