CS131_HomeWork_hw1

cs131 hw1

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• 参考资料：
  • wwdguu/CS131_homework
  • mikucy/CS131

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python numpy

• np.var() : 方差
• np.flip() : 矩阵翻转
• np.argmax(): 最大值的indices

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一、 Convolutions

1.1 commutative Property

推导两个式子相等：

令 ：

• x = m-i ==> i = m- x
• y = m- j ==> j = m- y

带入上式子为：

然后。令 x = i, y =j;即可；

1.2 Linear and Shift Invariance

推导：

1.3 实现

• 卷积需要翻转卷积核；

  1.3.1 convoution

• 使用四个循环
  `python

def conv_nested(image, kernel):

Hi, Wi = image.shape
Hk, Wk = kernel.shape
out = np.zeros((Hi, Wi))
padded = zero_pad(image,Hk//2,Wk//2)
kernel = np.flip(np.flip(kernel, 0), 1)  # 上下翻转，在左右翻转
### YOUR CODE HERE
for m in range(Hi):
    for n in range(Wi):
        for i in range(Hk):
            for j in range(Wk):
                    out[m,n]+=kernel[i,j]*padded[m+i,n+j]

return out

- padding添加：


```python

def zero_pad(image, pad_height, pad_width):
    H, W = image.shape
    out = np.zeros(shape=(H+pad_height*2,W+pad_width*2),dtype=np.float32)
    out[pad_height:pad_height+H,pad_width:pad_width+W]=image[:,:]
    return out

• 使用矩阵相乘：

def conv_fast(image, kernel):
    Hi, Wi = image.shape
    Hk, Wk = kernel.shape
    out = np.zeros((Hi, Wi))
    padd_H = Hk // 2
    padd_W = Wk // 2
    img_padd = zero_pad(image, padd_H, padd_W)
    # 卷积过程
    kernel = np.flip(np.flip(kernel, 0), 1)  # 上下翻转，在左右翻转
    for i in range(Hi):
        for j in range(Wi):

            out[i, j] = np.sum(img_padd[i:(i+Hk), j:(j+Wk)] * kernel)

    return out

part 2 Cross-correlation

• cross correlation和卷积一样，只是不需要翻转卷积核；

• cross correlation

def cross_correlation(f, g):
    Hi, Wi = f.shape
    Hk, Wk = g.shape
    out = np.zeros((Hi, Wi))
    padd_H = Hk // 2
    padd_W = Wk // 2
    img_padd = zero_pad(f, padd_H, padd_W)
    # 卷积过程
    for i in range(Hi):
        for j in range(Wi):
            out[i, j] = np.sum(img_padd[i:(i+Hk), j:(j+Wk)] * g)
    return out
`

• zero_mean_cross_correlation

def zero_mean_cross_correlation(f, g):

    out = None
    mean = np.mean(g)
    g = np.subtract(g, mean)
    out = cross_correlation(f,g)
    return out
`

• normalized_cross_correlation

def normalized_cross_correlation(f, g):
    # out = None
    Hk, Wk = f.shape
    Hg, Wg = g.shape
    paddedF = zero_pad(f, Hg // 2, Wg // 2)
    out = np.zeros_like(f)
    # g = np.flip(np.flip(g, 0), 1)
    g_mean = np.mean(g)
    g_delta = np.sqrt(np.var(g))
    g_t = (g - g_mean) / g_delta

    for m in range(Hk):
        for n in range(Wk):
            conv = paddedF[m:(m+Hg), n:(n+Wg)]
            f_mean = np.mean(conv)
            f_delta = np.sqrt(np.var(conv))
            f_t = (paddedF[m:(m+Hg), n:(n+Wg)] - f_mean)/f_delta
            out[m, n] = np.sum(f_t * g_t)

    print('end')

    return out

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