{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Bringing an Accurate Fresnel to Real-Time Rendering\n", "\n", "#### Laurent Belcour (Unity), Megane Bati (IOGS), and Pascal Barla (Inria)\n", "\n", "This notebook is the implementation & details guide for *Bringing an Accurate Fresnel to Real-Time Rendering: A Preintegrable Decomposition*. It provides all the details to reproduce our code and results for off-line rendering. For real-time rendering, we also provide a reference code to compute the directional albedo. Please refer to that code to reproduce the new split-sum integral form.\n", "\n", "You can freely copy/paste the python scripts here to reproduce our work. To work, our code requires that you have the python package below installed. You can install those packages using pip for example.\n", "\n", "Note that this notebook requires that you download our Fresnel python package fresnel_tools.py." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "# %matplotlib inline\n", "%matplotlib notebook\n", "import numpy as np\n", "import scipy.optimize\n", "from scipy.interpolate import interp2d\n", "import matplotlib as mpl\n", "from matplotlib.ticker import MaxNLocator\n", "import matplotlib.pyplot as plt\n", "import matplotlib.colors as colors\n", "from matplotlib.widgets import Slider\n", "from mpl_toolkits.mplot3d import Axes3D\n", "\n", "# Custom package\n", "import fresnel_tools as ft\n", "\n", "mpl.rcParams['font.family']=['serif']\n", "mpl.rcParams['font.serif']=['Linux Libertine'] + mpl.rcParams['font.serif']" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A Basis Decomposition for Fresnel\n", "\n", "In this work, we which to reproduce [Fresnel reflectance](https://en.wikipedia.org/wiki/Fresnel_equations#Power_(intensity)_reflection_and_transmission_coefficients) using a linear decomposition of data driven basis elements. That is, if the unpolarized reflected power of a planar surface (denoted Fresnel reflectance) is given by $F(\\theta)$, we will express it as:\n", "$$\n", "F(\\theta) = \\sum_i c_i b_i(\\theta),\n", "$$\n", "where $b_i(\\theta)$ are basis functions and $c_i$ are coefficients.\n", "\n", "Note that Fresnel reflectance is a wavelength depent function $F(\\theta; \\lambda)$. In such a case, we will incorporate wavelength dependence in the coefficients $c_i$, but not in the basis elements $b_i(\\theta)$:\n", "$$\n", "F(\\theta; \\lambda) = \\sum_i c_i(\\lambda) b_i(\\theta).\n", "$$\n", "In such form, the basis elements can be factored out of spectral preintegration (when one whish to extract RGB reflectance out of spectral data):\n", "$$\n", "\\int_\\lambda F(\\theta; \\lambda) s(\\lambda) \\mbox{d}\\lambda = \\sum_i \\left[ \\int_\\lambda c_i(\\lambda) s(\\lambda) \\mbox{d}\\lambda \\right] b_i(\\theta).\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## How Many Basis Elements?\n", "\n", "In this Section, we analyze how many basis elements are required to reproduce Fresnel curves with high acuracy. To do so, we extract the eigenvalues of a set of Fresnel curves and plot their relative value to the sum of all eigenvalues." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp =$('
