损失函数SSIM (structural similarity index) 的PyTorch实现
SSIM介绍
结构相似性指数（structural similarity index，SSIM）, 出自参考文献[1]，用于度量两幅图像间的结构相似性。和被广泛采用的L2 loss不同，SSIM和人类的视觉系统（HVS）类似，对局部结构变化的感知敏感。
SSIM分为三个部分：照明度、对比度、结构，分别如下公式所示：
将上面三个式子汇总到一起就是SSIM：
其中，上式各符号分别为图像x和y的均值、方差和它们的协方差，显而易见，不赘述。，为常数。一般默认，. L为像素值的动态范围，如8-bit深度的图像的L值为2^8-1=255.
更详细的说明可以参考维基百科[2].
Pytorch实现
SSIM值越大代表图像越相似，当两幅图像完全相同时，SSIM=1。所以作为损失函数时，应该要取负号，例如采用 loss = 1 - SSIM 的形式。由于PyTorch实现了自动求导机制，因此我们只需要实现SSIM loss的前向计算部分即可，不用考虑求导。（具体的求导过程可以参考文献[3]）
以下是代码实现，来源于github [4].
import torch
import torch.nn.functional as F
from math import exp
import numpy as np
# 计算一维的高斯分布向量
def gaussian(window_size, sigma):
gauss = torch.Tensor([exp(-(x - window_size//2)**2/float(2*sigma**2)) for x in range(window_size)])
return gauss/gauss.sum()
# 创建高斯核，通过两个一维高斯分布向量进行矩阵乘法得到
# 可以设定channel参数拓展为3通道
def create_window(window_size, channel=1):
_1D_window = gaussian(window_size, 1.5).unsqueeze(1)
_2D_window = _1D_window.mm(_1D_window.t()).float().unsqueeze(0).unsqueeze(0)
window = _2D_window.expand(channel, 1, window_size, window_size).contiguous()
return window
# 计算SSIM
# 直接使用SSIM的公式，但是在计算均值时，不是直接求像素平均值，而是采用归一化的高斯核卷积来代替。
# 在计算方差和协方差时用到了公式Var(X)=E[X^2]-E[X]^2, cov(X,Y)=E[XY]-E[X]E[Y].
# 正如前面提到的，上面求期望的操作采用高斯核卷积代替。
def ssim(img1, img2, window_size=11, window=None, size_average=True, full=False, val_range=None):
# Value range can be different from 255. Other common ranges are 1 (sigmoid) and 2 (tanh).
if val_range is None:
if torch.max(img1) > 128:
max_val = 255
else:
max_val = 1
if torch.min(img1) < -0.5:
min_val = -1
else:
min_val = 0
L = max_val - min_val
else:
L = val_range
padd = 0
(_, channel, height, width) = img1.size()
if window is None:
real_size = min(window_size, height, width)
window = create_window(real_size, channel=channel).to(img1.device)
mu1 = F.conv2d(img1, window, padding=padd, groups=channel)
mu2 = F.conv2d(img2, window, padding=padd, groups=channel)
mu1_sq = mu1.pow(2)
mu2_sq = mu2.pow(2)
mu1_mu2 = mu1 * mu2
sigma1_sq = F.conv2d(img1 * img1, window, padding=padd, groups=channel) - mu1_sq
sigma2_sq = F.conv2d(img2 * img2, window, padding=padd, groups=channel) - mu2_sq
sigma12 = F.conv2d(img1 * img2, window, padding=padd, groups=channel) - mu1_mu2
C1 = (0.01 * L) ** 2
C2 = (0.03 * L) ** 2
v1 = 2.0 * sigma12 + C2
v2 = sigma1_sq + sigma2_sq + C2
cs = torch.mean(v1 / v2) # contrast sensitivity
ssim_map = ((2 * mu1_mu2 + C1) * v1) / ((mu1_sq + mu2_sq + C1) * v2)
if size_average:
ret = ssim_map.mean()
else:
ret = ssim_map.mean(1).mean(1).mean(1)
if full:
return ret, cs
return ret
# Classes to re-use window
class SSIM(torch.nn.Module):
def __init__(self, window_size=11, size_average=True, val_range=None):
super(SSIM, self).__init__()
self.window_size = window_size
self.size_average = size_average
self.val_range = val_range
# Assume 1 channel for SSIM
self.channel = 1
self.window = create_window(window_size)
def forward(self, img1, img2):
(_, channel, _, _) = img1.size()
if channel == self.channel and self.window.dtype == img1.dtype:
window = self.window
else:
window = create_window(self.window_size, channel).to(img1.device).type(img1.dtype)
self.window = window
self.channel = channel
return ssim(img1, img2, window=window, window_size=self.window_size, size_average=self.size_average)
参考来源
[1] Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment: from error visibility to structural similarity[J]. IEEE transactions on image processing, 2004, 13(4): 600-612.
[2] https://en.wikipedia.org/wiki/Structural_similarity
[3] Zhao H, Gallo O, Frosio I, et al. Loss functions for neural networks for image processing[J]. arXiv preprint arXiv:1511.08861, 2015.
[4] https://github.com/jorge-pessoa/pytorch-msssim