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Constrained Least Squares
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Fundamentals of Digital Image and Video Processing
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Image Recovery: Part 1
In this module we study the problem of image and video recovery. Topics include: introduction to image and video recovery, image restoration, matrix-vector notation for images, inverse filtering, constrained least squares (CLS), set-theoretic restoration approaches, iterative restoration algorithms, and spatially adaptive algorithms.
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Aggelos K. KatsaggelosJoseph Cummings Professor
Department of Electrical Engineering and Computer Science
This CLS filter and it's variants is probably
one of the most common utilized restoration filter.
It simply turns an unconstrained least squares
of optimization problem into a constrained one.
It restricts the solution space of the least squares approach.
By incorporating into the problem prior knowledge about the solution.
A meaningful solution, therefore, results this way.
The mechanics of finding the solution to
the constrained problem are not a challenge.
They present some standard optimization steps.
The real challenge is how to best represent this prior knowledge.
And the determination of the significance or the
weight of this prior knowledge towards the final solution.
This weight is typically referred to as
a Lagrange multiplier or a regularization parameter.
And controls the relative importance of the fidelity to the
data term, and the prior knowledge term towards the final solution.
We will explain all these aspects in detail in this segment.
A way to improve upon the solution we obtained through the least squares filter.
Or the English filter, that we just described
is to impose a constrained on the solution.
So as before, we minimize this error term.
With respect to f but subject to the constraints shown here.
C's an operator that needs to be chosen.
It's by and large a high pass filter or operator.
In which case this constraint tells us that the energy as imposed by the norm.
Of the restored image at high frequencies is below a threshold here.
Which means that we don't allow
arbitrary high frequency fluctuations in the solution.
Or put it differently, this imposes a smoothness requirement on the solution.
So I have this term that reserves the fidelity
to the data.
And this standard expresses prior knowledge.
[SOUND].
And so the way to solve the constrained problem is to
relax it, and turned it, turn it into an unconstrained problem.
Through the introduction of a parameter, alpha here.
Which is referred to as either a regularization
parameter or Lagrange
multiplier.
The regularization is the process
according to which you try to have the solution depend continuously on the data.
Or not allowing the noise amplification that we saw in the previous examples.
So now we have to minimize the function inside the parenthesis.
We take the gradient of it, with respect to f, we set it equal to 0.
Something that you can do now, since you've done it for the least squares.
Plus I've given you all the formulas that
you need to carry out this gradient operation.
And if we do so, we'll see that the solution is given by this expression.
So f is the restored image, here's a
matrix that, again, I find it's generalized inverse.
If it's invertible, that's a regular inverse.
Times this vector here.
Another way to interpret this introduction of
the constrained is by looking at this matrix.
If H dispose H, is not invertible, then the
addition of this term might make the matrix invertible.
Or by and large, the condition number based on expose H
is large that's an expression of this reposeness as it's referred to.
And the addition of this term makes the condition number smaller.
And therefore, the problem behaves better.
One choice for C is to be a high pass filter such as the
2D Laplacian, and this is the impulse response of the 2D Laplacian.
We did mention it when we talked about enhancement and we're going
to study it a bit more, actually, carefully when we talk about segmentation.
So if both C and H are block circulant, then
I can take this expression to the discrete frequency domain.
And this is the expression of the constrained least squares
filter, under this case, again, of H and C block circulant.
And therefore I can bring everything into the discrete frequency domain.
Clearly, if alpha equals 0, then CLS, Constrained
Least Squares becomes least squares or inverse filter.
By and large the solution f is a function of alpha, clearly, right.
As can be seen from here.
And alpha should be chosen in such a way that C of
f of a, the norm of these squares should be less than epsilon.
So, assuming that we know epsilon, we have
to adjust alpha appropriately so the constraint is satisfied.
So let us see now how this filter performs
and how it compares with the, the least squares filter.
So again for the example of in considering.
This is the magnitude of the frequency response of the degradation system, is
just, the frequency response of this one dimensional motion blur.
We've seen this before.
This is u, this is v.
And this is the frequency response of the constraint filter.
So this is the magnitude of the 2D Laplacian.
It's a high pass filter.There's 0 here at the center.
It's the actually it's 0 at 0 if you look at the impulse response.
And the just grows to high frequencies so if you look at
you know the various frequencies like pie pie or minus pie pie.
It's achieves it's largest value which is equal to four.
Okay, so this is the filter we use
to carry out the constrained least squares filtering.
So, we see here that the denominator of the constrained least squares filter.
This is the value of alpha, 0.1.
So I add the magnitude of the low
pass degradation filter to the magnitude of the high pass regularization filter.
So this is the shape here, and the main characteristic is now.
I don't have small values anymore, very small values.
Actually, I don't have the exact 0s that I
had before in, right, that this, this function has.
And at high frequencies, you see that the values increase, right?
So, when I'm going to invert this now according to the least
squares filter, I'm, I'm not going to have 1 over very small values.
