SSIM介紹
結構相似性指數（structural similarity index，SSIM）, 出自參考文獻[1]，用於度量兩幅圖像間的結構相似性。和被廣泛採用的L2 loss不同，SSIM和人類的視覺系統（HVS）類似，對局部結構變化的感知敏感。
SSIM分爲三個部分：照明度、對比度、結構，分別如下公式所示：
將上面三個式子彙總到一起就是SSIM：
其中，上式各符號分別爲圖像x和y的均值、方差和它們的協方差，顯而易見，不贅述。
更詳細的說明可以參考維基百科[2].
Pytorch實現
SSIM值越大代表圖像越相似，當兩幅圖像完全相同時，SSIM=1。所以作爲損失函數時，應該要取負號，例如採用 loss = 1 - SSIM 的形式。由於PyTorch實現了自動求導機制，因此我們只需要實現SSIM loss的前向計算部分即可，不用考慮求導。（具體的求導過程可以參考文獻[3]）
以下是代碼實現，來源於github [4].
import torch
import torch.nn.functional as F
from math import exp
import numpy as np
# 計算一維的高斯分佈向量
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()
# 創建高斯核，通過兩個一維高斯分佈向量進行矩陣乘法得到
# 可以設定channel參數拓展爲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
# 計算SSIM
# 直接使用SSIM的公式，但是在計算均值時，不是直接求像素平均值，而是採用歸一化的高斯核卷積來代替。
# 在計算方差和協方差時用到了公式Var(X)=E[X^2]-E[X]^2, cov(X,Y)=E[XY]-E[X]E[Y].
# 正如前面提到的，上面求期望的操作採用高斯核卷積代替。
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)
參考來源
[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.