Convolution function in Python

Convolution is a mathematical function that combines two functions to produce another function. This resulting function captures the relevant characteristics of the input functions. It plays a crucial role in extracting the essential features of an image for a recognition task. We use a kernel in deep learning while applying convolution to an image. It acts as a filter that slides over the image to extract the relevant information. For example, the filter may be designed to blur the image, detect the edges, or it helps to highlight certain features.

Mathematical intuition

Mathematically, the convolution of an image with a kernel is represented as:

$O(x, y) = ∑∑ I(a, b) * K(x-a, y-b)$

Here:

$I$ represent the image.
$K$ is the kernel.
$O$ is the output resulting in a convolving image with the filer?
$∑∑$ makes sure that operation is performed over all possible positions of the kernel within the image.
$a$ and $b$ are the dimensions of the kernel.
$(x-a, y-b)$ is the relative positions of the kernel concerning the output pixel $(x, y)$ .

At each position, the coefficients of the kernel are multiplied with the corresponding pixel values from the input image, and then products are summed up to obtain the output value.

Implementation

So far, we have understood how the convolution function works. Now, let's implement a convolution in Python from scratch. As you know, an image is represented as an array, as shown in the code below.

import numpy as np
def Multiply(filter,I,row,col):
    #function for filter and I window multiplication
    result=0
    for i in range(3):
        for c in range(3):
            result=result+filter[i][c]*I[i+row][col+c]
    return result
def Convolution(fitler,I):
    Result = np.random.rand(3,4)
    for i in range(3):
        for c in range(4):
            Result[i][c]=Multiply(filter,I,i,c)
    return Result
if __name__=="__main__":
  filter=np.array([[-2,0,1],
              [-4,0,4],
              [-2,0,2]])
  I=np.array([[8,8,8,10,10,10],[8,8,8,10,10,10],[8,8,8,10,10,10],[9,9,9,9,9,9],[9,9,9,9,9,9]])
  result=Convolution(filter, I)
  print("The result of the convolution is : ")
  print(result)

Code explanation

Here is the explanation of the above code.

Lines 3–10: Multiply(filter, I, row, col) calculates the element-wise multiplication between the filter and a window of the input image I at a specific row and col. It iterates over the input image's filter and corresponding window, multiplies the values, and sums them up to obtain the result.
Lines 12–17: Convolution(filter, I) initializes the Result array to store the convolved output. It then iterates over the positions in the Result array and calls the Multiply function to calculate the convolution at each position.

Applications of a convolution function

In image processing and computer vision, a wide range and notable applications of a convolution function are discussed below.

It is used for filtering and smoothing operations. With the help of convolution, we can add or remove noise from the image.
Convolution is widely used for edge detection and feature extraction in computer vision models.
We use the convolution function to recognize and simulate the audio in speech recognition.
Convolution analyzes the relationship between signals and estimates the similarity or alignment between two-time series.

Conclusion

Convolution is an essential operation in the field of deep learning. It helps automate tasks by recognizing and extracting useful information from an image. In convolution, we can achieve edge detection, smoothness, and sharpening of an image by applying different kernels. In this Answer, we have implemented the convolution function in Python to gain insights and understand the concept better.

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