{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "1d23018e",
   "metadata": {},
   "source": [
    "# Update nonlinear Signal"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9c667fe",
   "metadata": {},
   "source": [
    "## Projection\n",
    "\n",
    "$S_{mk}' = \\sum_{n}D_{kn} \\cdot \\mathrm{sgn}(T_{mn}^{\\mathrm{exp}})\\cdot\\sqrt{|T_{mn}^{\\mathrm{exp}}|} \\dfrac{S_{mn}}{|S_{mn}|}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c951a5c",
   "metadata": {},
   "source": [
    "## Iteration  \n",
    "\n",
    "$S_{mk}' = S_{mk} - \\alpha\\eta\\cdot\\nabla_{mk}r$  \n",
    "\n",
    "<br>\n",
    "\n",
    "### Intensity\n",
    "\n",
    "$r = \\sum_n(T_{mn}^{\\mathrm{exp}} - \\mu T_{mn})^2$  \n",
    "\n",
    "$\\nabla_{mk}r = \\frac{dr}{dS_{mk}^*} = -2\\sum_n D_{kn}(\\frac{T_{mn}^{\\mathrm{exp}}}{\\mu} - T_{mn})\\cdot S_{mn}$  \n",
    "\n",
    "$H_{zz} = \\sum_n D_{kn}D_{nj} (2T_{mn} - \\frac{T_{mn}^{\\mathrm{exp}}}{\\mu})$  \n",
    "\n",
    "<br>\n",
    "\n",
    "### Magnitude\n",
    "\n",
    "$r=\\sum_n(\\sqrt{T_{mn}^{\\mathrm{exp}}} - \\mu |S_{mn}|)^2$  \n",
    "\n",
    "$\\nabla_{mk}r = \\frac{dr}{dS_{mk}^*} = -2\\sum_n D_{kn}(\\sqrt{T_{mn}^{\\mathrm{exp}}}\\cdot\\frac{S_{mn}}{\\mu|S_{mn}|} - S_{mn})$ \n",
    "\n",
    "$H_{zz} = \\sum_n D_{kn}D_{nj}$  "
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "name": "python"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}