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What's the relationship between sigma and radius? I've read that sigma is equivalent to radius, I don't see how sigma is expressed in pixels. Or is "radius" just a name for sigma, not related to pixels?
There are three things at play here. The variance, ($\sigma^2$), the radius, and the number of pixels. Since this is a 2-dimensional gaussian function, it ...
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You would generate bandlimited Gaussian noise by first generating white noise, then filtering it to the bandwidth that you desire. As an example:
% design FIR filter to filter noise to half of Nyquist rate
b = fir1(64, 0.5);
% generate Gaussian (normally-distributed) white noise
n = randn(1e4, 1);
% apply to filter to yield bandlimited noise
nb = filter(b,1,...
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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, keeping track of each result. Then, calculate the histogram of the result; you'll find that you got each of 1, 2, 3, 4, 5 and 6 approximately the same number of ...
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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 $$C_{\text{CI-AWGN}}=\frac{1}{2}\log_2\left(1+\frac{P}{N}\right)$$ is the capacity of the continuous-input channel under the power constraint
$$\mathsf{E}X^2\le P$$
The mutual information ...
13
An image "should not be blurred using a Gaussian Kernel" in general. This can be a safe bet for a lot of basic image processing needs, and a smoothing is almost mandatory when you want to control the information lost by the downsampling.
Blurring is (often, not always) another word for low-pass filtering. When an image contains high-frequency ...
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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 other side their distribution is considered normally distributed.
Perhaps the issue here is what ‘random’ means?
To be clear: ‘random’ and ‘normally-...
10
The parameter sigma is enough to define the Gaussian blur from a continuous point of view. In practice however, images and convolution kernels are discrete. How to choose an optimal discrete approximation of the continuous Gaussian kernel?
The discrete approximation will be closer to the continuous Gaussian kernel when using a larger radius. But this may ...
9
Just as a small addition to Jason's answer: usually you need to generate bandlimited noise with a given variance $\sigma^2$. You can add this code to the code given in Jason's answer:
var = 3.0; % just an example
scale = sqrt(var)/std(nb);
nb = scale*nb; % nb has variance 'var'
Note that you have to do the scaling after filtering, because in general ...
9
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 you $N\times M$ pixels, each needing $L^2$ multiply-accumulates. So you end up with $A_{2D}=L^2MN$ operations.
Now, if you can decompose that filter into an $...
8
Since the DFT is representable by multiplication with the Fourier matrix, your question is equivalent to asking what the eigenvectors of the Fourier matrix are.
Actually, Wikipedia provides the answer (http://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors).
However, since the eigenvalues ($1, -1, i, -i$) are not simple, the ...
8
The Deriche and van Vliet filters are heuristics. In both cases they choose the locations of poles and zeros to minimize either the RMS difference or the maximum difference of the filter's impulse response from a Gaussian.
Both filters are causal-anti-causal pairs. So I think they have no phase error or group delay, but you need to be able to run them ...
8
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 find varying recommended values. In GSM telephony for instance, a $BT=0.3$ is recommended. In satellite communications with GMSK, for near-earth missions the ...
8
According to (digital) sampling theorem, signals should be properly bandlimited, before they are (down) sampled.
A digital filter limits the bandwidth of the signal and makes it suitable for downsampling without aliasing.
A Gausssian kernel is very suitable as a lowpass filter, as it has a number of nice features. The Gaussian function is mathematically ...
8
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 of this central limit theorem, even if uniform p.d.f. random process goes into an LTI filter with a long and dense impulse response, what will come out tends ...
answered Nov 7 '18 at 2:26
robert bristow-johnson
15.7k3 gold badges27 silver badges65 bronze badges
8
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, subject of quantities of optimization and quantification/quantization schemes. In one 1D, the three most direct for a finite-length filter are illustrated below:...
7
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 exactly as integers, fixed point values, or floats). For example, say we wish to construct a 7x7 Gaussian Kernel we can do so using the 7th row of Pascal's triangle ...
7
For you questions:
1. After applying gaussian filter on a histogram, the pixel value of new histogram will be changed.
2. The sum of pixels in new histogram is almost impossible to remain unchanged.
Visually speaking, after your applying the gaussian filter (low pass), the histogram shall become more smooth than before. Thus, the new histogram is ...
7
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 waveform set $\left\lbrace \phi_n(t) \right\rbrace$ for modulation. In linear modulation, whom M-ary modution belongs to, $\phi_n(t) = \phi(t - nT)$ where $T$ ...
7
Short answer
10*log(bw/fs) to take into account the oversampling operation because the awgn() function specifies the signal-to-noise ratio per sample, in dB.
Longer answer
The discrete time AWGN model is
$$Y = X+N$$
where X is data from continuous time $X(t)$, N is noise sequence from AWGN process $N(t)$ and Y is receive symbols.
If $X(t)$ is ...
7
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 layer of randomness. This will be found to be futile.
Imagine that for each sample the noise is random in the sense that it comes from a distribution that is ...
6
Every time you generate discrete noise samples (Using MATLAB's randn / rand for instance) you actually generate a band limited noise.
All you need to do is the adjustment of the variance of the discrete samples to the variance of the "Continuous" noise those samples are allegedly taken from.
Given a continuous White Noise (Wide Sense) with variance $ \...
6
Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself?
Have a look at the Fourier Transfrom of a Gaussian Signal.
$$ \mathcal{F}_{x} \left\{ {e}^{-a {x}^{2} } \right\} \left( \omega \right) = \sqrt{\frac{\pi}{a}} {e}^{- {\pi}^{2} \frac{ {\omega}^{2} }{a} } $$
First, Gaussian Signal stays Gaussian under Fourier Transform.
As you can see, the ...
5
The filter you referenced is known as a Binomial filter. It is an approximation to a Gaussian, but for smaller filters it's a very crude one. The design is oriented more toward efficiency than accuracy. Rather than sampling a Gaussian directly, the idea behind the approximation is based on the Central Limit Theorem. In this case, it means that a small moving ...
5
I think you gave a nice summary of existing analytic solutions for discrete-time IIR filters. But I would also add Bessel filters to the list of filters approximating ideal frequency-selective filter characteristics. Its magnitude response does not show as sharp a transition as the other filter types of the same order, but this is the price you pay for an ...
5
In addition to Peter's answer, if you have a nonlinear system that is well-behaved in a sense of being only mildly nonlinear or at least exhibiting no discontinuities, special variants of the Kalman filter can still be applied.
Extended Kalman Filter
This filter linearizes the system at the current state of the system using a first order Taylor Series ...
5
Well, let's look at the two issues: 1) linearity and 2) Gaussianity.
Linearity
If you're imaging moving 3D objects (people) with a single camera, then you're working with a 2D projection of those 3D objects. That dimensionality reduction can cause non-linearities to appear.
Take a 2D to 1D example: an object moving in a circle in 2D. The object is ...
5
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 fft / fft2) and look on the absolute value.
Look for the -3dB point and you'll see.
There is also an intuitive explanation on the original article which say that blurring ...
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