That will just amplify noise and, and things of that nature.
So actually, this is the magnitude of the
least squares filter for this particular choice of alpha.
And of course for this particular C, we're talking about here.
So again this 0, 0 is in the center so
this is one over the sink with alpha equals 0.
And we see that the restoration filter's tapered off at
high frequencies, right it doesn't get amplified at high frequencies.
So it has kind of large values at mid frequencies and
then drops off as you go to, to high frequencies.
So this does not allow, again there
the noise amplification that we've seen before.
[SOUND] As in the previous slide, we show,
here, the denominator of the constrained least squares filter.
For a different alpha, alpha is 0.01, an
order of magnitude smaller than in the previous case.
So because of that, the shape of H u, v has not been altered that much.
However I don't have the exact 0s that I had before.
So the addition of C has made the
denominator of the constrained least squares filter better behaving.
So now the magnitude of the constrained least squares filter looks like this.
Again the main characteristic is that the high frequencies have tapered off.
They don't go up to high values and is responsible for noise amplification.
We show here three different restorations by the constrained least
squares filter for three different values of alpha, the regularization parameter.
So the general observation is that, as we explained already,
as alpha becomes smaller and smaller, we move this direction.
We move closer to the inverse filter and therefore we see
more noise present in the restored image more noise of the fixation.
Or going this way the as alpha increases the solution
is a smoother solution because we have those two terms.
Fidelity to the data and smoothness of the solution.
So a higher alpha gives more weight to the smoothness of the solution.
When I minimize it, therefore, I have to make
the norm of CF much smaller now as alpha increases.
And therefore, this results to a smoother solution.
So I have this knob, I have this parameter alpha that now I can adjust and control
the smoothness of the solution versus the noise amplification.
And if epsilon again is given in the constraint.
If epsilon is given, then I have to
adjust alpha so that this constraint is satisfied.
If epsilon is not given to us, then we just either
pick alpha by trial and error, similar to what we show here.
Or utilize a number of techniques that exist, such as cross-validation
techniques that will let us, choose the appropriate alpha.
Let
us now have a closer look at the effect of the regularization
parameter, which you called alpha, on the quality of the restored images.
Based on constrained least squares filter.
So it has been shown in this work shown here.
That, the error between the restored image,
which is a function of the regularization parameter.
And the original image
is given by this expression.
Don't worry about this expectation value.
This is an expression that holds through on the average.
So there are two terms.
A biased term, which is a function of the restored image.
And the variance term, which is also a function of the restored image.
Actually, the exact expressions for the variance and bias are shown here.
All the quantities in these expressions have been defined previously.
Sigma of N squared is the variance of the noise.
And the image dimensions are n by n.
If I plot these two quantities, the variance of
the bias for a specific image, actually linear was used.
The degradation was due to a 7 by 7 flat.
Impulse response, the 2D Laplacian was used for C.
And the [INAUDIBLE] signal-to-noise ratio was 20dB.
We see this plot here.
So, on the vertical axis, this error is shown.
While on the horizontal axis, the values of alpha are shown.
There's actually a very wide range of the values of alpha
going from 10 to the minus 14 to 10 to the 4th.
So there 18 orders of magnitude.
And actually the vertical scale goes from 10 the minus 6, to then 10 to the 12th.
So the main observation here is that, the biased
term, this term here, increases as a function of alpha.
So the larger the alpha, the larger the bias in the solution.
While the variance term decreases as a function of alpha.
We show here in solid line the sum
of these two terms because this expresses the error.
Our objective clearly is to minimize this
error between the restoration and the original image.
Therefore, we're interested in finding the minimum of this solid curve.
A point around here.
So then, the corresponding alpha should be the optimum one might say
alpha with respect to mean square error that one would like to use.
This function here could be rather flat for a range of values.
So it may not be a single point, but a range of values of alpha.
Let's say this one, for which the function of the
error is pretty flat, achieves the, the, the smallest values.
So this then shows that the solution is robust with respect to alpha.
Seems there might be a couple of other magnitudes of
values of alpha that will give, the smallest possible error.
So let's see how these terms, the variance of
the bias man, manifest themselves with a specific example.
So let's look at the effect of the regularization parameter
on the CLS solution, with the use of an example.
So, using the same example throughout this lecture.
So the blurred image is due to motion over eight pixels.
Blurred signal to noise ratio is 20 db.
And here is the restored image for alpha equals 0.01 and here is
the error the absolute value of the error between the original and the restored.
And since this is small, but in large it's mapped
linearly into this range 32 to 55 for visualization purposes.
There's the smallest alpha we'll use.
So, the image is rather noisy.
And then is we look at the arrow, it gives the appearance of noise.
So if you recall from the previous drawing we had there.
The figure at the small let's say value of alpha,
the variance is rather large and the bias is, is small.
And there's alpha increases, the bias increases, and the variance drops.
So, looking again at this error here, we can say that it it looks again like noise.
It has a large variance.
