Convolution

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!!).