


import numpy as np
from PIL import Image
def        __lowerCamelCase		(				UpperCAmelCase_		:	np.array		):




      """simple docstring"""





      a							,       a							,       a		:Optional[Any]											= rgb[:, :, 0], rgb[:, :, 1], rgb[:, :, 2]
      return 0.2989 * r + 0.5870 * g + 0.1140 * b
def        __lowerCamelCase		(				UpperCAmelCase_		:	np.array		):




      """simple docstring"""





      return (gray > 127) & (gray <= 255)
def        __lowerCamelCase		(				UpperCAmelCase_		:	np.array				,						UpperCAmelCase_		:	np.array		):




      """simple docstring"""





      a		:Optional[int]											= np.zeros_like(UpperCAmelCase_		)
      a		:List[Any]											= np.zeros(
          (image.shape[0] + kernel.shape[0] - 1, image.shape[1] + kernel.shape[1] - 1)		)

      # Copy image to padded image
      a		:str											= image

      # Iterate over image & apply kernel
      for x in range(image.shape[1]		):
            for y in range(image.shape[0]		):
                  a		:Optional[Any]											= (
                      kernel * image_padded[y : y + kernel.shape[0], x : x + kernel.shape[1]]
                  ).sum()
                  a		:Optional[Any]											= int(summation == 5		)
      return output


# kernel to be applied
snake_case       :					Dict        =  np.array([[0, 1, 0], [1, 1, 1], [0, 1, 0]])

if __name__ == "__main__":
     # read original image
     snake_case       :					List[Any]        =  np.array(Image.open(R'''..\image_data\lena.jpg'''))
     # Apply erosion operation to a binary image
     snake_case       :					Dict        =  erosion(grayabinary(rgbagray(image)), structuring_element)
     # Save the output image
     snake_case       :					Any        =  Image.fromarray(output).convert('''RGB''')
     pil_img.save('''result_erosion.png''')









import numpy as np
def        __lowerCamelCase		(				UpperCAmelCase_		:	Optional[Any]		):




      """simple docstring"""





      a							,       a		:Union[str, Any]											= a.shape
      a		:Any											= min(UpperCAmelCase_				,						UpperCAmelCase_		)
      a		:Tuple											= np.eye(UpperCAmelCase_		)
      a		:Any											= a.copy()

      for k in range(t - 1		):
            # select a column of modified matrix A':
            a		:List[str]											= r[k:, [k]]
            # construct first basis vector
            a		:List[str]											= np.zeros_like(UpperCAmelCase_		)
            a		:List[Any]											= 1.0
            # determine scaling factor
            a		:Union[str, Any]											= np.linalg.norm(UpperCAmelCase_		)
            # construct vector v for Householder reflection
            a		:List[str]											= x + np.sign(x[0]		) * alpha * ea
            v /= np.linalg.norm(UpperCAmelCase_		)

            # construct the Householder matrix
            a		:Union[str, Any]											= np.eye(m - k		) - 2.0 * v @ v.T
            # pad with ones and zeros as necessary
            a		:Tuple											= np.block([[np.eye(UpperCAmelCase_		), np.zeros((k, m - k)		)], [np.zeros((m - k, k)		), q_k]]		)

            a		:List[Any]											= q @ q_k.T
            a		:int											= q_k @ r

      return q, r


if __name__ == "__main__":
     import doctest

     doctest.testmod()





