Morphological Opening and Closing

mediumPROMorphological OperationsComputer Vision 2

Implement a function to compute morphological opening and closing on a binary image using a given structuring element. Morphological operations are essential in Computer Vision for image processing and analysis, as they allow for the removal of noise and the extraction of relevant features. The opening operation, defined as $Dilate(Erode(image, se), se)$ , removes small bright noise, while the closing operation, defined as $Erode(Dilate(image, se), se)$ , fills small dark holes.

The structuring element is centred on each pixel (its middle cell is the anchor). Erosion sets an output pixel to 1 only if every image pixel lying under a 1 of se is 1; dilation sets it to 1 if at least one image pixel lying under a 1 of se is 1. Pixels outside the image count as 0 for both operations, so erosion turns off any pixel whose neighbourhood reaches past the border. The output grids have the same size as image.

Apply erosion to the image using the structuring element to shrink the image.
Apply dilation to the eroded image using the structuring element to expand it, resulting in the opening operation.
Apply dilation to the original image using the structuring element to expand it.
Apply erosion to the dilated image using the structuring element to shrink it, resulting in the closing operation.

Dilate(Erode(image, se), se) = Opening \\ Erode(Dilate(image, se), se) = Closing

This technique is widely used in image denoising and object detection applications.

Example

Input

image = [[0,0,0,0],[0,1,0,0],[0,0,0,0],[0,0,0,0]]
se = [[0,1,0],[1,1,1],[0,1,0]]

Output

{'opening': [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], 'closing': [[0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]}

Reasoning

image is a $4 \times 4$ binary grid with one bright pixel at row 1, column 1 (from 0). se is a $3 \times 3$ cross: the centre and its four direct neighbours.
Opening, erosion first: a pixel stays 1 only if all 5 cross positions centred on it are bright. The lone pixel at $(1,1)$ has dark neighbours, so erosion leaves all 0, and dilating an empty grid keeps the opening all 0.
Closing, dilation first: a pixel turns on if its cross touches a bright pixel. That lights $(1,1)$ and its four neighbours $(0,1)$ , $(1,0)$ , $(1,2)$ , $(2,1)$ , a plus of 5 ones.
Eroding the plus: at $(1,1)$ all five cross cells are bright, so it stays 1. The cross at $(0,1)$ reaches row $-1$ , outside the image, which counts as 0; the other arms also touch a dark or outside pixel. Only $(1,1)$ remains.
Notice the opening erased the speck while the closing returned the original image. Because outside pixels count as 0, closing can also shrink bright regions that touch the border.

Constraints

image is a 2D binary list
se is a 2D binary structuring element with odd dimensions
Return dict with 'opening' and 'closing' keys

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Background Knowledge

Morphological operations are a set of image processing techniques used to analyze and manipulate the shape and structure of objects in an image. These operations are based on the concept of a structuring element (SE), which is a small binary image used to probe the input image. The two primary morphological operations are erosion and dilation. Erosion is the process of shrinking or thinning an object in an image by removing pixels from its boundary, while dilation is the process of expanding or thickening an object by adding pixels to its boundary. The equation for erosion can be represented as: $A \ominus B = \{z | (B)_{z} \subseteq A\}$ , where $A$ is the input image, $B$ is the structuring element, and $(B)_{z}$ is the translation of $B$ by $z$ . Similarly, the equation for dilation is: $A \oplus B = \{z | (B^{c})_{z} \cap A^{c} = \emptyset\}$ , where $B^{c}$ is the complement of $B$ and $A^{c}$ is the complement of $A$ .

The concepts of opening and closing are built upon these basic operations. Opening is defined as the dilation of the eroded image, i.e., $A \circ B = (A \ominus B) \oplus B$ . It is used to remove small bright noise or objects from an image. On the other hand, closing is defined as the erosion of the dilated image, i.e., $A \bullet B = (A \oplus B) \ominus B$ . It is used to fill small dark holes or gaps in an image. Understanding these definitions is crucial to solving the problem.

In the context of this problem, the given binary image and structuring element will be used to compute both opening and closing. The resulting images will be returned as a dictionary with keys 'opening' and 'closing'. It's essential to note that zero-padding will be used for out-of-bounds pixels, meaning that pixels outside the image boundaries will be considered as zeros.

Algorithm/Approach

The general approach to solve this problem involves applying the definitions of opening and closing directly. This means that for opening, we first need to erode the input image using the given structuring element and then dilate the result. For closing, we first need to dilate the input image using the structuring element and then erode the result. The key steps involve understanding how to apply erosion and dilation operations using the structuring element and then combining these operations to achieve opening and closing.

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