# Convolution vs Cross-correlation

• fog37
fog37
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?

Philip Koeck
Computer science news on Phys.org
fog37 said:
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...

fog37
fog37 said:
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...

Philip Koeck

## 1. What is the difference between convolution and cross-correlation?

Convolution and cross-correlation are both mathematical operations that involve two functions. The main difference between them is that in convolution, one of the functions is flipped before the operation is performed, while in cross-correlation, no flipping occurs.

## 2. When should I use convolution vs cross-correlation?

Convolution is commonly used in signal processing for tasks like filtering and feature extraction, while cross-correlation is often used in pattern recognition and image processing for tasks like template matching and alignment.

## 3. Are convolution and cross-correlation commutative?

No, convolution and cross-correlation are not commutative operations. This means that the order in which the functions are inputted into the operation matters, and switching the order will result in different outcomes.

## 4. Which operation is more computationally efficient: convolution or cross-correlation?

Typically, cross-correlation is more computationally efficient than convolution because it does not involve the additional step of flipping one of the functions. This can make a significant difference in performance when dealing with large datasets or real-time processing.

## 5. Can convolution and cross-correlation be used interchangeably?

While convolution and cross-correlation are closely related operations, they are not interchangeable. The differences in how they are computed and the outcomes they produce make them suitable for different applications. It is important to understand these differences and choose the appropriate operation for the task at hand.

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