19 votes
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Why should an image be blurred using a Gaussian Kernel before downsampling?

An image "should not be blurred using a Gaussian Kernel" in general. This however can be a safe bet for a lot of basic image processing needs, and a smoothing is almost mandatory when you ...
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14 votes

Why is Gaussian noise called so?

Noise is random, but like most random phenomena, it follows a certain pattern. Different patterns are given different names. Consider rolling a die. This is clearly random. Roll the die 1000 times, ...
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  • 13.9k
14 votes

Capacity of AWGN channel

Assuming a channel whose input at each time is a continuous random variable $X$ and its output is $Y=X+Z$, where $Z\sim\mathcal{N}(0,N)$ and $Z$ is independent of $X$, then $$C_{\text{CI-AWGN}}=\frac{...
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  • 4,115
11 votes

Why is random noise assumed to be normally distributed?

Starting at an even more basic level than the other (much smarter) answers, I'd like to pick up on this part of the question: This seems contradictory to me as on one side it is random then on the ...
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  • 211
9 votes
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Downsampling and Gaussian Filtering in the Context of Scale Space Pyramids

You're correct, it has to do with the Cut Off frequency of the Gaussian Blur Filter in its Frequency Domain. In order to see it, just apply a DFT (Using MATLAB it can be achieved by ...
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  • 41.4k
9 votes
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What is a $BT$ (Bandwidth-Time) product with reference to modulation?

The $BT$ product is the bandwidth-symbol time product where $B$ is the $-3\textrm{ dB}$(half-power) bandwidth of the pulse/filter and $T$ is the symbol duration. For different applications you will ...
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  • 3,282
9 votes
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Why does the separable filter reduce the cost of computing the operator?

Assume you have a $N\times M$ sized image. If you know take what is classically used, a square filter kernel, of let's say size $L\times L$, you'd need to convolve that with the picture – which gives ...
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9 votes

Why should an image be blurred using a Gaussian Kernel before downsampling?

According to (digital) sampling theorem, signals should be properly bandlimited, before they are (down) sampled. A practical digital filter approximately limits the bandwidth of the signal and makes ...
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9 votes
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Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?

Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself? Have a look at the Fourier Transfrom of a Gaussian Signal. $$ \mathcal{F}_{x} \left\{ {e}^{-a {x}^{2} } \right\} \left( \omega \...
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8 votes

How to Calculate Gaussian Kernel for a Small Support Size?

Gaussian Kernel is made by using the Normal Distribution for weighing the surrounding pixel in the process of Convolution. Since we're dealing with discrete signals and we are limited to finite length ...
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8 votes
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Gaussian Filter Close to Image Border

Pixels outside the image borders must be extrapolated. Now, you need to chose the model of your extrapolation. For instance, if you're working within the Discrete Fourier model a periodic ...
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  • 41.4k
8 votes
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Correct way to add AWGN to a signal

Short answer 10*log(bw/fs) to take into account the oversampling operation because the awgn() function specifies the signal-to-...
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  • 5,780
8 votes

Why is random noise assumed to be normally distributed?

normal distribution (i like to call it "gaussian") remains normal after addition of normally distributed numbers. so if gaussian goes into an LTI filter, a gaussian distribution comes out. but because ...
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8 votes
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How to approximate gaussian kernel for image blur

The continuous Gaussian, whatever its dimension (1D, 2D), is a very important function in signal and image processing. As most data is discrete, and filtering can be costly, it has been and still is, ...
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7 votes

Gaussian Blur - Standard Deviation, Radius and Kernel Size

It turns out that the rows of Pascal's Triangle approximate a Gaussian quite nicely and have the practical advantage of having integer values whose sum is a power of 2 (we can store these values ...
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  • 193
7 votes
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Multi Scale Shape and Detail Enhancement from Multi Light Image Collections

What the authors meant is to create a matrix of Weights, $ {U}^{\left( i, j \right)} $. It is a matrix of the size of the image. The given calculation is by the exponent of two terms (Each of them is ...
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  • 41.4k
7 votes

Capacity of AWGN channel

The capacity formula $$C = 0.5 \log (1+\frac{S}{N}) \tag{1}$$ is for discrete time channel. Assuming you have a sequence of data $\left\lbrace a_n \right\rbrace$ to send out, you need an orthonormal ...
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  • 5,780
7 votes

Why is random noise assumed to be normally distributed?

I'll try to clear one possible source of confusion. If picking each sample value from a single distribution feels "not random enough", then let's try to make things "more random" by adding another ...
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6 votes
Accepted

Simulating Range Bearing Sensor with MATLAB with Gaussian Noise (Generating Gaussian Colored Random Vector)

Few notes: First you should multiply the noise by the Standard Deviation (Root of the Variance for zero mean noise). You can do that by multiplying the Lower Cholesky Decomposition of matrix by a ...
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  • 41.4k
6 votes
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Why Does the Odd Multiple of $ \frac{\pi}{4} $ on Gaussian Cause Loss in Repeatability Under Image Rotations?

The answer boils down to 2 issues with the practical approximations of the Gaussian Kernel: Though the Gaussian Kernel is radially symmetric its discrete approximation has a rectangle support. ...
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6 votes
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Decorrelating Stationary Colored Gaussian Noise -- Effect On The Desired Signal

Yes, the "Whitening" is basically filtering the signal using an LTI System. The Result would be the signal $ \mathbf{s} $ filtered by the system which whitens the noise. In the framework of Matched ...
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  • 41.4k
6 votes

Additive White Gaussian Noise (AWGN) and Undecimated DWT

One property of Orthogonal Transformations is that White Noise stays White (Uncorrelated) under Orthogonal Transformations (One could say it's a property of White Noise). Usually this property ...
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  • 41.4k
6 votes

Parameters of Gaussian Kernel in the Context of Image Convolution

When dealing with Gaussian Blur in the Image Processing context the following holds: The Standard Deviation, $ \sigma $, is sometimes called radius. I think this ...
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6 votes
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How Is Laplacian of Gaussian (LoG) Implemented as Four 1D Convolutions?

I think Chris Luengo's answer is perfect. The trick is that you can calculate the 2nd derivative of the image (Using Finite Differences -> Convolution) and then blur it with Gaussian Filter. Since ...
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  • 41.4k
6 votes
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The Effect of the Standard Deviation ($ \sigma $) of a Gaussian Kernel when Smoothing a Gradients Image

Let's analyze it in 1D as the intuition is the same. First, let's have a look on a few different Gaussian Kernels: As expected, they are wider as the Standard Deviation (STD) increase. It means that ...
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  • 41.4k
6 votes
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Gaussian filter: The best parameters for an implementation

The effect is in sidelobe level and is evaluated with consideration to number of samples per symbol used, waveform quality and the spectral mask. Increasing the length of the filter in total number of ...
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  • 37.7k
5 votes

Is the discrete Gaussian kernel an eigenfunction of the DFT?

This answer complements @CedronDawg's answer which introduced this family of eigenvectors. More specifically, this answer presents three algorithms and a hybrid algorithm for generating for a given ...
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