In this class you will learn the basic principles and tools used to process images and videos, and how to apply them in solving practical problems of commercial and scientific interests. Digital images and videos are everywhere these days – in thousands of scientific (e.g., astronomical, bio-medical), consumer, industrial, and artistic applications. Moreover they come in a wide range of the electromagnetic spectrum - from visible light and infrared to gamma rays and beyond. The ability to process image and video signals is therefore an incredibly important skill to master for engineering/science students, software developers, and practicing scientists. Digital image and video processing continues to enable the multimedia technology revolution we are experiencing today. Some important examples of image and video processing include the removal of degradations images suffer during acquisition (e.g., removing blur from a picture of a fast moving car), and the compression and transmission of images and videos (if you watch videos online, or share photos via a social media website, you use this everyday!), for economical storage and efficient transmission. This course will cover the fundamentals of image and video processing. We will provide a mathematical framework to describe and analyze images and videos as two- and three-dimensional signals in the spatial, spatio-temporal, and frequency domains. In this class not only will you learn the theory behind fundamental processing tasks including image/video enhancement, recovery, and compression - but you will also learn how to perform these key processing tasks in practice using state-of-the-art techniques and tools. We will introduce and use a wide variety of such tools – from optimization toolboxes to statistical techniques. Emphasis on the special role sparsity plays in modern image and video processing will also be given. In all cases, example images and videos pertaining to specific application domains will be utilized.
Wiener Noise Smoothing Filter
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Do curso por Northwestern University
Fundamentals of Digital Image and Video Processing
906 classificações
Na lição
Image Recovery : Part 2
In this module we look at the problem of image and video recovery from a stochastic perspective. Topics include: Wiener restoration filter, Wiener noise smoothing filter, maximum likelihood and maximum a posteriori estimation, and Bayesian restoration algorithms.
Conheça os instrutores
Aggelos K. KatsaggelosJoseph Cummings Professor
Department of Electrical Engineering and Computer Science
Hello, in this segment we consider the noise smoothing problem.
That is the case when the impulses points of
the variance system is equal to the dimensional delta function.
In this case if we said the frequency response of the varying
filter equal to one and the expression of the winner restoration filter.
The Wiener noise smoothing filter results.
This does not become an enhancement problem
however, since the Wiener smoothing filter was derived
based on a modelling of the degradation
and the optimization of a specific objective function.
Therefore, under these conditions, it is an optimal noise smoothing filter.
We
also derive a specially adaptive version of the noise smoothing filter.
Each pixel now in the image is processed differently, based
on the special activity of the neighborhood it belongs to.
So, let's proceed with a material of this segment.
I can use the Wiener filter to perform noise smoothing.
So, in this case the impulse response of the degradation system is a delta.
A delta convolved with f will give me f, i, j here, so
therefore the degradation model now is sigma plus noise.
I observe an image, which equals the originally
merge plus the noise that was added to that.
So under the same assumptions as before, that is f and w are wide and stationary.
F and w are correlated and they are both zero mean.
I can obtain now the form of the Wiener noise smoothing filter.
Since here, h i j is a delta function, we know from
what we have learned before that the frequency
response of the filter is equal to one.
So substituting this in the expression we had for the Wiener restoration filter
we obtain just the frequency response of the noise smoothing Wiener filter.
So this exactly the point that was
made earlier on when we talked about enhancement.
We have again the signal plus noise
problem here, and when we talked about enhancement,
we'd argue that a low past type of filter would remove some of the noise.
And then we would try different shapes of the
filter and different extends of for a given shape,
and visually evaluate the results and pick the one
that would just be more suitable for our purposes.
Instead here we have a specific objective which is the
minimization of the mean squared error, mean squared estimation error.
And this objective gives rise to this
expression for the Wiener noise smoothing filter.
Having a dual use the Weiner noise smoothing filter.
Here's another interpretation of the Weiner restoration filter.
We show again here the degradation model.
We call g i j, the image at the output of the LSI degradation system,
noise is then added to it, and we observe y which therefore is equal to g plus w.
So y is the signal plus noise now model where the
signal g is not the original signal but instead the blend signal.
Now, we know that the power spectrum of g equals to this.
This is one of the first expressions we showed there, standard expressions
when we have random process at the input of an LSI system.
Okay, now with that, here is the frequency response of the
Wiener restoration filter that we saw a couple of times already.
Clearly I can multiply this by one, so I multiple by H omega1
omega2 divided by H omega1 omega2.
This is identity, well the exception of well H is zero,
the frequency for which is zero, but we'll assume that's perfectly fine.
And now I can combine this H complex conjugate with this H.
And the product will give me the magnitude squared of H omega1, omega2.
So then what I obtained I see here.
Here that H magnitude squared times P of ff is P of gg, so that's the numerator
and I can substitute in the denominator, so I have this expression now.
Right?
Again, I combined this H complex conjugate with this H, and
with this P of ff and I got P of gg.
And then this down here is also P of gg, omega1 omega2.
So we see that I can think of a
Wiener restoration filter as the product of two filters.
The first filter is just smoothing the noise.
Not with respect to the original signal but with respect to the blurred signal.
So first, we take care of getting rid of the noise.
And then, the noise free signal is inverse the convult here.
I apply an inverse filter for the deconvolution.
So this is an interpretation that justifies again first getting rid of the
noise and then getting rid of the blur,
but in a very specific way within the context of the Wiener restoration filter.
Actually people introduced all kinds of modifications to this concept here, which
of course, here is mathematically equivalent
to doing the Wiener restoration filter.
But I can alter the noise smoothing and the inverse
filter and end up with combinations of of other filters.
