SSIM是一種衡量兩幅圖片相似度的指標。
出處來自於2004年的一篇TIP，
標題為：Image Quality Assessment: From Error Visibility to Structural Similarity
地址為：https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1284395

與PSNR一樣，SSIM也經常用作影象質量的評價。

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先了解SSIM的輸入
SSIM的輸入就是兩張影象，我們要得到其相似性的兩張影象。其中一張是未經壓縮的無失真影象(即ground truth)，另一張就是你恢復出的影象。所以，SSIM可以作為super-resolution質量的指標。
假設我們輸入的兩張影象分別是x和y，那麼
$S$

S I M ( x , y ) = [ l ( x , y ) ] α [ c ( x , y ) ] β [ s ( x , y ) ] γ − − − ( 1 ) SSIM(x,y)=[l(x,y)]^\alpha[c(x,y)]^\beta[s(x,y)]^\gamma ---(1)
$\alpha>0$, $\beta>0$,and $\gamma>0$.
式1是SSIM的數學定義，其中：
$l(x,y)=\frac{2\mu_x\mu_y+c_1}{\mu_x^2+\mu_y^2+c_1},$
$c(x,y)=\frac{\sigma_{xy}+c_2}{\sigma_x^2+\sigma_y^2+c_2},$
$s(x,y)=\frac{\sigma_{xy}+c_3}{\sigma_x\sigma_y+c_3}$
其中l(x, y)是亮度比較，c(x,y)是對比度比較，s(x,y)是結構比較。 $\mu_x$和 $\mu_y$分別代表x,y的平均值， $\sigma_x$和 $\sigma_y$分別代表x,y的標準差。 $\sigma_{xy}$代表x和y的協方差。而 $c_1$， $c_2$， $c_3$分別為常數，避免分母為0帶來的系統錯誤。
在實際工程計算中，我們一般設定 $\alpha=\beta=\gamma=1$，以及 $c_3=c_2/2$，可以將SSIM簡化為下：

$SSIM(x, y)= \frac{(2\mu_x\mu_y+c_1)(\sigma_{xy}+c_2)}{(\mu_x^2+\mu_y^2+c_1)(\sigma_x^2+\sigma_y^2+c_2)}$
總結：

1. SSIM具有對稱性，即SSIM(x,y)=SSIM(y,x)
2. SSIM是一個0到1之間的數，越大表示輸出影象和無失真影象的差距越小，即影象質量越好。當兩幅影象一模一樣時，SSIM=1；

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如PSNR一樣，SSIM這種常用計算函式也被tensorflow收編了，我們只需在tf中呼叫ssim就可以了：

tf.image.ssim(x, y, 255)

原始碼如下：

def ssim(img1, img2, max_val):
  """Computes SSIM index between img1 and img2.

  This function is based on the standard SSIM implementation from:
  Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). Image
  quality assessment: from error visibility to structural similarity. IEEE
  transactions on image processing.

  Note: The true SSIM is only defined on grayscale.  This function does not
  perform any colorspace transform.  (If input is already YUV, then it will
  compute YUV SSIM average.)

  Details:
    - 11x11 Gaussian filter of width 1.5 is used.
    - k1 = 0.01, k2 = 0.03 as in the original paper.

  The image sizes must be at least 11x11 because of the filter size.

  Example:
  # Read images from file.
      im1 = tf.decode_png('path/to/im1.png')
      im2 = tf.decode_png('path/to/im2.png')
      # Compute SSIM over tf.uint8 Tensors.
      ssim1 = tf.image.ssim(im1, im2, max_val=255)

      # Compute SSIM over tf.float32 Tensors.
      im1 = tf.image.convert_image_dtype(im1, tf.float32)
      im2 = tf.image.convert_image_dtype(im2, tf.float32)
      ssim2 = tf.image.ssim(im1, im2, max_val=1.0)
      # ssim1 and ssim2 both have type tf.float32 and are almost equal.
    img1: First image batch.
    img2: Second image batch.
    max_val: The dynamic range of the images (i.e., the difference between the
      maximum the and minimum allowed values).

  Returns:
    A tensor containing an SSIM value for each image in batch.  Returned SSIM
    values are in range (-1, 1], when pixel values are non-negative. Returns
    a tensor with shape: broadcast(img1.shape[:-3], img2.shape[:-3]).
  """
   _, _, checks = _verify_compatible_image_shapes(img1, img2)
  with ops.control_dependencies(checks):
    img1 = array_ops.identity(img1)

  # Need to convert the images to float32.  Scale max_val accordingly so that
  # SSIM is computed correctly.
  max_val = math_ops.cast(max_val, img1.dtype)
  max_val = convert_image_dtype(max_val, dtypes.float32)
  img1 = convert_image_dtype(img1, dtypes.float32)
  img2 = convert_image_dtype(img2, dtypes.float32)
  ssim_per_channel, _ = _ssim_per_channel(img1, img2, max_val)
  # Compute average over color channels.
  return math_ops.reduce_mean(ssim_per_channel, [-1])

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參考：https://en.wikipedia.org/wiki/Structural_similarity