# Gaussian blur fourier transform

In electronics and signal processinga Gaussian filter is a filter whose impulse response is a Gaussian function or an approximation to it, since a true Gaussian response is physically unrealizable. Gaussian filters have the properties of having no overshoot to a step function input while minimizing the rise and fall time. This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay. It is considered the ideal time domain filter, just as the sinc is the ideal frequency domain filter. Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.

These equations can also be expressed with the standard deviation as parameter. However, since it decays rapidly, it is often reasonable to truncate the filter window and implement the filter directly for narrow windows, in effect by using a simple rectangular window function.

In other cases, the truncation may introduce significant errors. Better results can be achieved by instead using a different window function ; see scale space implementation for details. Filtering involves convolution. The filter function is said to be the kernel of an integral transform. The Gaussian kernel is continuous. Most commonly, the discrete equivalent is the sampled Gaussian kernel that is produced by sampling points from the continuous Gaussian.

An alternate method is to use the discrete Gaussian kernel  which has superior characteristics for some purposes. Unlike the sampled Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete diffusion equation. Since the Fourier transform of the Gaussian function yields a Gaussian function, the signal preferably after being divided into overlapping windowed blocks can be transformed with a Fast Fourier transformmultiplied with a Gaussian function and transformed back.

This is the standard procedure of applying an arbitrary finite impulse response filter, with the only difference that the Fourier transform of the filter window is explicitly known. Due to the central limit theoremthe Gaussian can be approximated by several runs of a very simple filter such as the moving average. The simple moving average corresponds to convolution with the constant B-spline a rectangular pulseand, for example, four iterations of a moving average yields a cubic B-spline as filter window which approximates the Gaussian quite well.

Borrowing the terms from statistics, the standard deviation of a filter can be interpreted as a measure of its size. The cut-off frequency of a Gaussian filter might be defined by the standard deviation in the frequency domain yielding. The response value of the Gaussian filter at this cut-off frequency equals exp However, it is more common to define the cut-off frequency as the half power point: where the filter response is reduced to 0.

Butterworth filter. These values are quite close to 1. Note that standard deviations do not sum up, but variances do. It remains to be seen where the advantage is over using a gaussian rather than a poor approximation. When applied in two dimensions, this formula produces a Gaussian surface that has a maximum at the origin, whose contours are concentric circles with the origin as center.

A two dimensional convolution matrix is precomputed from the formula and convolved with two dimensional data. Each element in the resultant matrix new value is set to a weighted average of that elements neighborhood.

The focal element receives the heaviest weight having the highest Gaussian value and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as brightness or a color intensity, and the overall effect is called Gaussian blur.

The Gaussian filter is non-causal which means the filter window is symmetric about the origin in the time-domain. This makes the Gaussian filter physically unrealizable. This is usually of no consequence for applications where the filter bandwidth is much larger than the signal.

In real-time systems, a delay is incurred because incoming samples need to fill the filter window before the filter can be applied to the signal. While no amount of delay can make a theoretical Gaussian filter causal because the Gaussian function is non-zero everywherethe Gaussian function converges to zero so rapidly that a causal approximation can achieve any required tolerance with a modest delay, even to the accuracy of floating point representation.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. This is to ensure that spurious high-frequency information does not appear in the downsampled image aliasing. Gaussian blurs have nice properties, such as having no sharp edges, and thus do not introduce ringing into the filtered image.

Gaussian blur is a low-pass filterattenuating high frequency signals. Its amplitude Bode plot the log scale in the frequency domain is a parabola. In other words, how much does it reduce the standard deviation of pixel values in the picture? The center element at [4, 4] has the largest value, decreasing symmetrically as distance from the center increases. The element 0. A Gaussian blur effect is typically generated by convolving an image with an FIR kernel of Gaussian values. In the first pass, a one-dimensional kernel is used to blur the image in only the horizontal or vertical direction.

In the second pass, the same one-dimensional kernel is used to blur in the remaining direction. The resulting effect is the same as convolving with a two-dimensional kernel in a single pass, but requires fewer calculations.

