卷积运算和相关运算在深度学习领域属于常见的运算，本文将从二者计算的公式和实现代码进行分析。
卷积运算
给定一个图像$X\in \mathbb{R}^{M\times N}$，和滤波器$W\in \mathbb{R}^{m\times n}$，一般$m \ll M $，$n \ll N $，其卷积为：
$$y_{ij} = \sum_{u=1}^m \sum_{v=1}^n w_{uv} \cdot x_{i-u+1, j-v+1}$$
- 代码实现
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20def conv(image, kernel):
""" An implementation of convolution filter. This function uses element-wise multiplication and np.sum() to efficiently compute weighted sum of neighborhood at each
pixel.
Args:
image: numpy array of shape (Hi, Wi)
kernel: numpy array of shape (Hk, Wk)
Returns:
out: numpy array of shape (Hi, Wi)
"""
Hi, Wi = image.shape
Hk, Wk = kernel.shape
out = np.zeros((Hi, Wi))
image = np.pad(image, (Hk//2, Wk//2), 'constant', constant_values=0)
print(image.shape)
kernel = np.flip(np.flip(kernel, 1), 0)
for i in range(Hi):
for j in range(Wi):
out[i, j] = sum(
sum(kernel * image[i:i + Hk, j:j + Wk]))
return out
相关运算
因为卷积运算需要对卷积核进行翻转，然而翻转对于深度卷积神经网络来说并不必须，所以通常深度学习框架，如Pytorch，Tensorflow采用相关运算取代卷积运算来避免不必要的运算复杂度的增加。
同样给定一个图像$X\in \mathbb{R}^{M\times N}$，和滤波器$W\in \mathbb{R}^{m\times n}$，一般$m \ll M $，$n \ll N $，其互相关为：
$$y_{ij} = \sum_{u=1}^m \sum_{v=1}^n w_{uv} \cdot x_{i+u-1, j+v-1}$$
- 代码实现上面的代码中，我利用前面的卷积运算，经过两轮的翻转，卷积核与最初的输入值保持一致，这样也就实现了相关系数的计算。
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12def cross_correlation(f, g):
""" Cross-correlation of f and g
Args:
f: numpy array of shape (Hf, Wf)
g: numpy array of shape (Hg, Wg)
Returns:
out: numpy array of shape (Hf, Wf)
"""
out = None
g = np.flip(np.flip(g, 0), 1)
out = conv(f, g)
return out