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Artist Friendly Metallic Fresnel. JCGT 2014.\n", " http://jcgt.org/published/0003/04/03/paper.pdf\n", " \n", " More details on it later.\n", "'''\n", "def RGtoNK2(r, g):\n", " if r == 0.0:\n", " return 1.0, 0.0\n", " n = g*(1.0-r)/(1.0+r) + (1.0-g)*(1.0+np.sqrt(r))/(1.0-np.sqrt(r))\n", " k2 = 0.0 if g<=0.0 else (r*(1.0+n)*(1.0+n) - (n-1.0)*(n-1.0)) / (1.0-r)\n", " return n, k2\n", "\n", "def RGtoNK(r, g):\n", " n, k2 = RGtoNK2(r, g)\n", " return n, np.sqrt(k2)\n", "\n", "'''\n", " We define the curves with respect to the elevation.\n", " For real-time implementation, it makes more sense to define\n", " since it avoids an acos call.\n", "'''\n", "cT = np.linspace(0.0, 1.0, nb_points, endpoint=True)\n", "T = np.arccos(cT)\n", "\n", "subrange = np.linspace(0, nb_points-1, 20, dtype=int)\n", "IOR = np.linspace(1.1, 3.0, nb_IOR)\n", "KAPPA = np.zeros((nb_IOR, ))\n", "curves = np.ndarray( (IOR.shape[0], nb_points) )\n", "\n", "'''\n", " We use a small percentage (1%) of conducting materials (g=1)\n", " and a large percentage (99%) of dieletrics to build the basis.\n", " This helps to better reconstruct both of them.\n", "'''\n", "alpha = 0.01\n", "for k in range(1, nb_IOR):\n", " r = k / nb_IOR\n", " \n", " if r <= alpha:\n", " r = r/alpha\n", " r = (r) / (1.0-1.0E-5) + 1.0E-5\n", " g = 1.0\n", " IOR[k], KAPPA[k] = RGtoNK(r, g)\n", " else:\n", " r = (r - alpha)/(1.0-alpha)\n", " IOR[k], KAPPA[k] = 1.01+(3.0-1.01)*r,0.0\n", "\n", "'''\n", " Build the Fresnel reflectance curves from the set of IOR\n", " set above. We remove F(0deg) from the curves and rescale\n", " it to map between 0 to 1.\n", "''' \n", "for k,ior in enumerate(IOR):\n", " F = ft.fresnel(cT, ior, KAPPA[k])\n", " R0 = ft.fresnel(1.0, ior, KAPPA[k])\n", " curves[k,:] = (F - R0) / (1.0-R0)\n", "\n", "u, s, basis = np.linalg.svd(curves)\n", "\n", "# Define the zeroth and first componnents\n", "basis0 = np.ones_like(basis[0, :])\n", "basis1 = np.abs(basis[0, :])\n", "basis1 = basis1 / max(basis1)\n", "\n", "for k,ior in enumerate(IOR):\n", " F = ft.fresnel(cT, ior, KAPPA[k])\n", " R0 = ft.fresnel(1.0, ior, KAPPA[k])\n", " curves[k,:] = (F - R0)\n", " curves[k,:] = curves[k,:]/(1.0-R0) - basis1\n", " \n", "u, s, basis = np.linalg.svd(curves)\n", "\n", "# Define the second and third componnents\n", "basis2 = basis[0] / np.abs(basis[0]).max()\n", "basis3 = basis[1] / np.abs(basis[1]).max()\n", "\n", "i1 = np.argmax( basis2 )\n", "t1 = T[ i1 ]\n", "ct1 = cT[ i1 ]\n", "print('Critical angle for basis2 = ', ct1, t1, 180.0/np.pi*t1, i1)\n", "\n", "i2 = np.argmax( +basis3 )\n", "i3 = np.argmax( -basis3 )\n", "i2 = i2 if abs(i2-i1) > abs(i3-i1) else i3\n", "t2 = T[ i2 ]\n", "ct2 = cT[ i2 ]\n", "print('Critical angle for basis3 = ', ct2, t2, 180.0/np.pi*t2, i2)\n", "\n", "'''\n", " Define the projection operator that maps a complex IOR (eta+i*kappa)\n", " to the coefficients of the decomposition.\n", "'''\n", "def fresnel2coeffs(eta12, kappa=0.0):\n", " coeffs = np.zeros((4))\n", " coeffs[0] = ft.fresnel(1.0, eta12, kappa)\n", " coeffs[1] = 1.0 - coeffs[0] if coeffs[0] else 0.0\n", "\n", " dF = np.zeros((1,2))\n", " dF[0,0] = ft.fresnel(ct1, eta12, kappa) - (coeffs[0] + coeffs[1]*basis1[i1])\n", " dF[0,1] = ft.fresnel(ct2, eta12, kappa) - (coeffs[0] + coeffs[1]*basis1[i2])\n", " \n", " F12 = np.zeros((2,2))\n", " F12[0,0] = basis2[i1]\n", " F12[0,1] = basis2[i2]\n", " F12[1,0] = basis3[i1]\n", " F12[1,1] = basis3[i2]\n", " A = np.linalg.inv(F12)\n", " \n", " coeffs[2:4] = np.dot(dF, A)\n", " return coeffs\n", "\n", "\n", "'''\n", " Display the basis elements and range of value for the coefficients for various\n", " values of eta and a dieletric material.