Discretization is typically achieved by sampling the Gaussian filter kernel at discrete points, normally at positions corresponding to the midpoints of each pixel. This reduces the computational cost but, for very small filter kernels, point sampling the Gaussian function with very few samples leads to a large error. In these cases, accuracy is maintained at a slight computational cost by integration of the Gaussian function over each pixel's area. This will cause a darkening or brightening of the image.Specifically, it is the function F defined by. Instead of F x one also writes W [ f ] x. Note that F x need not exist for every real number xwhen the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation or, equivalently, the diffusion equation with constant diffusion coefficient. By using values of t different from 1, we can define the generalized Weierstrass transform of f.

The generalized Weierstrass transform provides a means to approximate a given integrable function f arbitrarily well with analytic functions.

Weierstrass used this transform in his original proof of the Weierstrass approximation theorem. The generalization W t mentioned below is known in signal analysis as a Gaussian filter and in image processing when implemented on R 2 as a Gaussian blur. As mentioned above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial is a polynomial of the same degree, and in fact same leading coefficient the asymptotic growth is unchanged.

This can be shown by exploiting the fact that the generating function for the Hermite polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform. The function e ax is thus an eigenfunction of the Weierstrass transform. This is, in fact, more generally true for all convolution transforms. In particular, by choosing a negative, it is evident that the Weierstrass transform of a Gaussian function is again a Gaussian function, but a "wider" one.

The Weierstrass transform assigns to each function f a new function F ; this assignment is linear. Both of these facts are more generally true for any integral transform defined via convolution.

This is the formal statement of the "smoothness" of F mentioned above. If f is integrable over the whole real axis i. This expresses the physical fact that the total thermal energy or heat is conserved by the heat equation, or that the total amount of diffusing material is conserved by the diffusion equation. There is a formula relating the Weierstrass transform W and the two-sided Laplace transform L. If we define. In terms of signal analysisthis suggests that if the signal f contains the frequency b i.

This has the consequence that higher frequencies are reduced more than lower ones, and the Weierstrass transform thus acts as a low-pass filter. This can also be shown with the continuous Fourier transformas follows.

The Fourier transform analyzes a signal in terms of its frequencies, transforms convolutions into products, and transforms Gaussians into Gaussians. The Weierstrass transform is convolution with a Gaussian and is therefore multiplication of the Fourier transformed signal with a Gaussian, followed by application of the inverse Fourier transform.Sign in to comment.

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Desiree on 12 Apr at Vote 0. Edited: Desiree on 12 Apr at I need to sample m points uniformly at random from the Fourier transform of the Gaussian Kernel. Thanks in advance! Answers 0. See Also. Tags fourier transform gaussian kernel samples. Opportunities for recent engineering grads. Apply Today. An Error Occurred Unable to complete the action because of changes made to the page.There are two reasons for this.

First, it is mathematically advanced and second, the resulting images, which do not resemble the original image, are hard to interpret. Nevertheless, utilizing Fourier Transforms can provide new ways to do familiar processing such as enhancing brightness and contrast, blurring, sharpening and noise removal. But it can also provide new capabilities that one cannot do in the normal image domain. These include deconvolution also known as deblurring of typical camera distortions such as motion blur and lens defocus and image matching using normalized cross correlation.

It is the goal of this page to try to explain the background and simplified mathematics of the Fourier Transform and to give examples of the processing that one can do by using the Fourier Transform.

## Blurring for Beginners

My thanks to Sean Burke for his coding of the original demo and to ImageMagick's creator for integrating it into ImageMagick. Both were heroic efforts. It is recommened that you compile a personal HDRI version if you want to make the most of these techniques.

The Fourier Transform An image normally consists of an array of 'pixels' each of which are defined by a set of values: red, green, blue and sometimes transparency as well.

But for our purposes here we will ignore transparency. Thus each of the red, green and blue 'channels' contain a set of 'intensity' or 'grayscale' values. This is known as a raster image ' in the spatial domain '. This is just a fancy way of saying, the image is defined by the 'intensity values' it has at each 'location' or 'position in space'.

