This content originally appeared on DEV Community and was authored by vivekvohra
CONVOLUTION
CONVOLUTION is a mathematical operator “*”.Graphically, it expresses how the 'shape' of one function is modified by the other.
It is applicable only to Linear Time Invariant Systems.
Time Invariant: systems where a shift in input results in an identical shift in output.eg:
Linear:In simple terms it is a graph between output and input which must be a straight line through the origin without having saturation or dead time.
For discrete systems, the formula is :
https://youtu.be/QmcoPYUfbJ8?si=2CCf64_x9fGl7jx5
In this video, we understand convolution using a great analogy. Suppose we want to find how much smoke burning several matchsticks produce. Here we take a time frame of 5 minutes. Let's define 2 functions:
Smoke function: S(t) - It describes the amount of smoke produced by a single matchstick across time. It might be an exponential decay graph.
Firework function: F(t)- It describes the number of matchsticks lit per minute.
Let it be a linear function. y = x
- t=0 1 match was lit
- t=2 2 matches were lit: total smoke = smoke from 2 sticks at present = 2*S(0) (because for matchsticks at this time they just started burning) + smoke of prev. 1 stick after 1 sec. of burning = 1*S(1).
| at minute 0 | 1*S(0) |
|---|---|
| at minute 1 | 2*S(0)+ 1*S(1) |
| at minute 2 | 3*S(0) + prev(t + 1) |
Thus we can conclude that it indeed follows the above equation.
We can visualize convolution as a sliding window:
Now if are 1st function is very large as compared to our 2nd function
Our visualization will be somewhat like this:
In 2-D version(Image processing)
or using Alternate illustration :
In the 2-D function, we can solve this Matrix convolution at (2,2) by flipping and multiplying.
i.e. = (i*1)+(h*2)+(g*3)+(f*4)+ (e*5) + (d*6) + (c*7) + (b*8) + (a*9)
In Image Processing, the 2nd function is known as Kernel which is in this case 3x3 matrix
[1/9, 1/9, 1/9],
[1/9, 1/9, 1/9],
[1/9, 1/9, 1/9]
We could have solved our Blur function by taking our 2nd function such that it sum of all nodes is 1 and then doing the convolution. (Remember while multiplying we have to flip the Matrix!!).
This content originally appeared on DEV Community and was authored by vivekvohra
vivekvohra | Sciencx (2025-03-07T20:14:09+00:00) Convolution. Retrieved from https://www.scien.cx/2025/03/07/convolution/
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