📘

Morphological Opening and Closing

MediumMorphological Operations 📚 Computer 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.

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:

The given image is a binary image with a single bright pixel (1) surrounded by dark pixels (0), and the structuring element se is a 3x3 matrix.
To compute the opening, we first apply erosion to the image using se, which removes the single bright pixel since it's smaller than the structuring element, resulting in an all-dark image.
To compute the closing, we first apply dilation to the image using se, which expands the single bright pixel, but since it's still surrounded by dark pixels, the dilated image will have the bright pixel and some neighboring pixels turned bright. Then, we apply erosion to this dilated image, which removes the newly added bright pixels, leaving only the original single bright pixel.
The final output is a dictionary with the opening as an all-dark image and the closing as the original image, since the small bright pixel is preserved after the closing operation: {'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]]}

Constraints:

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

More from Computer Vision 2

1D Bilateral FilterHard Image Sharpening with Unsharp MaskMedium Gaussian BlurMedium Gaussian Kernel GenerationEasy 2D Image ConvolutionEasy

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.

👑

Continue the full explanation

You're reading the free preview. Unlock the complete walkthrough, the code editor, test runner and reference solution with Premium.

🔒

Editor locked

The code editor is locked for Pro problems. It is only available for free problems. Please upgrade to gain access to the code editor for all problems.

solution.py

Python 3.13

Test Results

0/0

Run code to see test results.