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Dirac delta function

The Dirac delta function is a rather special function defined as follows.

δ (x) = \lim_{u \to 0} \frac{1}{| u | \sqrt{π}} e^{- \frac{x^{2}}{u^{2}}}

Or, less formally,

δ (x) = {\begin{matrix} \infty si x = 0 \\ 0 sinon \end{matrix}

It is the identity of the convolution operator.

Link with image processing

In image processing, the convolution operator is a powerful tool for applying filters to images.

An example of the use of this filtering is the creation of what is called a Gaussian blur (or Gaussian smoothing).

Intuitively, when we convolve an image with a Gaussian filter (whose values follow a Gaussian function of center the center of the filter), we obtain a new image whose each pixel is worth a weighted average of the pixels of the original image.

This makes the image appear blurred. By changing the parameters of the Gaussian function used, the smoothing of the image is more or less accentuated.

Let's imagine that the filter follows the delta function of Dirac ("normalized"). In this case we only average a single pixel, that is to say that the blur becomes null and the result of the convolution is simply the original image.

This is a way to understand that the delta function is the identity of the convolution operator.

Link with probability theory

Let $X_{n} \sim 𝒩 (μ, \frac{1}{n})$ . Denoting $f_{n}$ the density function of $X_{n}$ .

f_{n} (x) = \frac{\sqrt{n}}{\sqrt{2 π}} e^{- \frac{n {(x - μ)}^{2}}{2}}

Then,

f_{n} (x) \overset{n \to \infty}{\to} δ (x - μ)

This function fulfills the properties of a probability density function.

Using the central limit theorem, let $X_{1}, X_{2}, . . ., X_{n} \overset{iid}{\sim} 𝒩 (μ, 1)$ , and denoting their average by $X$ .

X \sim 𝒩 (μ, \frac{1}{n}), n \to \infty

Now, let $Y$ be a random variable with the same support as $X$ , and let $S = Y + X$ . We then have when $n$ goes to infinity:

f_{S} (s) = (f_{Y} * f_{X}) (s) = (f_{Y} * δ (\cdot - μ)) (s) = f_{Y} (s - μ)