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‎slides/2_26/11-Convolution.key

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‎slides/2_26/11-Convolution.pdf

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‎slides/2_26/channels.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"# Multiple Input and Output Channels\n",
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"\n",
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"**Multiple Input Channels**\n",
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"\n",
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"![Cross-correlation with 2 input channels](http://www.d2l.ai/_images/conv_multi_in.svg)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "1"
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}
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},
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"outputs": [],
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"source": [
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"import d2l\n",
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"from mxnet import nd\n",
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"\n",
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"def corr2d_multi_in(X, K):\n",
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" # First, traverse along the 0th dimension (channel dimension) of X and K. \n",
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" # Then, add them together by using * \n",
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" return nd.add_n(*[d2l.corr2d(x, k) for x, k in zip(X, K)])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"We can construct the input array `X` and the kernel array `K` of the above diagram to validate the output of the cross-correlation operation."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "2"
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\n",
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"[[ 56. 72.]\n",
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" [104. 120.]]\n",
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"<NDArray 2x2 @cpu(0)>"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"X = nd.array([[[0, 1, 2], [3, 4, 5], [6, 7, 8]],\n",
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" [[1, 2, 3], [4, 5, 6], [7, 8, 9]]])\n",
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"K = nd.array([[[0, 1], [2, 3]], [[1, 2], [3, 4]]])\n",
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"\n",
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"corr2d_multi_in(X, K)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"**Multiple Output Channels**\n",
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"\n",
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"For multiple output channels we simply generate multiple outputs and then stack them together. "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "3"
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}
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},
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"outputs": [],
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"source": [
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"def corr2d_multi_in_out(X, K):\n",
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" # Traverse along the 0th dimension of K, and each time, perform cross-correlation \n",
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" # operations with input X. All of the results are merged together using the stack function.\n",
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" return nd.stack(*[corr2d_multi_in(X, k) for k in K])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We construct a convolution kernel with 3 output channels by concatenating the kernel array `K` with `K+1` (plus one for each element in `K`) and `K+2`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "4"
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(3, 2, 2, 2)"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"K = nd.stack(K, K + 1, K + 2)\n",
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"K.shape"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"We can have multiple input and output channels."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "5"
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}
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"(2, 3, 3)\n",
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"(3, 2, 2, 2)\n",
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"\n",
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"[[[ 56. 72.]\n",
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" [104. 120.]]\n",
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"\n",
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" [[ 76. 100.]\n",
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" [148. 172.]]\n",
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"\n",
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" [[ 96. 128.]\n",
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" [192. 224.]]]\n",
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"<NDArray 3x2x2 @cpu(0)>\n"
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]
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}
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],
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"source": [
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"print(X.shape)\n",
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"print(K.shape)\n",
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"print(corr2d_multi_in_out(X, K))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## $1\\times 1$ Convolutions\n",
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"\n",
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"![1x1 convolutions](http://www.d2l.ai/_images/conv_1x1.svg)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "6"
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}
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},
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"outputs": [],
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"source": [
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"def corr2d_multi_in_out_1x1(X, K):\n",
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" c_i, h, w = X.shape\n",
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" c_o = K.shape[0]\n",
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" X = X.reshape((c_i, h * w))\n",
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" K = K.reshape((c_o, c_i))\n",
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" Y = nd.dot(K, X) # Matrix multiplication in the fully connected layer.\n",
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" return Y.reshape((c_o, h, w))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"This is equivalent to cross-correlation with an appropriately narrow $1\\times 1$ kernel."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"attributes": {
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"classes": [],
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"id": "",
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"n": "7"
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}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"X = nd.random.uniform(shape=(3, 3, 3))\n",
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"K = nd.random.uniform(shape=(2, 3, 1, 1))\n",
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"\n",
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"Y1 = corr2d_multi_in_out_1x1(X, K)\n",
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"Y2 = corr2d_multi_in_out(X, K)\n",
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"\n",
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"(Y1 - Y2).norm().asscalar() < 1e-6"
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]
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}
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],
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"metadata": {
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"celltoolbar": "Slideshow",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"name": "ipython",
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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‎slides/2_26/channels.pdf

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