{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "2cecfdc1",
   "metadata": {},
   "source": [
    "# Ptychographic Iterative Engine (Gradients and Hessians)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7dda1a25",
   "metadata": {},
   "source": [
    "## PIE - Gradient"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0ccdb3f",
   "metadata": {},
   "source": [
    "PIE ultimately assumes independent Object and Probe. This is respected in these derivatives.  \n",
    "\n",
    "$S_{mk} = O_k\\cdot P_{mk}$\n",
    "\n",
    "$P_{mk} = G(A_{mk})$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a429270f",
   "metadata": {},
   "source": [
    "### With respect to Object\n",
    "\n",
    "Same for all nonlinear methods  \n",
    "\n",
    "$\\frac{dL_m}{dO_k^*} = -1\\cdot P_{mk}^*\\cdot \\Delta S_{mk}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "faa41ea8",
   "metadata": {},
   "source": [
    "### With respect to Probe and Probe-Pulse\n",
    "\n",
    "$\\frac{dL_m}{dA_k^*} = \\left(\\frac{dA_{mj}}{dA_k}\\right)^*\\left(\\frac{dP_{mj}}{dA_{mj}}\\right)^*\\frac{dL_m}{dP_{mj}^*}$  \n",
    "\n",
    "$\\frac{dL_m}{dP_{mk}^*} = -1\\cdot O_k^*\\cdot \\Delta S_{mk}$  \n",
    "\n",
    "<br>\n",
    "\n",
    "$\\frac{dA_{mj}}{dA_k} = \\sum_{n}D_{kn}D_{nj}\\cdot e^{1j\\phi_{mn}}\\qquad$ (in the exponent $j$ refers to the imaginary unit)  \n",
    "\n",
    "<br>\n",
    "\n",
    "$\\frac{dP_{mj}}{dA_{mj}}$ depends on the nonlinear method.  \n",
    "\n",
    "\n",
    "1. SHG  \n",
    "\n",
    "    $\\frac{dP_{mj}}{dA_{mj}} = 1$\n",
    "\n",
    "\n",
    "2. THG  \n",
    "\n",
    "    $\\frac{dP_{mj}}{dA_{mj}} = 2A_{mj}$\n",
    "\n",
    "\n",
    "3. PG  \n",
    "\n",
    "    $\\frac{dP_{mj}}{dA_{mj}} = A_{mj}^*$\n",
    "\n",
    "\n",
    "4. SD  \n",
    "\n",
    "    $\\frac{dP_{mj}}{dA_{mj}^*} = 2A_{mj}^*\\qquad$ (the other wirtinger derivative is zero)\n",
    "\n",
    "\n",
    "5. nth-HG  \n",
    "\n",
    "    $\\frac{dP_{mj}}{dA_{mj}} = (n-1)\\cdot A_{mj}^{n-2}$   \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "For SD the wirtinger derivative of probe with respect to the modified probe pulse is zero. Thus one has to use a different chain-rule composition:  \n",
    "\n",
    "$\\frac{dL_m}{dA_k^*} = \\left(\\frac{dA_{mj}}{dA_k}\\right)^*\\frac{dP_{mj}}{dA_{mj}^*}\\frac{dL_m}{dP_{mj}}$  \n",
    "\n",
    "$\\frac{dL_m}{dP_{mj}} = \\left(\\frac{dL_m}{dP_{mj}^*}\\right)^*$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae157cac",
   "metadata": {},
   "source": [
    "### With respect to Object (Chirp-Scan)  \n",
    "\n",
    "In a Chirp-Scan the object itself is modified along woth the probe. \n",
    "\n",
    "$S_{mk} = O_{mk}\\cdot P_{mk}$\n",
    "\n",
    "$\\frac{dL_m}{dO_k^*} = \\left(\\frac{dO_{mj}}{dO_k}\\right)^*\\frac{dL_m}{dO_{mj}^*}$  \n",
    "\n",
    "$\\frac{dL_m}{dO_{mk}^*} = -1\\cdot P_{mk}^*\\cdot \\Delta S_{mk}$  \n",
    "\n",
    "\n",
    "\n",
    "$\\frac{dO_{mj}}{dO_k} = \\sum_{n}D_{kn}D_{nj}\\cdot e^{1j\\phi_{mn}}\\qquad$ (in the exponent $j$ refers to the imaginary unit)  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff7b2bc9",
   "metadata": {},
   "source": [
