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In this lecture, we are going to continue our discussion of convolution by looking at a totally different

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perspective on how it works.

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This is very helpful for understanding convolution, but it doesn't teach you anything new mechanically.

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So this lecture is optional if you want to move on and just learn about how to make scenes.

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Something I mentioned very often is victimization.

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We like to victimize operations in code because using numpy functions is a lot more efficient than writing

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your own python for loops.

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The pattern you're looking for when you want to vector arise in operation is usually of the form of

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a dot product.

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A dot product is an element y's multiplication and then the summation of those results.

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So whenever you see the sum over ei of I times by that's a dot products.

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This also applies to matrix multiplication, since if A and B are matrices, then the matrix multiplication

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of A and B is just a dot b.

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You'll notice that for convolution we yet again have something similar.

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The only difference is that there are two summations.

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However, this is not really a relevant detail since the outcome is still the same.

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Instead of summing over one axis or something over two axes.

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But it's still an element.

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Why some in add?

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The question is why is the DOT product important?

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One definition of the DOT product other than the element why some in ad is that it's the magnitude of

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aid multiplied by the magnitude of B multiplied by the cosine of the angle between A and B.

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We sometimes call this the cosine similarity or cosine distance, depending on the sign that you use.

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So how does this work?

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Geometrically.

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Consider just the angle for a moment.

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If the angle between the two vectors is zero, then the cosine of that angle is one.

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That's the maximum value of the cosine.

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Now imagine that the angle between two vectors is 90 degrees.

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Then the cosine of that angle is zero.

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Finally imagine that the angle between a two vectors is 180 degrees.

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This is basically as far apart as possible.

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Then the cosine of that angle is minus one, which is the minimum value of the cosine.

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So if you're just using raw cosine, then it's a similarity.

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The larger the number, the closer the two vectors are.

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The smaller the number, the further away they are.

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The maximum value when A and B are parallel is one.

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The minimum value when A and B are anti parallel is minus one.

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And if A and B are orthogonal, then the cosine similarity is just zero.

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Okay.

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So why is that important?

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Consider now how you would find that the cosine of the angle between two vectors that's just a be divided

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by the magnitude of a and the magnitude of B.

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So I just rearrange the equation that we had before.

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Now let's compare this to another popular measurement, the Pearson correlation.

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The Pearson correlation is defined as what you see here.

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But notice how similar this is to cosine similarity.

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The only difference is that the Pearson correlation uses mean subtraction.

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And so now you have two hints that convolution is really correlation.

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The first one from before was that we were actually doing what is called the cross correlation.

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And second, now you have that the DOT product is actually very similar to the Pearson correlation.

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So I hope that by this point you are convinced that the DOD product, while it seems like an abstract

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concept, can be thought of as a correlation measure.

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It tells me how correlated is the first thing with the second thing.

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If they are highly, positively correlated, then the DOT products should be large and positive.

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That means two vectors pointing in nearly the same direction.

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If they are highly negatively correlated, then the DOT product should be large and negative.

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That means the two vectors are pointing in nearly opposite direction.

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Finally, if the two vectors are orthogonal or at right angles, then the DOT product should be zero.

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The reason why this is important is you don't have to think of a filter as an abstract concept.

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In fact, it's just a pattern finder.

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This actually makes the term filter make a lot more sense.

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It filters out everything not related to the pattern contained in the filter by setting them to zero

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and keeps everything that is related to the pattern.

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So what convolution is doing is it's passing this filter along each point on the original input image

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and sliding it along.

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At each point it asks Is the pattern, here is the pattern, here is the pattern here and so forth.

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Then it gives us a high number in the positions where the pattern is found and then a small number where

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the pattern is not found.

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And thus, this is your first alternative perspective on convolution.

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It's just a sliding pattern finder that passes through an entire image looking for a particular pattern.