From 7e1ead372d73032557aca646929d2098c5a5895d Mon Sep 17 00:00:00 2001 From: dangqingqing Date: Fri, 20 Jan 2017 20:09:37 +0800 Subject: [PATCH] Refine conv in recognize digits --- recognize_digits/README.md | 41 ++++++++++++++++---------- recognize_digits/image/conv_layer.png | Bin 255659 -> 254303 bytes 2 files changed, 25 insertions(+), 16 deletions(-) diff --git a/recognize_digits/README.md b/recognize_digits/README.md index 7776da3..7dff29b 100644 --- a/recognize_digits/README.md +++ b/recognize_digits/README.md @@ -53,7 +53,6 @@ Softmax回归模型采用了最简单的两层神经网络,即只有输入层 1. 经过第一个隐藏层,可以得到 $ H_1 = \phi(W_1X + b_1) $,其中$\phi$代表激活函数,常见的有sigmoid、tanh或ReLU等函数。 2. 经过第二个隐藏层,可以得到 $ H_2 = \phi(W_2H_1 + b_2) $。 3. 最后,再经过输出层,得到的$Y=softmax(W_3H_2 + b_3)$,即为最后的分类结果向量。 - 图3为多层感知器的网络结构图,图中权重用黑线表示、偏置用红线表示、+1代表偏置参数的系数为1。

@@ -63,14 +62,36 @@ Softmax回归模型采用了最简单的两层神经网络,即只有输入层 ### 卷积神经网络(Convolutional Neural Network, CNN) +在多层感知器模型中,将图像展开成一维向量输入到网络中,忽略了图像的位置和结构信息,而卷积神经网络能够更好的利用图像的结构信息。[LeNet-5](http://yann.lecun.com/exdb/lenet/)是一个最简单的卷积神经网络。图6显示了其结构:输入的二维图像,先经过两次卷积层到池化层,再经过全连接层,最后使用softmax分类作为输出层。下面我们主要介绍卷积层和池化层。 + +

+
+图6. LeNet-5卷积神经网络结构
+

+ #### 卷积层 + +卷积层是卷积神经网络的核心基石。在图像识别里我们提到的卷积是二维卷积,即离散二维滤波器(也称作卷积核)与二维图像做卷积操作,简单的讲是二维滤波器滑动到二维图像上所有位置,并在每个位置上与该像素点及其领域像素点做内积。卷积操作被广泛应用与图像处理领域,不同卷积核可以提取不同的特征,例如边沿、线性、角等特征。在深层卷积神经网络中,通过卷积操作可以提取出图像低级到复杂的特征。 +