Image Processing with Fourier Transform

But an image can also be represented in another way, known as the image's ' frequency domain '. In this domain, each image channel is represented in terms of sinusoidal waves.

## How to Blur an Image with a Fourier Transform in Matlab – Part I [Revised*]

In such a ' frequency domain ', each channel has 'amplitude' values that are stored in locations based not on X,Y 'spatial' coordinates, but on X,Y 'frequencies'. Since this is a digital representation, the frequencies are multiples of a 'smallest' or unit frequency and the pixel coordinates represent the indices or integer multiples of this unit frequency.

This follows from the principal that "any well-behaved function can be represented by a superposition combination or sum of sinusoidal waves". In other words, the 'frequency domain' representation is just another way to store and reproduce the 'spatial domain' image. But how can an image be represented as a 'wave'? To prevent any chance of distortions resulting from saving FFT images, It is best not to save them to disk at all, but hold them in memory while you process the image.

This format can also save multiple images in the one file. However it can only store one image per file. The " TIFF " file format can also be used though is not as acceptable for web browsers, though it does allow multiple images per file. Magnitude Only. Phase Only. Alternatively you can use the following small shell script, to calculate a log scaling factor to use for the specific magnitude image.

However remember that you can not use a spectrum image, for the inverse " -ift " transform as it will produce an overly bright image. For example Real Only. Imaginary Only.

### Gaussian filter

To see this single pixel more clearly lets also magnify that area of the image For example lets replace that 'dark pink' DC pixel with some other color such as the more orange color 'tomato' The unusual creation of the gradient image in the above is necessary to ensure that the resulting sine wave image tiles perfectly across the image.

Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing.

It only takes a minute to sign up. Let's say you have an original image and a version of the same image that may have been convoluted with a Gaussian blur. How could you demonstrate that the Gaussian blur has been applied and calculate the blur radius? If you have the blurred and unblurred images, what I think you are asking is how you can recover the point spread function PSF. In the absence of noise, this is theoretically possible by considering the operation in the frequency domain.

The blurred was introduced by a convolution of the unblurred image with a Gaussian kernel. In Fourier space, convolution is applied by multiplication of the image with the kernel, so if you have two of the operands the source and the resultit should be possible to recover the blur kernel by division of the respective Fourier transforms.

This is effectively what some deconvolution operations attempt to do when the PSF is unknown or is a mixture of defocus and motion blur e. Where G is the Fourier transform of the blurred image, H is the blurring function, F is the source image and N is additive noise. Assuming the noise is negligible unrealistic in most real world situationsthe blur kernel can be recovered by calculating. From this you could measure the radius.

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 am not quite sure that I understand how Gaussian blur works.

So one of my questions is if I have understood it correctly? For 2d I take RGB of each pixel in the image and apply the filter to it by multiplying RGB of the pixel and the surrounding pixels with the associated filter position. These are then summed to be the new pixel RGB values. For 1d I apply the filter first horizontally and then vetically, which should give the same result if I understand things correctly. Is this result exactly the same result as when the 2d filter is applied?

Another question I have is about how the algorithm can be optimized. I have read that the Fast Fourier Transform is applicable to Gaussian blur.

But I can't figure out how to relate it. Can someone give me a hint in the right direction? Yes, the 2D Gaussian kernel is separable so you can just apply it as two 1D kernels. Note that you can't apply these operations "in place" however - you need at least one temporary buffer to store the result of the first 1D pass. FFT-based convolution is a useful optimisation when you have large kernels - this applies to any kind of filter, not just Gaussian. Just how big "large" is depends on your architecture, but you probably don't want to worry about using an FFT-based approach for anything smaller than, say, a 49x49 kernel.

The general approach is:. Note that if you're applying the same filter to more than one image then you only need to FFT the padded kernel once.

You still have at least two FFTs to perform per image though one forward and one inversewhich is why this technique only becomes a computational win for large-ish kernels. Learn more.