Convolution Neural Network (CNN) 原理与实现

本文结合Deep learning的一个应用，Convolution Neural Network 进行一些基本应用，参考Lecun的Document 0.1进行部分拓展，与结果展示（in python）。

分为以下几部分：

1. Convolution（卷积）

2. Pooling（降采样过程）

3. CNN结构

4.  跑实验

下面分别介绍。

PS：本篇blog为ese机器学习短期班参考资料（20140516课程），本文只是简要讲最naive最simple的思想，重在实践部分，原理课上详述。

1. Convolution（卷积）

类似于高斯卷积，对imagebatch中的所有image进行卷积。对于一张图，其所有feature map用一个filter卷成一张feature map。 如下面的代码，对一个imagebatch（含两张图）进行操作，每个图初始有3张feature map(R,G,B), 用两个9*9的filter进行卷积，结果是，每张图得到两个feature map。

卷积操作由theano的conv.conv2d实现，这里我们用随机参数W，b。结果有点像edge detector是不是？

Code: （详见注释）

 # -*- coding: utf-8 -*-
 """
 Created on Sat May 10 18:55:26 2014

 @author: rachel

 Function: convolution option of two pictures with same size (width,height)
 input: 3 feature maps (3 channels <RGB> of a picture)
 convolution: two 9*9 convolutional filters
 """

 from theano.tensor.nnet import conv
 import theano.tensor as T
 import numpy, theano


 rng = numpy.random.RandomState(23455)

 # symbol variable
 input = T.tensor4(name = 'input')

 # initial weights
 w_shape = (2,3,9,9) #2 convolutional filters, 3 channels, filter shape: 9*9
 w_bound = numpy.sqrt(3*9*9)
 W = theano.shared(numpy.asarray(rng.uniform(low = -1.0/w_bound, high = 1.0/w_bound,size = w_shape),
                                 dtype = input.dtype),name = 'W')

 b_shape = (2,)
 b = theano.shared(numpy.asarray(rng.uniform(low = -.5, high = .5, size = b_shape),
                                 dtype = input.dtype),name = 'b')

 conv_out = conv.conv2d(input,W)

 #T.TensorVariable.dimshuffle() can reshape or broadcast (add dimension)
 #dimshuffle(self,*pattern)
 # >>>b1 = b.dimshuffle('x',0,'x','x')
 # >>>b1.shape.eval()
 # array([1,2,1,1])
 output = T.nnet.sigmoid(conv_out + b.dimshuffle('x',0,'x','x'))
 f = theano.function([input],output)





 # demo
 import pylab
 from PIL import Image
 #minibatch_img = T.tensor4(name = 'minibatch_img')

 #-------------img1---------------
 img1 = Image.open(open('//home//rachel//Documents//ZJU_Projects//DL//Dataset//rachel.jpg'))
 width1,height1 = img1.size
 img1 = numpy.asarray(img1, dtype = 'float32')/256. # (height, width, 3)

 # put image in 4D tensor of shape (1,3,height,width)
 img1_rgb = img1.swapaxes(0,2).swapaxes(1,2).reshape(1,3,height1,width1) #(3,height,width)


 #-------------img2---------------
 img2 = Image.open(open('//home//rachel//Documents//ZJU_Projects//DL//Dataset//rachel1.jpg'))
 width2,height2 = img2.size
 img2 = numpy.asarray(img2,dtype = 'float32')/256.
 img2_rgb = img2.swapaxes(0,2).swapaxes(1,2).reshape(1,3,height2,width2) #(3,height,width)



 #minibatch_img = T.join(0,img1_rgb,img2_rgb)
 minibatch_img = numpy.concatenate((img1_rgb,img2_rgb),axis = 0)
 filtered_img = f(minibatch_img)


 # plot original image and two convoluted results
 pylab.subplot(2,3,1);pylab.axis('off');
 pylab.imshow(img1)

 pylab.subplot(2,3,4);pylab.axis('off');
 pylab.imshow(img2)

 pylab.gray()
 pylab.subplot(2,3,2); pylab.axis("off")
 pylab.imshow(filtered_img[0,0,:,:]) #0:minibatch_index; 0:1-st filter

 pylab.subplot(2,3,3); pylab.axis("off")
 pylab.imshow(filtered_img[0,1,:,:]) #0:minibatch_index; 1:1-st filter

 pylab.subplot(2,3,5); pylab.axis("off")
 pylab.imshow(filtered_img[1,0,:,:]) #0:minibatch_index; 0:1-st filter

 pylab.subplot(2,3,6); pylab.axis("off")
 pylab.imshow(filtered_img[1,1,:,:]) #0:minibatch_index; 1:1-st filter
 pylab.show()

2. Pooling（降采样过程）

最常用的Maxpooling. 解决了两个问题：

1. 减少计算量

2. 旋转不变性 （原因自己悟）

     PS：对于旋转不变性，回忆下SIFT，LBP：采用主方向；HOG：选择不同方向的模版

Maxpooling的降采样过程会将feature map的长宽各减半。（下面结果图中没有体现出来，python自动给拉到一样大了，但实际上像素数是减半的）

Code: （详见注释）

 # -*- coding: utf-8 -*-
 """
 Created on Sat May 10 18:55:26 2014

 @author: rachel

 Function: convolution option
 input: 3 feature maps (3 channels <RGB> of a picture)
 convolution: two 9*9 convolutional filters
 """

 from theano.tensor.nnet import conv
 import theano.tensor as T
 import numpy, theano


 rng = numpy.random.RandomState(23455)

