old_contents: def sat(x):
      """Summed area table / integral image.
  
      The integral image contains the sum of all elements above and to the
      left of it, i.e.:
  
      .. math::
  
         S[m, n] = \sum_{i \leq m} \sum_{j \leq n} X[i, j]
  
      Parameters
      ----------
      X : ndarray of uint
          Input image.
  
      Returns
      -------
      S : ndarray
          Summed area table.
  
      References
      ----------
      .. [1] F.C. Crow, "Summed-area tables for texture mapping,"
             ACM SIGGRAPH Computer Graphics, vol. 18, 1984, pp. 207-212.
  
      """
      return x.cumsum(1).cumsum(0)
  
  def sat_sum(sat, r0, c0, r1, c1):
      """Using a summed area table / integral image, calculate the sum
      over a given window.
  
      Parameters
      ----------
      sat : ndarray of uint64
          Summed area table / integral image.
      r0, c0 : int
          Top-left corner of block to be summed.
      r1, c1 : int
          Bottom-right corner of block to be summed.
  
      Returns
      -------
      S : int
          Sum over the given window.
  
      """
      S = 0
  
      S += sat[r1, c1]
  
      if (r0 - 1 >= 0) and (c0 - 1 >= 0):
          S += sat[r0 - 1, c0 - 1]
  
      if (r0 - 1 >= 0):
          S -= sat[r0 - 1, c1]
  
      if (c0 - 1 >= 0):
          S -= sat[r1, c0 - 1]
  
      return S
  
new_contents: def sat(x):
      """Summed area table / integral image.
  
      The integral image contains the sum of all elements above and to the
      left of it, i.e.:
  
      .. math::
  
         S[m, n] = \sum_{i \leq m} \sum_{j \leq n} X[i, j]
  
      Parameters
      ----------
      X : ndarray
          Input image.
  
      Returns
      -------
      S : ndarray
          Summed area table.
  
      References
      ----------
      .. [1] F.C. Crow, "Summed-area tables for texture mapping,"
             ACM SIGGRAPH Computer Graphics, vol. 18, 1984, pp. 207-212.
  
      """
      return x.cumsum(1).cumsum(0)
  
  def sat_sum(sat, r0, c0, r1, c1):
      """Use a summed area table / integral image to sum over a given window.
  
      Parameters
      ----------
      sat : ndarray
          Summed area table / integral image.
      r0, c0 : int
          Top-left corner of block to be summed.
      r1, c1 : int
          Bottom-right corner of block to be summed.
  
      Returns
      -------
      S : int
          Sum over the given window.
  
      """
      S = 0
  
      S += sat[r1, c1]
  
      if (r0 - 1 >= 0) and (c0 - 1 >= 0):
          S += sat[r0 - 1, c0 - 1]
  
      if (r0 - 1 >= 0):
          S -= sat[r0 - 1, c1]
  
      if (c0 - 1 >= 0):
          S -= sat[r1, c0 - 1]
  
      return S
  
subject: Update sat docs after feedback from Brian Holt.