29.01.2021

# Gaussian blur sigma

By Akizahn

If you code Computer Graphics stuff, or if you work in any field of science, then you are necessarily familiar with the Gaussian function a. Normal distributionGaussian point-spread function…. The Gaussian function is commonly used as a convolution kernel in Digital Image Processing to blur an image. Computing a convolution is generally very slow, so choosing a convolution kernel that is as small as possible is always desirable.

Changing the mean shifts the function to the left or right, and changing the standard deviation stretches or compresses the bell-shaped curve, but always leaving its surface integral in [-INF. Gaussian 1D. Wikipedia has a fantastic explanation of this in the Percentile page. In 2D, the Gaussian function can be described as the product of two perpendicular 1D Gaussians, and due to radial symmetry, the same principle applies:.

Gaussian 2D. The difference between using an infinite or a size-limited Gaussian kernel is negligible to the naked eye. As you can see, the difference can only be slightly noticed around areas of very high power. The base image used for this example intentionally features a very high dynamic range of approx. Infinite vs. In 2D, the Gaussian function can be described as the product of two perpendicular 1D Gaussians, and due to radial symmetry, the same principle applies: Gaussian 2D This knowledge is very valuable when building a Gaussian-based blur convolution kernel.You will find many algorithms using it before actually processing the image.

The size of the kernel and the standard deviation. Create a vector of equally spaced number using the size argument passed.

We will see the function definition later. In order to set the sigma automatically, we will use following equation: This will work for our purpose, where filter size is between :. As you are seeing the sigma value was automatically set, which worked nicely.

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This simple trick will save you time to find the sigma for different settings. Here is the dorm function. Just calculated the density using the formula of Univariate Normal Distribution. We will create the convolution function in a generic way so that we can use it for other operations.

This is not the most efficient way of writing a convolution function, you can always replace with one provided by a library. However the main objective is to perform all the basic operations from scratch.

I am not going to go detail on the Convolution or Cross-Correlation operation, since there are many fantastic tutorials available already. Here we will only focus on the implementation. The function has the image and kernel as the required parameters and we will also pass average as the 3rd argument. The average argument will be used only for smoothing filter. Since our convolution function only works on image with single channel, we will convert the image to gray scale in case we find the image has 3 channels Color Image.

Then plot the gray scale image using matplotlib. We want the output image to have the same dimension as the input image. In order to do so we need to pad the image.

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Here we will use zero paddingwe will talk about other types of padding later in the tutorial. In the the last two lines, we are basically creating an empty numpy 2D array and then copying the image to the proper location so that we can have the padding applied in the final output.

In the below image we have applied a padding of 7, hence you can see the black border. This will be done only if the value of average is set True. This is because we have used zero padding and the color of zero is black. You can implement two different strategies in order to avoid this. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment. This site uses Akismet to reduce spam.

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When applying a Gaussian blur to an image, typically the sigma is a parameter examples include Matlab and ImageJ. How does one know what sigma should be? Is there a mathematical way to figure out an optimal sigma? In my case, i have some objects in images that are bright compared to the background, and I need to find them computationally.

I am going to apply a Gaussian filter to make the center of these objects even brighter, which hopefully facilitates finding them. How can I determine the optimal sigma for this? There's no formula to determine it for you; the optimal sigma will depend on image factors - primarily the resolution of the image and the size of your objects in it in pixels. Also, note that Gaussian filters aren't actually meant to brighten anything; you might want to look into contrast maximization techniques - sounds like something as simple as histogram stretching could work well for you.

Since you're working with images, bigger sigma also forces you to use a larger kernel matrix to capture enough of the function's energy. For your specific case, you want your kernel to be big enough to cover most of the object so that it's blurred enoughbut not so large that it starts overlapping multiple neighboring objects at a time - so actually, object separation is also a factor along with size.

Since you mentioned MATLAB - you can take a look at various gaussian kernels with different parameters using the fspecial 'gaussian', hsize, sigma function, where hsize is the size of the kernel and sigma is, well, sigma. Try varying the parameters to see how it changes.

Gaussian Blur - Image Processing Algorithm

Optimal sigma for Gaussian filtering of an image? Ask Question. Asked 9 years, 9 months ago. Active 8 years, 2 months ago. Viewed 29k times. Dima By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information.

I just want to know the role of sigma or variance at Gaussian smoothing. I mean, what happens by increasing the value of sigma for the same window size. It would be really helpful if somebody provide some nice literature about it. I already tried few but couldn't find what i am looking for. By increasing sigmawe are allowing some higher frequencies I think it should be done in the following steps, first from the signal processing point of view:.

Now putting all together! When we apply a Gaussian filter to an image, we are doing a low pass filtering. But as you know this happen in the discrete domain image pixels.

So we have to quantize our Gaussian filter in order to make a Gaussian kernel. In the quantization step, as the Gaussian filter GF has a small sigma it has the steepest pick. So the more weights will be focused in the center and the less around it. In the sense of natural image statistics!

The scientists in this field of studies showed that our vision system is a kind of Gaussian filter in the responses to the images. Now see a specific point in that seen. This is the Sigma appear here.

Viewed 10k times.In image processinga Gaussian blur also known as Gaussian smoothing is the result of blurring an image by a Gaussian function named after mathematician and scientist Carl Friedrich Gauss. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. Gaussian smoothing is also used as a pre-processing stage in computer vision algorithms in order to enhance image structures at different scales—see scale space representation and scale space implementation.

Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform.

By contrast, convolving by a circle i. Since the Fourier transform of a Gaussian is another Gaussian, applying a Gaussian blur has the effect of reducing the image's high-frequency components; a Gaussian blur is thus a low pass filter. The Gaussian blur is a type of image-blurring filter that uses a Gaussian function which also expresses the normal distribution in statistics for calculating the transformation to apply to each pixel in the image. The formula of a Gaussian function in one dimension is. When applied in two dimensions, this formula produces a surface whose contours are concentric circles with a Gaussian distribution from the center point. Values from this distribution are used to build a convolution matrix which is applied to the original image. This convolution process is illustrated visually in the figure on the right.

Each pixel's new value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight having the highest Gaussian value and neighboring pixels receive smaller weights as their distance to the original pixel increases. This results in a blur that preserves boundaries and edges better than other, more uniform blurring filters; see also scale space implementation.

In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. Thus contributions from pixels outside that range can be ignored. In addition to being circularly symmetric, the Gaussian blur can be applied to a two-dimensional image as two independent one-dimensional calculations, and so is termed separable filter. That is, the effect of applying the two-dimensional matrix can also be achieved by applying a series of single-dimensional Gaussian matrices in the horizontal direction, then repeating the process in the vertical direction.

Applying successive Gaussian blurs to an image has the same effect as applying a single, larger Gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied.

Because of this relationship, processing time cannot be saved by simulating a Gaussian blur with successive, smaller blurs — the time required will be at least as great as performing the single large blur. Gaussian blurring is commonly used when reducing the size of an image. When downsampling an image, it is common to apply a low-pass filter to the image prior to resampling.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. From Wikipedia Gaussian Blur it said that. Applying multiple, successive Gaussian blurs to an image has the same effect as applying a single, larger Gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied.

It can be shown using some simple convolution theory. However, there is an easier way using some simple probability theory. Recall that the sum of two independent random variables gives a random variable with a density equal to the convolution of the two randomc variables being summed. The multivariate generalization is also true. See also: . Sign up to join this community. The best answers are voted up and rise to the top. 