 # symbol variable
 input = T.tensor4(name = 'input')

 # initial weights
 w_shape = (2,3,9,9) #2 convolutional filters, 3 channels, filter shape: 9*9
 w_bound = numpy.sqrt(3*9*9)
 W = theano.shared(numpy.asarray(rng.uniform(low = -1.0/w_bound, high = 1.0/w_bound,size = w_shape),
                                 dtype = input.dtype),name = 'W')

 b_shape = (2,)
 b = theano.shared(numpy.asarray(rng.uniform(low = -.5, high = .5, size = b_shape),
                                 dtype = input.dtype),name = 'b')

 conv_out = conv.conv2d(input,W)

 #T.TensorVariable.dimshuffle() can reshape or broadcast (add dimension)
 #dimshuffle(self,*pattern)
 # >>>b1 = b.dimshuffle('x',0,'x','x')
 # >>>b1.shape.eval()
 # array([1,2,1,1])
 output = T.nnet.sigmoid(conv_out + b.dimshuffle('x',0,'x','x'))
 f = theano.function([input],output)





 # demo
 import pylab
 from PIL import Image
 from matplotlib.pyplot import *

 #open random image
 img = Image.open(open('//home//rachel//Documents//ZJU_Projects//DL//Dataset//rachel.jpg'))
 width,height = img.size
 img = numpy.asarray(img, dtype = 'float32')/256. # (height, width, 3)


 # put image in 4D tensor of shape (1,3,height,width)
 img_rgb = img.swapaxes(0,2).swapaxes(1,2) #(3,height,width)
 minibatch_img = img_rgb.reshape(1,3,height,width)
 filtered_img = f(minibatch_img)


 # plot original image and two convoluted results
 pylab.figure(1)
 pylab.subplot(1,3,1);pylab.axis('off');
 pylab.imshow(img)
 title('origin image')

 pylab.gray()
 pylab.subplot(2,3,2); pylab.axis("off")
 pylab.imshow(filtered_img[0,0,:,:]) #0:minibatch_index; 0:1-st filter
 title('convolution 1')

 pylab.subplot(2,3,3); pylab.axis("off")
 pylab.imshow(filtered_img[0,1,:,:]) #0:minibatch_index; 1:1-st filter
 title('convolution 2')

 #pylab.show()




 # maxpooling
 from theano.tensor.signal import downsample

 input = T.tensor4('input')
 maxpool_shape = (2,2)
 pooled_img = downsample.max_pool_2d(input,maxpool_shape,ignore_border = False)

 maxpool = theano.function(inputs = [input],
                           outputs = [pooled_img])

 pooled_res = numpy.squeeze(maxpool(filtered_img))
 #pylab.figure(2)
 pylab.subplot(235);pylab.axis('off');
 pylab.imshow(pooled_res[0,:,:])
 title('down sampled 1')

 pylab.subplot(236);pylab.axis('off');
 pylab.imshow(pooled_res[1,:,:])
 title('down sampled 2')

 pylab.show()

3. CNN结构

想必大家随便google下CNN的图都滥大街了，这里拖出来那时候学CNN的时候一张图，自认为陪上讲解的话画得还易懂（<!--囧-->）

废话不多说了，直接上Lenet结构图：（从下往上顺着箭头看，最下面为底层original input）

4. CNN代码

去资源里下载吧，我放上去了喔~（in python）

这里贴少部分代码，仅表示建模的NN：

 rng = numpy.random.RandomState(23455)

     # transfrom x from (batchsize, 28*28) to (batchsize,feature,28,28))
     # I_shape = (28,28),F_shape = (5,5),
     N_filters_0 = 20
     D_features_0= 1
     layer0_input = x.reshape((batch_size,D_features_0,28,28))
     layer0 = LeNetConvPoolLayer(rng, input = layer0_input, filter_shape = (N_filters_0,D_features_0,5,5),
                                 image_shape = (batch_size,1,28,28))
     #layer0.output: (batch_size, N_filters_0, (28-5+1)/2, (28-5+1)/2) -> 20*20*12*12

     N_filters_1 = 50
     D_features_1 = N_filters_0
     layer1 = LeNetConvPoolLayer(rng,input = layer0.output, filter_shape = (N_filters_1,D_features_1,5,5),
                                 image_shape = (batch_size,N_filters_0,12,12))
     # layer1.output: (20,50,4,4)

     layer2_input = layer1.output.flatten(2) # (20,50,4,4)->(20,(50*4*4))
     layer2 = HiddenLayer(rng,layer2_input,n_in = 50*4*4,n_out = 500, activation = T.tanh)

     layer3 = LogisticRegression(input = layer2.output, n_in = 500, n_out = 10)

layer0, layer1 ：分别是卷积+降采样

layer2+layer3：组成一个MLP（ANN）

训练模型：

 cost = layer3.negative_log_likelihood(y)
 params = layer3.params + layer2.params + layer1.params + layer0.params
 gparams = T.grad(cost,params)

 updates = []
 for par,gpar in zip(params,gparams):
     updates.append((par, par - learning_rate * gpar))

 train_model = theano.function(inputs = [minibatch_index],
                               outputs = [cost],
                               updates = updates,
                               givens = {x: train_set_x[minibatch_index * batch_size : (minibatch_index+1) * batch_size],
                                         y: train_set_y[minibatch_index * batch_size : (minibatch_index+1) * batch_size]})

根据cost（最上层MLP的输出NLL），对所有层的parameters进行训练

剩下的具体见代码和注释。

PS：数据为MNIST所有数据

final result：
Optimization complete. Best validation score of 0.990000 % obtained at iteration 122500, with test performance 0.950000 %