SSIM(结构相似性)-数学公式及python实现

SSIM是一种衡量两幅图片相似度的指标。
出处来自于2004年的一篇TIP，
标题为：Image Quality Assessment: From Error Visibility to Structural Similarity
地址为：https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1284395

与PSNR一样，SSIM也经常用作图像质量的评价。

先了解SSIM的输入
SSIM的输入就是两张图像，我们要得到其相似性的两张图像。其中一张是未经压缩的无失真图像(即ground truth)，另一张就是你恢复出的图像。所以，SSIM可以作为super-resolution质量的指标。
假设我们输入的两张图像分别是x和y，那么
$SSIM(x,y)=[l(x,y)]^\alpha[c(x,y)]^\beta[s(x,y)]^\gamma ---(1)$
$\alpha>0$ , $\beta>0$ ,and $\gamma>0$ .
式1是SSIM的数学定义，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)是亮度比较，c(x,y)是对比度比较，s(x,y)是结构比较。 $\mu_x$ 和 $\mu_y$ 分别代表x,y的平均值， $\sigma_x$ 和 $\sigma_y$ 分别代表x,y的标准差。 $\sigma_{xy}$ 代表x和y的协方差。而 $c_1$ ， $c_2$ ， $c_3$ 分别为常数，避免分母为0带来的系统错误。
在实际工程计算中，我们一般设定 $\alpha=\beta=\gamma=1$ ，以及 $c_3=c_2/2$ ，可以将SSIM简化为下：

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

SSIM具有对称性，即SSIM(x,y)=SSIM(y,x)
SSIM是一个0到1之间的数，越大表示输出图像和无失真图像的差距越小，即图像质量越好。当两幅图像一模一样时，SSIM=1；

如PSNR一样，SSIM这种常用计算函数也被tensorflow收编了，我们只需在tf中调用ssim就可以了：

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

源代码如下：

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])

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

SSIM(结构相似性)-数学公式及python实现

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