    "## PIE - Pseudo Hessian"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9362e91d",
   "metadata": {},
   "source": [
    "### With respect to Object\n",
    "\n",
    "Same for all nonlinear methods  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{dO_k^*} = \\frac{S_{mn}}{2|S_{mn}|} \\cdot D_{kn} P_{mk}^*$  \n",
    "\n",
    "$\\frac{d}{dO_j}\\left(\\frac{d|S_{mn}|}{dO_k}\\right)^* = \\left(\\frac{d}{dO_j^*}\\left(\\frac{d|S_{mn}|}{dO_k^*}\\right)^*\\right)^* =  \\frac{1}{4}D_{kn}D_{nj} P_{mk}^*P_{mj}\\cdot\\frac{1}{|S_{mn}|}$  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b659f1a5",
   "metadata": {},
   "source": [
    "### With respect to Probe and Probe-Pulse  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{dA_k^*} = \\left(\\frac{dA_{mj}}{dA_k}\\right)^*\\left(\\frac{dP_{mj}}{dA_{mj}}\\right)^*\\frac{d|S_{mn}|}{dP_{mj}^*}$  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{dP_{mk}^*} = \\frac{S_{mn}}{2|S_{mn}|} \\cdot D_{kn} O_k^*$  \n",
    "\n",
    "$\\frac{dP_{mj}}{dA_{mj}}$ depends on the nonlinear method. Same as in gradient.  \n",
    "\n",
    "\n",
    "$\\frac{dA_{mj}}{dA_k} = \\sum_{n}D_{kn}D_{nj}\\cdot e^{1j\\phi_{mn}}\\qquad$ (in the exponent $j$ refers to the imaginary unit)  \n",
    "\n",
    "<br>\n",
    "\n",
    "$\\frac{d}{dA_j}\\left(\\frac{d|S_{mn}|}{dA_k}\\right)^* = \\frac{dA_{mu}}{dA_k}\\left(\\frac{dA_{mi}}{dA_j}\\right)^*\\frac{dP_{mu}}{dA_{mu}}\\left(\\frac{dP_{mi}}{dA_{mi}}\\right)^*\\frac{d}{dP_{mu}}\\left(\\frac{d|S_{mn}|}{dP_{mi}}\\right)^*$\n",
    "\n",
    "$\\frac{d}{dP_{mj}}\\left(\\frac{d|S_{mn}|}{dP_{mk}}\\right)^* = \\left(\\frac{d}{dP_{mj}}\\left(\\frac{d|S_{mn}|}{dP_{mk}}\\right)^*\\right)^* = \\frac{1}{4}D_{kn}D_{nj} O_k^*O_j\\cdot\\frac{1}{|S_{mn}|}$  \n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "For SD the wirtinger derivative of probe with respect to the modified probe pulse is zero. Thus one has to use a different chain-rule composition:  \n",
    "\n",
    "\n",
    "$\\frac{d}{dA_j}\\left(\\frac{d|S_{mn}|}{dA_k}\\right)^* = \\frac{dA_{mu}}{dA_k}\\left(\\frac{dA_{mi}}{dA_j}\\right)^*\\frac{dP_{mu}^*}{dA_{mu}}\\left(\\frac{dP_{mi}^*}{dA_{mi}}\\right)^*\\frac{d}{dP_{mu}^*}\\left(\\frac{d|S_{mn}|}{dP_{mi}^*}\\right)^*$  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{dP_{mi}^*}=\\left(\\frac{d|S_{mn}|}{dP_{mi}}\\right)^*$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "977e01e7",
   "metadata": {},
   "source": [
    "### With respect to Object (Chirp-Scan)  \n",
    "\n",
    "$\\frac{d|S_{mn}|}{dA_k^*} = \\left(\\frac{dA_{mj}}{dA_j}\\right)^*\\frac{d|S_{mn}|}{dA_{mj}^*}$\n",
    "\n",
    "$\\frac{d|S_{mn}|}{dA_{mk}^*} = \\frac{S_{mn}}{2|S_{mn}|} \\cdot D_{kn} P_{mk}^*$  \n",
    "\n",
    "<br>\n",
    "\n",
    "$\\frac{d}{dA_j}\\left(\\frac{d|S_{mn}|}{dA_k}\\right)^* = \\frac{dA_{mu}}{dA_k}\\left(\\frac{dA_{mi}}{dA_j}\\right)^*\\frac{d}{dA_{mu}}\\left(\\frac{d|S_{mn}|}{dA_{mi}}\\right)^*$\n",
    "\n",
    "$\\frac{d}{dA_{mj}}\\left(\\frac{d|S_{mn}|}{dA_{mk}}\\right)^* = \\left(\\frac{d}{dA_{mj}^*}\\left(\\frac{d|S_{mn}|}{dA_{mk}^*}\\right)^*\\right)^* =  \\frac{1}{4}D_{kn}D_{nj} P_{mk}^*P_{mj}\\cdot\\frac{1}{|S_{mn}|}$  \n"
   ]
  }
 ],
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