README.ipynb

{
 "cells": [
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   "metadata": {
    "tags": []
   },
   "source": [
    "# Dilation Rate Gridding Problem and How to Solve It With the Fibonacci Sequence"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9102a0b-4a54-4ef6-8ecb-c86ca0d13987",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3b1e235b-d870-4515-98ea-12bf20df0fab",
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   "source": [
    "In convolutional neural networks, it is assumed that the receptive field must cover a sufficiently large area of the input image, so that the last layers of the network can have a global overview of the elements present in it."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55263b74-5d2d-46e8-aa18-5a0973611434",
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   "source": [
    "On the figure below, we see how stacking two 3x3 convolutions will result in a 5x5 receptive field with 18 parameters (3x3 + 3x3 = 18):"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dde0078f-c54a-4021-a1dc-2aa0d218c9b1",
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   "source": [
    "[![](receptive-field.png)](https://towardsdatascience.com/a-guide-for-building-convolutional-neural-networks-e4eefd17f4fd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5caa9dc5-d160-49b1-ab62-e40d66676076",
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   "source": [
    "There's usually 3 common ways to increase the receptive field:\n",
    "1) more layers\n",
    "2) pooling with strides\n",
    "3) dilated convolutions."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39040a63-c476-4f42-97c2-cb9283312c3b",
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   "source": [
    "The most used technique is to add a pooling layer. It's indeed very effective at increasing the receptive field without any added computational cost.\n",
    "\n",
    "However, pooling can cause several issues such as [breaking the Shannon-Nyquist sampling theorem](https://richzhang.github.io/antialiased-cnns/). It also, by definition, provokes a loss of data which can be undesirable when we need to preserve the original image resolution."
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "The other technique uses dilated convolutions:"
   ]
  },
  {
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   "id": "b68b1c9a-e6e5-488d-a16f-eae3016e6d44",
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   "source": [
    "![](dilated-conv.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93fb2429-6cf7-4f22-a7e6-34b0a0d8b4fe",
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   "source": [
    "Dilated convolutions preserve the number of parameters and the resolution, while effectively increasing the receptive field.\n",
    "\n",
    "However, they also come with an interesting issue: [**the gridding problem**](https://arxiv.org/abs/1702.08502)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e724947-3ab2-4e5e-9d62-55d25b8fc0d7",
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   "source": [
    "## Problematic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f747f684-f408-46ba-a35d-d723eaebcf3f",
   "metadata": {},
   "source": [
    "Let's do some visualizations using Python and PyTorch to see how the gridding problem appears."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d82b4868-12f3-4628-bf96-28595533935e",
   "metadata": {},
   "source": [
    "Some imports:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "0b0f8dd6-595e-4ec0-a917-9a0b6d1f3b00",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import io\n",
    "\n",
    "import imageio\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import PIL\n",
    "import torch"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d49836db-6e79-4d85-9e3e-968e4d92335d",
   "metadata": {},
   "source": [
    "Settings for plotting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9884634d-f4a3-469a-981f-aa43cac67d81",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[\"figure.facecolor\"] = (1.0, 1.0, 1.0, 1)\n",
    "plt.rcParams[\"figure.figsize\"] = [3, 3]\n",
    "plt.rcParams[\"figure.dpi\"] = 150\n",
    "plt.rcParams[\"legend.fontsize\"] = \"large\"\n",
    "plt.rcParams[\"axes.titlesize\"] = \"large\"\n",
    "plt.rcParams[\"axes.labelsize\"] = \"large\"\n",
    "plt.rcParams[\"xtick.labelsize\"] = \"small\"\n",
    "plt.rcParams[\"ytick.labelsize\"] = \"small\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3d492fe-d470-4e96-8b25-75acb4f0e689",
   "metadata": {},
   "source": [
    "Below, we create a function that generates a neural network with $N$ convolution layers with respective dilation rates defined as a parameter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "01ababbc-954f-4aee-9f0c-de872fe19a58",
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate_network(dilation_rates: list):\n",
    "    layers = []\n",
    "    for dilation in dilation_rates:\n",
    "        conv = torch.nn.Conv2d(\n",
    "            in_channels=1,\n",
    "            out_channels=1,\n",
    "            kernel_size=3,\n",
    "            stride=1,\n",
    "            padding=\"same\",\n",
    "            dilation=dilation,\n",
    "            bias=False,\n",
    "            padding_mode=\"zeros\",\n",
    "        )\n",
    "        conv.weight.data.fill_(1)\n",
    "        layers.append(conv)\n",
    "    network = torch.nn.Sequential(*layers)\n",
    "    return network"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1ab34cd-9ecd-46bf-8d32-eaa1645f3b98",
   "metadata": {},
   "source": [
    "We also need another function that:\n",
    "1) initializes the network\n",
    "2) creates the input image\n",
    "3) computes the output\n",
    "4) returns the gradient of the output center pixel with respect to the input image."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8be0d18e-37db-429a-9505-d5ae91f72a99",
   "metadata": {},
   "outputs": [],
   "source": [
    "def inference_and_return_grad(pixels: int, dilation_rates: list):\n",
    "    net = generate_network(dilation_rates)\n",
    "\n",
    "    input_image = torch.zeros((1, 1, pixels, pixels))\n",
    "    input_image.requires_grad_(True)\n",
    "\n",
    "    output = net(input_image)\n",
    "\n",
