Zernike Coefficients Ellipstical Gaussian.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 = 0\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": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cond_eval (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "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=10000)[1]\n",
    "    else\n",
    "        0\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "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": 15,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "gr()\n",
    "maxn = 20\n",
    "\n",
    "heatmap(hcat([[c[1]^2 for c in [cond_eval(n, m) for n in range(0, 2*maxn, step=1)]] for m in range(0, maxn, step=1)]...))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coeffs1[1] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
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       "  423.958,783.716 433.799,1445.05 443.64,969.691 453.48,1445.28 463.321,1123.49 473.162,1440.14 483.003,1240.44 492.844,1442.22 502.684,1322.67 512.525,1443.65 \n",
       "  522.366,1376.35 532.207,1444.57 542.048,1408.95 551.888,1445.11 561.729,1427.4 571.57,1445.42 581.411,1437.14 591.252,1445.58 601.092,1441.95 610.933,1445.66 \n",
       "  620.774,1444.16 630.615,1445.69 640.456,1445.12 650.296,1445.71 660.137,1445.5 669.978,1445.71 679.819,1445.64 689.66,1445.71 699.501,1445.69 709.341,1445.72 \n",
       "  719.182,1445.71 729.023,1445.72 738.864,1445.71 748.705,1445.72 758.545,1445.72 768.386,1445.72 778.227,1445.72 788.068,1445.72 797.909,1445.72 807.749,1445.72 \n",
       "  817.59,1445.72 827.431,1445.72 837.272,1445.72 847.113,1445.72 856.953,1445.72 866.794,1445.72 876.635,1445.72 886.476,1445.72 896.317,1445.72 906.157,1445.72 \n",
       "  915.998,1445.72 925.839,1445.72 935.68,1445.72 945.521,1445.72 955.361,1445.72 965.202,1445.72 975.043,1445.72 984.884,1445.72 994.725,1445.72 1004.57,1445.72 \n",
       "  1014.41,1445.72 1024.25,1445.72 1034.09,1445.72 1043.93,1445.72 1053.77,1445.72 1063.61,1445.72 1073.45,1445.72 1083.29,1445.72 1093.13,1445.72 1102.97,1445.72 \n",
       "  1112.81,1445.72 1122.66,1445.72 1132.5,1445.72 1142.34,1445.72 1152.18,1445.72 1162.02,1445.72 1171.86,1445.72 1181.7,1445.72 1191.54,1445.72 1201.38,1445.72 \n",
       "  1211.22,1445.72 1221.06,1445.72 1230.9,1445.72 1240.74,1445.72 1250.59,1445.72 1260.43,1445.72 1270.27,1445.72 1280.11,1445.72 1289.95,1445.72 1299.79,1445.72 \n",
       "  1309.63,1445.72 1319.47,1445.72 1329.31,1445.72 1339.15,1445.72 1348.99,1445.72 1358.83,1445.72 1368.68,1445.72 1378.52,1445.72 1388.36,1445.72 1398.2,1445.72 \n",
       "  1408.04,1445.72 1417.88,1445.72 1427.72,1445.72 1437.56,1445.72 1447.4,1445.72 1457.24,1445.72 1467.08,1445.72 1476.92,1445.72 1486.76,1445.72 1496.61,1445.72 \n",
       "  1506.45,1445.72 1516.29,1445.72 1526.13,1445.72 1535.97,1445.72 1545.81,1445.72 1555.65,1445.72 1565.49,1445.72 1575.33,1445.72 1585.17,1445.72 1595.01,1445.72 \n",
       "  1604.85,1445.72 1614.7,1445.72 1624.54,1445.72 1634.38,1445.72 1644.22,1445.72 1654.06,1445.72 1663.9,1445.72 1673.74,1445.72 1683.58,1445.72 1693.42,1445.72 \n",
       "  1703.26,1445.72 1713.1,1445.72 1722.94,1445.72 1732.79,1445.72 1742.63,1445.72 1752.47,1445.72 1762.31,1445.72 1772.15,1445.72 1781.99,1445.72 1791.83,1445.72 \n",
       "  1801.67,1445.72 1811.51,1445.72 1821.35,1445.72 1831.19,1445.72 1841.03,1445.72 1850.87,1445.72 1860.72,1445.72 1870.56,1445.72 1880.4,1445.72 1890.24,1445.72 \n",
       "  1900.08,1445.72 1909.92,1445.72 1919.76,1445.72 1929.6,1445.72 1939.44,1445.72 1949.28,1445.72 1959.12,1445.72 1968.96,1445.72 1978.81,1445.72 1988.65,1445.72 \n",
       "  1998.49,1445.72 2008.33,1445.72 2018.17,1445.72 2028.01,1445.72 2037.85,1445.72 2047.69,1445.72 2057.53,1445.72 2067.37,1445.72 2077.21,1445.72 2087.05,1445.72 \n",
       "  2096.89,1445.72 2106.74,1445.72 2116.58,1445.72 2126.42,1445.72 2136.26,1445.72 2146.1,1445.72 2155.94,1445.72 2165.78,1445.72 2175.62,1445.72 2185.46,1445.72 \n",
       "  2195.3,1445.72 2205.14,1445.72 2214.98,1445.72 2224.83,1445.72 2234.67,1445.72 2244.51,1445.72 2254.35,1445.72 2264.19,1445.72 2274.03,1445.72 2283.87,1445.72 \n",
       "  2293.71,1445.72 \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip870)\" d=\"\n",
       "M2000.39 216.178 L2283.21 216.178 L2283.21 95.2176 L2000.39 95.2176  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip870)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  2000.39,216.178 2283.21,216.178 2283.21,95.2176 2000.39,95.2176 2000.39,216.178 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip870)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  2023.57,155.698 2162.65,155.698 \n",
       "  \"/>\n",
       "<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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