Implement a morphological operation to apply erosion to a given binary image using a structuring element. The goal is to produce a new binary image where each pixel is 1 only if all corresponding pixels in the image covered by the structuring element are also 1.

The concept of erosion is fundamental in morphological operations, which are used to manipulate and analyze the shape and structure of objects in images. Erosion is a process that shrinks or reduces the size of objects in an image by removing pixels from the object boundaries. This is achieved by applying a structuring element, a small binary image, to the original image. The structuring element is used to probe the image, and for each pixel, it checks if all the pixels covered by the structuring element are 1.

To perform erosion, the following steps are taken:

Center the structuring element at each pixel in the image.
Check if all the pixels in the image covered by the structuring element are 1.
If all the pixels are 1, the corresponding pixel in the output image is set to 1; otherwise, it is set to 0.

f(x, y) = \min_{(s, t) \in S} g(x + s, y + t)

This technique is widely used in image processing and computer vision applications to remove noise and extract features from images.

The structuring element se is centered at the middle pixel, which is at position $(1, 1)$ , since the element has 3 rows and 3 columns.
To apply erosion, we slide the structuring element over the entire image, checking if all positions where se has a 1, the corresponding image pixels are also 1. If they are, the output pixel at that position is set to 1.
For the given image and structuring element, the only position where all overlapping pixels match is at the center of the image, resulting in a single 1 at the output position $(2, 2)$ , since the image is 0-padded out-of-bounds.
The resulting output after applying the erosion operation is [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]].

image is a 2D binary list (0 or 1)
se (structuring element) is a 2D binary list with odd dimensions
Return eroded binary image (same size as input)

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.

📘

Binary Image Erosion

EasyMorphological Operations

To perform erosion, the following steps are taken:

Center the structuring element at each pixel in the image.
Check if all the pixels in the image covered by the structuring element are 1.
If all the pixels are 1, the corresponding pixel in the output image is set to 1; otherwise, it is set to 0.

f(x, y) = \min_{(s, t) \in S} g(x + s, y + t)

This technique is widely used in image processing and computer vision applications to remove noise and extract features from images.

Example:

Input:

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

Output:

[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]

Reasoning:

The structuring element se is centered at the middle pixel, which is at position $(1, 1)$ , since the element has 3 rows and 3 columns.
To apply erosion, we slide the structuring element over the entire image, checking if all positions where se has a 1, the corresponding image pixels are also 1. If they are, the output pixel at that position is set to 1.
For the given image and structuring element, the only position where all overlapping pixels match is at the center of the image, resulting in a single 1 at the output position $(2, 2)$ , since the image is 0-padded out-of-bounds.
The resulting output after applying the erosion operation is [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]].

Constraints:

image is a 2D binary list (0 or 1)
se (structuring element) is a 2D binary list with odd dimensions
Return eroded binary image (same size as input)

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.