    "    # we return the gradient of the output pixel center\n",
    "    # with respect to the input image\n",
    "    center_pixel = output[0, 0, pixels // 2, pixels // 2]\n",
    "    center_pixel.backward()\n",
    "    return input_image.grad"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d5e8626-07e8-40a2-8022-2e859cad0c6a",
   "metadata": {},
   "source": [
    "We define the input as a 21x21 pixels image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "516b532d-5a64-4e39-95c2-fe88b2f046fe",
   "metadata": {},
   "outputs": [],
   "source": [
    "pixels = 21"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2e9ae00-01a0-4781-9479-2f10bbb61d10",
   "metadata": {},
   "source": [
    "We create a function to help with plotting the gradient.\n",
    "\n",
    "For all visualizations below:\n",
    "- white pixels have no impact on the output center pixel\n",
    "- blue pixels have an impact on the output center pixel, the darker the blue, the greater the impact."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "86eff179-274e-47bc-a7b9-b96578e450e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_grad(grad, title, show=True):\n",
    "    grad = grad[0, 0]\n",
    "    ax = plt.imshow(grad, cmap=\"Blues\", interpolation=\"none\")\n",
    "    plt.title(\"Dilation rates: \" + title)\n",
    "    if show:\n",
    "        plt.show()\n",
    "    else:\n",
    "        return ax"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84b762b6-6c94-4ba5-8ba6-2f20a015bffc",
   "metadata": {},
   "source": [
    "We start with plotting the gradient of the output center pixel w.r.t. the input for a 3-layers convolutional network without dilation ($dilation=1$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0b138936-da3f-4256-91f3-ccb70ae67073",
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 450x450 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grad(\n",
    "    inference_and_return_grad(pixels, dilation_rates=(1, 1, 1)), title=\"(1, 1, 1)\"\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6a26905-b086-4ed9-b699-2cd331c870f6",
   "metadata": {},
   "source": [
    "Looks good to me, it's a classic 3-layers CNN with 3x3 kernels..."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd047b7b-9d77-4e3b-b912-69371e15a245",
   "metadata": {},
   "source": [
    "Now, let's try to reproduce the gridding problem with a 4-layers CNN and a dilation rate of 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3d672c70-c384-4b0e-b33a-ab6d342222b1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 450x450 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grad(\n",
    "    inference_and_return_grad(pixels, dilation_rates=(2, 2, 2, 2)), title=\"(2, 2, 2, 2)\"\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c319fa08-a4d1-436c-9bac-9afbfc6592fe",
   "metadata": {},
   "source": [
    "We can see that the output center pixel has a big receptive field encompassing the whole image, which is the purpose of dilated convolutions.\n",
    "\n",
    "However, we also clearly notice a uniform grid of pixels not contributing to the output center pixel, which is the unwanted gridding problem."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "027cde2b-6fe9-431d-b28f-369ee628e411",
   "metadata": {},
   "source": [
    "In some neural network architectures, we sometimes see successive dilation rates growing exponentially:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "186e0cad-4e77-48f9-8560-0ce791fcc121",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 450x450 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grad(\n",
    "    inference_and_return_grad(pixels, dilation_rates=(1, 2, 4)), title=\"(1, 2, 4)\"\n",
    ");"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce43c8cf-7832-478d-a638-0e70a59f70bb",
   "metadata": {},
   "source": [
    "This shouldn't be used since we can see that some pixels contribute more to the output center pixel than the input center pixel itself."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a13a7906-6661-4fbc-b2b4-dfeed7672acd",
   "metadata": {},
   "source": [
    "## Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00db3385-6348-46c2-9616-98f0156f1de3",
   "metadata": {},
   "source": [
    "The key to solve the gridding problem is to use the Fibonacci sequence.\n",
    "\n",
    "We can see that it effectivily solves the gridding issue:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "14c4c736-0bb5-45f5-a0a6-206b2980ce13",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 450x450 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grad(\n",
    "    inference_and_return_grad(pixels, dilation_rates=(1, 1, 2, 3, 5)),\n",
    "    title=\"(1, 1, 2, 3, 5)\",\n",
    ");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "a73d8f80-167f-482d-ac1f-47cebe71da06",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fibonacci_sequence(seq_len: int):\n",
    "    fib_seq = [1, 1]\n",
    "    for _ in range(seq_len - 2):\n",
    "        fib_seq.append(fib_seq[-1] + fib_seq[-2])\n",
    "    return fib_seq"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5053e40-78f2-4fe8-afa8-fe0c738d3ebe",
   "metadata": {},
   "source": [
    "We can visualize the impact on the gradient of adding several conv layers with dilation rates following the Fibonacci sequence:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e2945468-0605-4f74-b731-16626442b818",
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "writer = imageio.get_writer(\"anim.gif\", mode=\"I\", duration=1)\n",
    "\n",
    "nb_layers_end = 6\n",
    "fib_seq = fibonacci_sequence(nb_layers_end)\n",
    "\n",
    "for nb_layers in range(1, nb_layers_end + 1):\n",
    "    ax = plot_grad(\n",
    "        inference_and_return_grad(pixels, dilation_rates=fib_seq[:nb_layers]),\n",
    "        show=False,\n",
    "        title=str(fib_seq[:nb_layers]),\n",
    "    )\n",
    "    buf = io.BytesIO()\n",
    "    ax.figure.savefig(buf, dpi=150)\n",
    "    buf.seek(0)\n",
    "    pil_img = PIL.Image.open(buf)\n",
    "    writer.append_data(np.array(pil_img))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0135628-01a3-4c56-8842-fde2b3c9bff8",
   "metadata": {
    "tags": []
   },
   "source": [
    "![](anim.gif)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}