Let us look now at the derivation
of a spatially adaptive Wiener noise smoothing filter.
This is the degradation model, it's signal plus noise.
The observed image y equals the original image f
plus the noise w that has been added to it.
We assume that the noise is zero mean and that
it is y, therefore it's spectral density equals it's variance.
So, as we have seen for this model, the
frequency response of the Wiener filter is given by
this expression, where I have already substituted the spectral
density of the noise equal to sigma w squared.
We are going to develop a filter that has this step shown here.
So, the observed image y, from it the mean m of f will be subtracted, then the
filter will develop the Wiener filter with impulse response r i j will be applied.
The mid will be added back to the image and
therefore the smoothed image is denoted by p i j.
Maybe, motivated by the exact result we show that got into which the
Weiner restoration filter is the concatenation of
Weiner smoothing filter and then inverse filter.
One can find in the literature a number of
results where two restoration filters are combined in different ways.
One such result is shown here.
It's referred to as the geometric mean filter, and we see that is the combination
of the generalized inverse filter and the Weiner restoration filter.
So alpha equals, a half is the geometric mean of
the two but alpha in general is between 0 into 1.
For alpha equals 0 we have the Wiener filter for alpha equal 1 if we
have the inverse filter, and of course
any combination for the various values of alpha.
In addition if you notice here there's a gamma parameter.
Weighting this stabilizing term in the denominator of the Weiner filter.
So this now starts feeling more like an enhancement problem, since these
two parameters in this particular case, alpha and gamma need to be
specified by the user based on the visual quality of the restored image.
And again this is just one example of the
many such examples one could find in the literature.
We want to modify the Wiener noise smoothing filter so
that it adapts to the local characteristics of the image.
Towards this task we divide the image into stationary regions.
And then for each region, we assume that the following model holds for the image.
So, f i j is equal to the local mean
plus the standard deviation
multiplied by a noise term, and for this noise term, we assume that it's mean is
equal to zero while its variance is equal to one.
Empirical evidence suggests that it is a reasonable model
for typical images and typical process, image processing applications.
Now for each of these regions, I can write
down the frequency response of the noise smoothing filter.
P of f f based on this model is equal to sigma f squared.
So, we see that the filter here
is constant, does not depend on omega1, omega2.
So, if I take this back to the time domain, the
impulse response of the Weiner noise smoothing filter is equal to this.
So, the constant gave rise to a delta in the special domain.
Let's look now at the model we've been using towards this noise smoothing filter.
So we take the input image, subtract the mean, process it
through the filter, add back the mean to get the output.
So the equation therefore is that P of i j equals i minus the mean convolved
with the impulse response of the noise smoothing filter and adding back the mean.
So if I substitute here for r i j equal to
we have already found over here since I convolved with the delta.
It means that the signal goes through and therefore, this is what I obtain.
So, this the output of the noise smoothing filter for a specific stationary region.
Now, if I shrink the region to the pixel.
Then the local variance becomes a function of i
j and so does the local mean, and this
gives rise to the special adaptive Wiener noise smoothing
filter which you will see in the next slide.
So based on the analysis and the result we obtained in the previous slide.
If each region, the image was divided into is shrunk down to a single pixel, then the
local variance of the image becomes a function
of i j, and so does the local mean.
After rearranging terms, we find that the output of the noise
smoothing filter at location i j is a convex combination of the input
at the same location and the local mean of the image.
So, this is the expression of
the specially adaptive Wiener noise smoothing filter.
It's a different filter in essence that is
applied to each and every pixel in the image.
So, let's look at some special cases of this filter.
So first of all, if there is no noise added to the data, then
the output image, it's simply equal to the input image, which makes perfect sense.
The input image is noise free.
Therefore, there is absolutely no need to do any filtering to the image.
If I look at the edge ridges of the image, at those locations, then the
local variance of the image is much greater than the variance of the noise.
And if I take this into account and look at this equation.
The first term here is approximately equal
to one while the second coefficient is approximately
equal to zero, and therefore the output of
the filter is approximately equal to the input.
And this also makes perfect sense based on
the arguments I made at an earlier point that
based on properties of the human visual system, the
masking properties, noise is not visible at the edges.
Therefore, noise goes through the filter at the edge location.
This is not visible.
And this also allows us to leave the edges unfiltered.
So therefore we obtain sharp edges, and the noise
is not visible, so this is a desirable result.
Now if we look at the flat regions, then the local
variance of the image is much smaller than the noise variance.
And then if I look at the coefficient, the first one is approximately equal to zero.
While the second one approximately equal to one,
and therefore the output is a smoothed version of
the input and the output that, that pixel location
is equal to the local mean of the input.
I want to comment now on computing some of these parameters here.
Actually I, I called m of f the local mean of the input, and this is
correct because the noise is zero mean, because
of the modelling assumptions early on, and therefore this approximately holds true.
Now, as far as finding the variance of the noise,
if I first of all the, the model I've been using is that y equals f plus w.
I omitted the i j here.
And to do the fact that signal and noise aren't correlated.
I do have that sigma y squared equals sigma f squared plus sigma w squared.
So if I look at the flat region of the image then
at that region sigma of f is.
Where it's approximately equal to zero and therefore, what
I measure if I loc, measure the local variance.
It's approximately equal to the variance of the noise.
So, this is one way to obtain an estimate of the variance
of the noise and then if I look at any other region.
I can obtain the variance of the image as the
variance of the data minus the variance of the noise.
So since I know the variance of the noise I have already calculate it.
I measure this I can find this way the local variance of the image.
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