<blockquote>
  <p>If I did two Gaussian blurs of size N, would that be the same mathematically as doing one Gaussian blur of size 2N?</p>
</blockquote>

<p>Almost. Applying two Gaussian blurs <em>is</em> equivalent to doing one Gaussian blur, but with a slightly different size calculation.</p>

<p>Applying a Gaussian blur to an image means doing a <a href="https://en.wikipedia.org/wiki/Convolution" rel="noreferrer">convolution</a> of the Gaussian with the image. Convolution is associative: Applying two Gaussian blurs to an image is equivalent to convolving the two Gaussians with each other, and convolving the resulting function with the image.</p>

<p>As it happens, the convolution of two Gaussians with each other is another Gaussian, whose variance is the sum of variances of the two original Gaussians. A Gaussian of N pixel width has variance <span class="math-container">$N^2$</span>; applying it twice is equivalent to a Gaussian with variance <span class="math-container">$2\cdot N^2$</span>, which corresponds to a width of <span class="math-container">$\sqrt{2}\cdot N$</span> pixels.</p>

<p>In the same vein, applying a Gaussian of width <span class="math-container">$2\cdot N$</span> is equivalent to applying a Gaussian of width <span class="math-container">$N$</span> four times.</p>

CC BY-SA 3.0
2015-11-03T17:12:34.340Z
N.B. this fact is also used when deriving theoretical bounds for [anisotropic diffusion](https://en.wikipedia.org/wiki/Anisotropic_diffusion).
geometrian
CC BY-SA 3.0
2015-08-15T20:08:25.447Z
Does this mean that applying a Gaussian blur 4 times would give twice the size, as required?
trichoplax is on Codidact now
CC BY-SA 3.0
2015-08-15T20:12:56.647Z
Yes, absolutely!
Benedikt Bitterli
CC BY-SA 3.0
2015-08-15T20:19:31.133Z
That covers the question entirely then - would it be worth mentioning how to get twice the size in the question so we can tidy away the comments?
trichoplax is on Codidact now
CC BY-SA 3.0
2015-08-16T01:01:10.063Z
If it takes 4 to double the size, what size would 3  represent?
Alan Wolfe
CC BY-SA 3.0
2015-08-16T01:08:55.603Z
I found the answer is N*sqrt(3). It turns out that the total Blur amount of multiple blurs is equal to the square root of the sum of the sizes squared. Wikipedia says that: https://en.wikipedia.org/wiki/Gaussian_blur
Alan Wolfe
CC BY-SA 4.0
2015-08-15T19:40:53.283Z
2020-02-25T08:43:59.617Z
2020-02-25T08:43:59.617Z
Benedikt Bitterli
<p>To make Gaussian blurring a 2d image faster, I know that you can do one axis and then the other.</p>

<p>I'm wondering though, if I did two Gaussian blurs of size $N$, would that be the same mathematically as doing one Gaussian blur of size $2N$?</p>

<real-time><post-processing><gaussian-blur>
Is doing multiple Gaussian blurs the same as doing one larger blur?
Alan Wolfe
# Is doing multiple Gaussian blurs the same as doing one larger blur?

(Asked by: username_0 on 2015-08-15)

To make Gaussian blurring a 2d image faster, I know that you can do one axis and then the other.

I'm wondering though, if I did two Gaussian blurs of size $N$, would that be the same mathematically as doing one Gaussian blur of size $2N$?

**Tags:** real-time, post-processing, gaussian-blur


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## 1 answers

### Answer by username_1 on 2015-08-15. Score: 16 | Accepted answer

> If I did two Gaussian blurs of size N, would that be the same mathematically as doing one Gaussian blur of size 2N?

Almost. Applying two Gaussian blurs *is* equivalent to doing one Gaussian blur, but with a slightly different size calculation.

Applying a Gaussian blur to an image means doing a [convolution](https://en.wikipedia.org/wiki/Convolution) of the Gaussian with the image. Convolution is associative: Applying two Gaussian blurs to an image is equivalent to convolving the two Gaussians with each other, and convolving the resulting function with the image.

As it happens, the convolution of two Gaussians with each other is another Gaussian, whose variance is the sum of variances of the two original Gaussians. A Gaussian of N pixel width has variance $N^2$; applying it twice is equivalent to a Gaussian with variance $2\cdot N^2$, which corresponds to a width of $\sqrt{2}\cdot N$ pixels.

In the same vein, applying a Gaussian of width $2\cdot N$ is equivalent to applying a Gaussian of width $N$ four times.

> **Comments:**
> **username_3**: Does this mean that applying a Gaussian blur 4 times would give twice the size, as required?  
> **username_1**: Yes, absolutely!  
> **username_3**: That covers the question entirely then - would it be worth mentioning how to get twice the size in the question so we can tidy away the comments?  
> **username_0**: If it takes 4 to double the size, what size would 3  represent?  
> **username_0**: I found the answer is N*sqrt(3). It turns out that the total Blur amount of multiple blurs is equal to the square root of the sum of the sizes squared. Wikipedia says that: https://en.wikipedia.org/wiki/Gaussian_blur  
> **username_2**: N.B. this fact is also used when deriving theoretical bounds for [anisotropic diffusion](https://en.wikipedia.org/wiki/Anisotropic_diffusion).  

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