> For the complete documentation index, see [llms.txt](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/di-si-men-ke-juan-ji-shen-jing-wang-luo-convolutional-neural-networks/convolutional-neural-networks/special-applications/410-feng-ge-dai-jia-han-shu-ff08-style-cost-function.md).

# 4.9 风格代价函数（Style cost function）

利用CNN网络模型，图片的风格可以定义成第$$l$$层隐藏层不同通道间激活函数的乘积（相关性）

[![](https://github.com/fengdu78/deeplearning_ai_books/raw/master/images/efa0f6e81320966647658cba96ff28ee.png)](https://github.com/fengdu78/deeplearning_ai_books/blob/master/images/efa0f6e81320966647658cba96ff28ee.png)

选取第$$l$$层隐藏层，各通道使用不同颜色标注。因为每个通道提取图片的特征不同，比如1通道（红色）提取的是图片的垂直纹理特征，2通道（黄色）提取的是图片的橙色背景特征。那么这两个通道的相关性越大，表示原始图片及既包含了垂直纹理也包含了该橙色背景；相关性越小，表示原始图片并没有同时包含这两个特征。即计算不同通道的相关性，反映了原始图片特征间的相互关系，从某种程度上刻画了图片的“风格”

![](/files/-Le0cUOoS6tLVjGbkYWq)

[![](https://github.com/fengdu78/deeplearning_ai_books/raw/master/images/e3d74c1ce2393ae4e706a1cc4024f311.png)](https://github.com/fengdu78/deeplearning_ai_books/blob/master/images/e3d74c1ce2393ae4e706a1cc4024f311.png)

接下来定义图片的风格矩阵（style matrix）为：

$$
G\_{kk^{'}}^{\[l]} = \sum\_{i = 1}^{n\_{H}^{\[l]}}{\sum\_{j = 1}^{n\_{W}^{\[l]}}{a\_{i, j,k}^{\[l]}a\_{i, j, k^{'}}^{\[l]}}}
$$

$$a\_{i, j, k}^{\[l]}$$为隐藏层$$l$$中$$(i,j,k)$$位置的激活项，$$i$$，$$j$$，$$k$$分别代表该位置的高度、宽度以及对应的通道数，k，$$k^{'}$$分别表示不同通道。风格矩阵$$G\_{kk^{'}}^{\[l]}$$计算第$$l$$层隐藏层不同通道对应的所有激活函数输出和，$$l$$层风格图像的矩阵$$G^{\[l]}$$是一个$$n\_{c} \times n\_{c}$$的矩阵：

![](/files/-Le0cUOsGx7rAKDyufOY)

若两个通道之间相似性高，则对应的$$G\_{kk^{'}}^{\[l]}$$较大；若两个通道之间相似性低，则对应的$$G\_{kk^{'}}^{\[l]}$$较小

风格矩阵$$G\_{kk'}^{[l](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/di-si-men-ke-juan-ji-shen-jing-wang-luo-convolutional-neural-networks/convolutional-neural-networks/special-applications/S)}$$表征了风格图片$$S$$第$$l$$层隐藏层的“风格”。生成图片$$G$$也有$$G\_{kk'}^{[l](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/di-si-men-ke-juan-ji-shen-jing-wang-luo-convolutional-neural-networks/convolutional-neural-networks/special-applications/G)}$$，$$G\_{kk'}^{[l](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/di-si-men-ke-juan-ji-shen-jing-wang-luo-convolutional-neural-networks/convolutional-neural-networks/special-applications/S)}$$与$$G\_{kk'}^{[l](https://baozoulin.gitbook.io/neural-networks-and-deep-learning/di-si-men-ke-juan-ji-shen-jing-wang-luo-convolutional-neural-networks/convolutional-neural-networks/special-applications/G)}$$越相近，则表示$$G$$的风格越接近$$S$$。即$$J^{\[l]}\_{style}(S,G)$$定义为：

$$
J\_{style}^{\[l]}(S,G) = \frac{1}{4 \times {n\_C}^2 \times (n\_H \times n\_W)^2} \sum *{i=1}^{n\_C}\sum*{j=1}^{n\_C}(G^{(S)}*{ij} - G^{(G)}*{ij})^2
$$

然后使用梯度下降算法，不断迭代修正$$G$$的像素值，使$$J^{\[l]}\_{style}(S,G)$$不断减小

为了提取更多的“风格”，可以使用多层隐藏层，然后相加，表达式为：

$$
J\_{style}(S,G)=\sum\_l\lambda^{\[l]}\cdot J^{\[l]}\_{style}(S,G)
$$

$$\lambda^{\[l]}$$表示累加过程中各层$$J^{\[l]}\_{style}(S,G)$$的权重系数，为超参数

最终的cost function为：

$$
J(G)=\alpha \cdot J\_{content}(C,G)+\beta \cdot J\_{style}(S,G)
$$

之后用梯度下降法，或者更复杂的优化算法来找到一个合适的图像$$G$$，并计算$$J(G)$$的最小值，这样将能够得到非常好看的结果


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