-
+
图4. 卷积层图片

-卷积层是卷积神经网络的核心基石。该层的参数由一组可学习的过滤器(也叫作卷积核)组成。在前向过程中,每个卷积核在输入层进行横向和纵向的扫描,与输入层对应扫描位置进行卷积,得到的结果加上偏置并用相应的激活函数进行激活,结果能够得到一个二维的激活图(activation map)。每个特定的卷积核都能得到特定的激活图(activation map),如有的卷积核可能对识别边角,有的可能识别圆圈,那这些卷积核可能对于对应的特征响应要强。 -图4是卷积层的一个动态图。由于3D量难以表示,所有的3D量(输入的3D量(蓝色),权重3D量(红色),输出3D量(绿色))通过将深度在行上堆叠来表示。如图4,输入层是$W_1=5,H_1=5,D_1=3$,我们常见的彩色图片其实就是类似这样的输入层,彩色图片的宽和高对应这里的$W_1$和$H_1$,而彩色图片有RGB三个颜色通道,对应这里的$D_1$;卷积层的参数为$K=2,F=3,S=2,P=1$,这里的$K$是卷积核的数量,如图4中有$Filter W_0$和$Filter W_1$两个卷积核,$F$对应卷积核的大小,图中$W0$和$W1$在每一层深度上都是$3\times3$的矩阵,$S$对应卷积核扫描的步长,从动态图中可以看到,方框每次左移或下移2个单位,$P$对应Padding扩展,是对输入层的扩展,图中输入层,原始数据为蓝色部分,可以看到灰色部分是进行了大小为1的扩展,用0来进行扩展;图4的动态可视化对输出层结果(绿色)进行迭代,显示每个输出元素是通过将突出显示的输入(蓝色)与滤波器(红色)进行元素相乘,将其相加,然后通过偏置抵消结果来计算的。 +图4给出一个卷积计算过程的示例图,输入图像大小为$H=5,W=5,D=3$,即$5x5$大小的3通道(RGB,也称作深度)彩色图像。这个示例图中包含2(用$K$表示)组卷积核,即图中$Filter W_0$和$Filter W_1$,在卷积计算中,通常对不同的输入通道采用不同的卷积核,在图示例中每组卷积核又包含3($D$)个$3x3$(用$FXF$表示)大小的卷积核。另外,这个示例中卷积核在图像的水平方向($W$方向)和垂直方向($H$方向)的滑动步长为2(用$S$表示);对输入图像周围各填充1(用$P$表示)个0,即图中输入层原始数据为蓝色部分,灰色部分是进行了大小为1的扩展,用0来进行扩展。经过卷积操作得到输出为$3x3x2$(用$H_{o}xW_{o}xK$表示)大小的特征图,即$3x3$大小的2通道特征图,其中$H_o$计算公式为:$H_o = (H - F + 2*P)/S + 1$,$W_o$同理。 而输出特征图中的每个像素,是每组滤波器与输入图像每个特征图的内积再求和,再加上偏置($b_o$),偏置通常对于每个输出特征图是共享的。例如图中输出特征图`o[:,:,0]`中的$-9$计算如下: + +$$-9 = \sum x[4:6,4:6,0] * W[:,:,0]] + \sum x[4:6,4:6,1] * W[:,:,1]] + \sum x[4:6,4:6,2] * W[:,:,2]] + b_0\\ +\sum x[4:6,4:6,0] * W[:,:,0]] = 2*1 + 2*(-1) + 0*1 + 0*0 + 2*(-1) + 0*1 + 0*0 + 0*0 + 0*0 = -2 \\ +\sum x[4:6,4:6,1] * W[:,:,1]] = 2*(-1) + 2*(-1) + 0*0 + 2*0 + 2*(-1) + 0*(-1) + 0*0 + 0*1 + 0*1 = -6 \\ +\sum x[4:6,4:6,2] * W[:,:,2]] = 0*0 + 0*1 + 0*1 + 2*(-1) + 1*0 + 0*1 + 0*1 + 0*0 + 0*1 = -2$$ + +在卷积操作中卷积核是可学习的参数,经过上面示例介绍,每层卷积的参数大小为$DxFxFxK$。在多层感知器模型中,神经元通常是全部连接,参数较多。而卷积层的参数较少,这也是由卷积层的主要特性即局部连接和共享权重所决定。 + +- 局部连接:每个神经元仅与输入神经元的一块区域连接,这块局部区域称作感受野(receptive field)。在图像卷积操作中,即神经元在空间维度(spatial dimension,即上图示例H和W所在的平面)是局部连接,但在深度上是全部连接。对于二维图像本身而言,也是局部像素关联较强。这种局部连接保证了学习后的过滤器能够对于局部的输入特征有最强的响应。局部连接的思想,也是受启发于生物学里面的视觉系统结构,视觉皮层的神经元就是局部接受信息的。 + +- 权重共享:计算同一个深度切片的神经元时采用的滤波器是共享的。例如图4中计算$o[:,:,0]$的每个每个神经元的滤波器均相同,都为$W_0$,这样可以很大程度上减少参数。共享权重在一定程度上讲是有意义的,例如图片的底层边缘特征与特征在图中的具体位置无关。但是在一些场景中是无意的,比如输入的图片是人脸,眼睛和头发位于不同的位置,希望在不同的位置学到不同的特征 (参考[斯坦福大学公开课]( http://cs231n.github.io/convolutional-networks/))。请注意权重只是对于同一深度切片的神经元是共享的,在卷积层,通常采用多组卷积核提取不同特征,即对应不同深度切片的特征,不同深度切片的神经元权重是不共享。另外,偏重对同一深度切片的所有神经元都是共享的。 + +通过介绍卷积计算过程及其特性,可以看出卷积是线性操作,并具有平移不变性(shift-invariant),平移不变性即在图像每个位置执行相同的操作。卷积层的局部连接和权重共享使得需要学习的参数大大减小,这样也有利于训练较大卷积神经网络。 #### 池化层

@@ -79,18 +100,6 @@ Softmax回归模型采用了最简单的两层神经网络,即只有输入层

池化是非线性下采样的一种形式,主要作用是通过减少网络的参数来减小计算量,并且能够在一定程度上控制过拟合。通常在卷积层的后面会加上一个池化层。池化包括最大池化、平均池化等。其中最大池化是用不重叠的矩形框将输入层分成不同的区域,对于每个矩形框的数取最大值作为输出层,如图5所示。 -#### LeNet-5网络 -

-
-图6. LeNet-5卷积神经网络结构
-

- -[LeNet-5](http://yann.lecun.com/exdb/lenet/)是一个最简单的卷积神经网络。图6显示了其结构:输入的二维图像,先经过两次卷积层到池化层,再经过全连接层,最后使用softmax分类作为输出层。卷积的如下三个特性,决定了LeNet-5能比同样使用全连接层的多层感知器更好地识别图像: - -- 神经元的三维特性: 卷积层的神经元在宽度、高度和深度上进行了组织排列。每一层的神经元仅仅与前一层的一块小区域连接,这块小区域被称为感受野(receptive field)。 -- 局部连接:CNN通过在相邻层的神经元之间实施局部连接模式来利用空间局部相关性。这样的结构保证了学习后的过滤器能够对于局部的输入特征有最强的响应。堆叠许多这样的层导致非线性“过滤器”变得越来越“全局”。这允许网络首先创建输入的小部分的良好表示,然后从它们组合较大区域的表示。 -- 共享权重:在CNN中,每个滤波器在整个视野中重复扫描。 这些复制单元共享相同的参数化(权重向量和偏差)并形成特征图。 这意味着给定卷积层中的所有神经元检测完全相同的特征。 以这种方式的复制单元允许不管它们在视野中的位置都能检测到特征,从而构成平移不变性的性质。 - 更详细的关于卷积神经网络的具体知识可以参考[斯坦福大学公开课]( http://cs231n.github.io/convolutional-networks/ )和[图像分类](https://github.com/PaddlePaddle/book/blob/develop/image_classification/README.md)教程。 ### 常见激活函数介绍 diff --git a/recognize_digits/image/conv_layer.png b/recognize_digits/image/conv_layer.png index e794e3d3cc8b7d1b9f86afb14a1967ff15aea54f..67d2e3337c94dc3effbf210416062401601d3ec8 100644 GIT binary patch literal 254303 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