Convolution vs Cross-correlation

[fog37]

TL;DR

understand the difference between convolution and cross-correlation results

Hello,
Convolution is essentially superposition. Conceptually, a copy of the same mask/filter is essentially placed at every point in the signal (1D, 2D, ect.). Once all these convolution masks are in place, we just compute the sum and get the convolved signal. The integral formula for convolution, which implies rotating by 180 degrees the mask, is less intuitive.

Let's now talk about cross-correlation. In this case, we don't place the mask at EVERY point in the input signal but simply slide the mask around and calculate the inner product with the overlapped area and move on to the next input signal area....The correlation map looks therefore very different from a convolution operation. The integral formula for cross- correlation seems very similar (the mask is not rotated) to the convolution integral but the concept is quite different.

Concetually, convolution places the mask everywhere in the input signal, does the inner product, and sum everything up. Cross-correlation simply perform the inner product area by area...

What actually occurs in deep learning and CNN (convolutional neural networks) is really cross-correlation and not convolution, correct?

 

[Philip Koeck]

TL;DR Summary: understand the difference between convolution and cross-correlation results

Hello,
Convolution is essentially superposition. Conceptually, a copy of the same mask/filter is essentially placed at every point in the signal (1D, 2D, ect.). Once all these convolution masks are in place, we just compute the sum and get the convolved signal. The integral formula for convolution, which implies rotating by 180 degrees the mask, is less intuitive.

Let's now talk about cross-correlation. In this case, we don't place the mask at EVERY point in the input signal but simply slide the mask around and calculate the inner product with the overlapped area and move on to the next input signal area....The correlation map looks therefore very different from a convolution operation. The integral formula for cross- correlation seems very similar (the mask is not rotated) to the convolution integral but the concept is quite different.

Concetually, convolution places the mask everywhere in the input signal, does the inner product, and sum everything up. Cross-correlation simply perform the inner product area by area...

What actually occurs in deep learning and CNN (convolutional neural networks) is really cross-correlation and not convolution, correct?

I don't know what CNN does, but in image processing CC is used to compare two functions in n dimensions: How similar are they and in what position to each other do they match best?

Convolution is used to produce a weighted sum of copies of the one function placed in positions (and with the respective weights) given by the other function. Obvioulsy this is quite messy unless the first function is limited in extent and the second function is a sum of delta distributions sufficiently far apart to avoid extensive overlap between the copies.
The minus in the argument makes sure the copies of the first function are correctly oriented.

This doesn't sound right to me or at least I don't understand it:

Concetually, convolution places the mask everywhere in the input signal, does the inner product, and sum everything up. Cross-correlation simply perform the inner product area by area...

 

[Philip Koeck]

TL;DR Summary: understand the difference between convolution and cross-correlation results

Hello,
Convolution is essentially superposition. Conceptually, a copy of the same mask/filter is essentially placed at every point in the signal (1D, 2D, ect.). Once all these convolution masks are in place, we just compute the sum and get the convolved signal. The integral formula for convolution, which implies rotating by 180 degrees the mask, is less intuitive.

Let's now talk about cross-correlation. In this case, we don't place the mask at EVERY point in the input signal but simply slide the mask around and calculate the inner product with the overlapped area and move on to the next input signal area....The correlation map looks therefore very different from a convolution operation. The integral formula for cross- correlation seems very similar (the mask is not rotated) to the convolution integral but the concept is quite different.

Concetually, convolution places the mask everywhere in the input signal, does the inner product, and sum everything up. Cross-correlation simply perform the inner product area by area...

What actually occurs in deep learning and CNN (convolutional neural networks) is really cross-correlation and not convolution, correct?

One more thing I should point out since you talk about masks and filters.
The latter are usually symmetrical. In that case it makes no difference whether you convolute or cross-correlate with them.

 

[fog37]

In both convolution and correlation, the filter and the mask are generally smaller in size than the input signal.
True, if the filter is symmetric, convolution and correlation will give the same result.

But correlation is about finding similarity while convolution is more about processing of the input signal by convolution filter (to blur, enhance, etc.). Convolution is when an input passes through a system. The system's output is determined by the convolution filter, also known as the impulse response of the system/filter....

I was not completely correct in the way I described the mechanics of correlation. Even in the case of correlation, the correlation mask is translated around and its entries multiply the input signal. The only difference between correlation and convolution is really the 180 flip of the mask...