\n", "'''\n", "plt.subplots(figsize=(9,4))\n", "handles = []\n", "labels = []\n", "\n", "plt.grid()\n", "handle0, = plt.plot(T, basis0)\n", "handle1, = plt.plot(T, basis1)\n", "handle2, = plt.plot(T, basis2)\n", "handle3, = plt.plot(T, basis3)\n", "plt.legend([handle0, handle1, handle2, handle3], [r'$b_0(\\theta)$', r'$b_1(\\theta)$', r'$b_2(\\theta)$', r'$b_3(\\theta)$'])\n", "plt.xlabel(r'angle of incidence $\\theta$')\n", "plt.axis([0, 0.5*np.pi, -0.9, 1.1])\n", "plt.tight_layout()\n", "plt.show()\n", "# plt.savefig('../figures/basis.pdf')" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $(' ');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas =$('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp =$('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "'''\n", " Display the fitting of dieletric reflectances\n", "'''\n", "plt.subplots(figsize=(9,4))\n", "plt.grid()\n", "handles = []\n", "labels = []\n", "\n", "for k,ior in enumerate([1.0, 1.01, 1.1, 1.5, 2.0]):\n", " F = ft.fresnel(cT, ior)\n", " handle, = plt.plot(T, F)\n", " handles.append(handle)\n", " \n", " coeffs = fresnel2coeffs(ior)\n", " P = coeffs[0] + coeffs[1]*basis1 + coeffs[2]*basis2 + coeffs[3]*basis3\n", " plt.scatter(T[subrange], P[subrange], color=handle.get_color(), marker='o')\n", " plt.plot(T, P, dashes=[1, 3], color=handle.get_color())\n", " \n", " L2 = np.linalg.norm(F-P)\n", " labels.append('IOR = ' + str(ior) + ', L2 = ' + f'{L2:.2f}')\n", "\n", "\n", "plt.legend(handles, labels, loc='upper left')\n", "plt.title('Reconstruct dielectrics $F(\\\\theta)$ with our decomposition')\n", "plt.axis([0, 0.5*np.pi, -0.01, 1.0])\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $(' ');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas =$('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp =$('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "'''\n", " Interactive editing of the reflectance curve using the coefficients\n", "'''\n", "# Params for sliders \n", "HSLIDERS = 2; VSLIDERS = 1\n", "INIT_VAL = [1.1, 0.0]; MIN_VAL = [1.01, 0.0]; MAX_VAL = [4.0, 4.0]\n", "COLOR = [\"lightgrey\", \"mediumaquamarine\"]\n", "LABELS = [\"$\\\\eta$\", \"$\\\\kappa$\"]\n", "LENGTH, X, Y, BOTTOM = ft.init(HSLIDERS, VSLIDERS) \n", " \n", "# Start plot\n", "START = 0.0; #START = np.pi/4.0-0.001\n", "\n", "fig, ax = plt.subplots(figsize=(9,4))\n", "plt.grid()\n", "\n", "plt.axis([START, np.pi/2.0, 0, 1])\n", "# plt.title(\"Fresnel along the isoline\")\n", "plt.xlabel(\"incidence angle $\\\\theta$\")\n", "plt.ylabel(\"Reflectance $F(\\\\theta)$\")\n", "plt.tight_layout()\n", "plt.subplots_adjust(bottom=0.25)\n", "\n", "# Some interesting points\n", "P = ft.fresnel(cT, INIT_VAL[0], INIT_VAL[1])\n", "fresnel, = plt.plot(T, P, 'tab:blue', linestyle='--')\n", "coeffs = fresnel2coeffs(INIT_VAL[0], INIT_VAL[1])\n", "P = coeffs[0] + coeffs[1]*basis1 + coeffs[2]*basis2 + coeffs[3]*basis3\n", "points, = plt.plot(T, P, 'tab:blue')\n", "P = coeffs[0] + (1.0-coeffs[0])*np.power(1.0-cT, 5.0)\n", "schlick, = plt.plot(T, P, 'orange')\n", "plt.legend(['Reference Fresnel', 'Our Decomposition', 'Schlick\\'s Fresnel'])\n", "\n", "# Add sliders\n", "sliders = []\n", "sliders.append(ft.add_slider(X[0], Y[0], INIT_VAL[0], LABELS[0], COLOR[0], LENGTH, MIN_VAL[0], MAX_VAL[0]))\n", "sliders.append(ft.add_slider(X[0], Y[1], INIT_VAL[1], LABELS[1], COLOR[1], LENGTH, MIN_VAL[1], MAX_VAL[1]))\n", "\n", "def update_coeffs(val):\n", " n = sliders[0].val\n", " k = sliders[1].val\n", " \n", " P = ft.fresnel(cT, n, k)\n", " fresnel.set_ydata(P)\n", " \n", " coeffs = fresnel2coeffs(n, k)\n", " P = coeffs[0] + coeffs[1]*basis1 + coeffs[2]*basis2 + coeffs[3]*basis3\n", " points.set_ydata(P)\n", " \n", " P = coeffs[0] + (1.0-coeffs[0])*np.power(1.0-cT, 5.0)\n", " schlick.set_ydata(P)\n", " \n", " \n", " fig.canvas.draw_idle()\n", "\n", "sliders[0].on_changed(update_coeffs)\n", "sliders[1].on_changed(update_coeffs)\n", "\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gulbransen Parametrization\n", "\n", "The Gulbransen parametrization map a couple $(r, g) \\in [0,1]^2$ denoted *reflectivity* ($r$) and *edge-tint* ($g$) to the complex Index of Refraction $\\eta + i\\,\\kappa$. This parametrization tries to provide an artistic mapping in a closed range that behave more linearly than the complex IOR. The parametrization is the following:\n", "\n", "\\begin{align}\n", " \\eta &= g \\dfrac{1-r}{1+r} + \\left(1-g\\right)\\dfrac{1+\\sqrt{r}}{1-\\sqrt{r}} \\\\\n", " \\kappa &= \\sqrt{\\dfrac{1}{1-r}\\left( r \\, \\left( 1+n \\right)^2 - \\left( n-1 \\right)^2 \\right)}\n", "\\end{align}\n", "\n", "See [JCGT 2014 *Artist Friendly Metallic Fresnel*](http://jcgt.org/published/0003/04/03/paper.pdf)" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "scrolled": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $(' ');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas =$('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp =$('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, axs = plt.subplots(1,2, figsize=(8,4))\n", "\n", "nb_POINTS = 100\n", "nb_RG = 50\n", "NEAR_ONE = 1.0 - 1.0E-5\n", "\n", "'''\n", " Generate two lines: one for the perfect dielectrics (g=0),\n", " and one for the maximum edge tint (g=1).\n", "'''\n", "R = np.linspace(0.00, NEAR_ONE, nb_POINTS)\n", "c_diel = np.zeros((nb_POINTS, 4))\n", "c_cond = np.zeros((nb_POINTS, 4))\n", "for i,r in enumerate(R):\n", " [eta,kappa] = RGtoNK(r,0.0)\n", " c_diel[i,:] = fresnel2coeffs(eta, 0.0)\n", " [eta,kappa] = RGtoNK(r,1.0)\n", " c_cond[i,:] = fresnel2coeffs(eta, kappa)\n", " \n", "'''\n", " Generate a grid of point, that provide the meshing of the\n", " space between the two curves using Gulbransen's\n", " parametrization.\n", "'''\n", "R = np.linspace(0.00, NEAR_ONE, nb_RG)\n", "G = np.linspace(0.00, NEAR_ONE, nb_RG)\n", "coeffs = np.zeros((nb_RG, nb_RG, 4))\n", "for i, r in enumerate(R):\n", " for j, g in enumerate(G):\n", " [eta,kappa] = RGtoNK(r,g)\n", " coeffs[i,j,:] = fresnel2coeffs(eta, kappa)\n", "\n", "ax = axs[0]\n", "ax.scatter(coeffs[:,:,0], coeffs[:,:,2], c=coeffs[:,:,0], s=0.1)\n", "ax.plot(c_diel[:,0], c_diel[:,2], c='red')\n", "ax.plot(c_cond[:,0], c_cond[:,2], c='k')\n", "# ax.plot(c_zero[:,0], c_zero[:,2], c='gray')\n", "ax.set_xlabel('$c_0$')\n", "ax.set_ylabel('$c_2$')\n", "ax.grid()\n", "ax.axis([0,1, -0.35, 0.35])\n", "ax.legend(['$g=0$', '$g=1$'], loc='upper right')\n", "ax.set_title('Gulbransen parametrization')\n", "\n", "'''\n", " Generate a grid of point, that provide the meshing of the\n", " space between the two curves using a remapped Gulbransen's\n", " parametrization.\n", "'''\n", "R = np.linspace(0.00, NEAR_ONE, nb_RG)\n", "g = np.linspace(0.00, NEAR_ONE, nb_RG)\n", "coeffG = np.zeros((nb_RG, nb_RG, 4))\n", "for i,r in enumerate(R):\n", " for j,g in enumerate(G):\n", " # empirical remapping\n", " g = 1.0-pow(1.0-g,3)\n", " [eta,k2] = RGtoNK2(r,g)\n", " coeffG[i,j,:] = fresnel2coeffs(eta, np.sqrt(k2))\n", "\n", "ax = axs[1]\n", "ax.scatter(coeffG[:,:,0], coeffG[:,:,2], c=coeffG[:,:,0], s=0.1)\n", "ax.plot(c_diel[:,0], c_diel[:,2], c='red')\n", "ax.plot(c_cond[:,0], c_cond[:,2], c='k')\n", "ax.set_xlabel('$c_0$')\n", "ax.set_ylabel('$c_2$')\n", "ax.grid()\n", "ax.axis([0,1, -0.35, 0.35])\n", "ax.legend(['$g=0$', '$g=1$'], loc='upper right')\n", "ax.set_title('Remapping using $\\\\bar{g}$')\n", "\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This remapping of the Gulbransen parametrization is still not satisfactory: we can see compressions close to the $g = 0$ and $g= 1$ curves.\n", "\n", "Such compressions are also visible when visualizing the manifold in 3D." ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
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