It doesn't reveal much about the structure of the image may be a little bit.
So if we increase out for now and to look at
the restoration, we see as we know by now a smoother image.
The noises reduced, but looking at the error the variance
of these error image is reduced but the bias has increased.
And by bias it means that now the error has has some structure.
It tells you something about the original image and actually what it
tells you is the error is dominant at the edges of the image.
At the regions of high frequency.
Because what happens, as alpha increases.
You give more preference, more weight to the smoothness.
You prefer the smooth solution, which means you filter out
some of the high frequency of the image, as well.
The image becomes smoother.
And therefore, the error shows here.
Its largest shows the, the edge configuration of the image.
If you consider even a high alpha
equals one, then the restoration is even smoother.
And this is the error term.
No noise in it, you might say, but the edges are pronounced.
The, the bias of this image is, is
greatest out, out of all the restorations we had.
[BLANK_AUDIO].
The choice of the regularization parameter alpha is of the utmost importance.
Since as we saw, determines the tradeoff
between noise amplification and smoothness of the solution.
We briefly discuss here some of the methods to determine it.
So if the noise variance is known then this problem is solved.
We minimize the stabilizing functional subject to the constraint that,
this, discrepancy is equal to epsilon square the noise variance.
Then, we choose alpha so that y minus H f,
which is a function of alpha, squared is approximately equal to epsilon squared.
So one can derive here iterative ways to adjust alpha so that this is satisfied.
Now if the noise variance is not known, one can
use visual inspection that is the human the loop approach.
The human operator assuming that he or she
has some knowledge about the structure of the image.
Say it's an image of a natural scene, can choose
alpha so as to produce the most desirable or useful solution.
Now if an iterative algorithm is used, to implement the CLS filter,
then, as we saw, the number of iterations can be used as a means of regradization.
So a second method, if the noise variance is not known, is the L-curve.
If we plot here, in a log log scale,
the fidelity to the data term.
And the log of the stabilizing term
then as a function of alpha this curve is going to have a shape like this.
So here alpha is two small
and therefore, it's under regularized solution.
While at this end so as alpha changes I traverse this curve here.
Alpha is too large
which means that we over regularized the solution.
So this point here, this looks like an L-curve.
And this point, therefore, is a transition
between and under regularized and an over regularized solution.
And therefore it seems like a reasonable choice for, for alpha.
So this is the alpha star that we are after.
And the third approach I would like to mention
here, where again the noise variance is not known.
So for all these approaches.
Noise variance
unknown is the generalized cross-validation approach.
The general idea behind it is to minimize the set of prediction errors.
That is to choose alpha so that the regularized solution
obtained with a data point removed predicts this missing point well.
When averaged over all ways of removing a point.
Specific expressions result in the, is quite
some detail in this reference here about it.
So here is a representative list of ways of choosing the regularization parameter.
And when we cover Bayesian techniques, we'll see that they
provide a means to estimate the regularization parameter as well.
We want to briefly discuss here, extensions or generalizations
of the constrained least squares filter, we discussed so far.
We show the expression for the filter here.
We minimize the sum of these quadratic terms.
The fidelity to the data term, and the destabilizing function of it.
So, both of these are L2 norms, or quadratic terms.
And as we saw in the case, we can obtain a
close form expression for the solution that minimizes this sum of quadratics.
In general area restoration can be
obtained by minimizing this function up here.
Where j 1 represents the distance measured between the data and
the restored image after it has been processed by the system.
And J 2 is the general stabilizing term.
So in general both J 1 and J 2 may be non-quadratic function of, of f.
In this case, no closed form expression for the solution can be found.
As was the case with CLS.
And in this case, one has to result to iteratively optimization solutions.
The successful approximations iteration we just
covered might present one possible choice.
But of course, there are a
number of methods available from optimization theory.
We will not discuss any of them here since this is outside the scope of this course.
So here's some examples, and rather
popular choices of these J1, J2 functionals.
So the first one is the maximum entropy regularization term.
So this is the expression for the negative of the entropy.
And instead of maximizing the entropy, we minimize the negative of the entropy.
We'll talk about the entropy concept when we cover compression.
But in general is a measure of
the uncertainty or the randomness in the image.
Therefore an image with maximum entropy is a smooth image.
The entropy's a non-quadratic term and therefore a non-linear
optimization problem results that needs to be solved iteratively.
This is a regularization term or an an approach
that maximental be approached that has been utilized quite widely.
The t tv or total variation equalization is another popular approach.
Delta of f here is the gradient of the image.
And Is is the Ith component, or the gradient of the Ith pixel.
So we assume that the nth pixel is very much, as was previously the case.
And therefore, the total variation of the signal, is
the total amount of change the signal goes through.
And is therefore a measure of the signal variability.
We want to control such variability and therefore a smoothness
requirement is important to solution by minimizing the total variation.
The resulting problem is also non linear and therefore
either of the optimization methods need to be deployed.
The main idea of the non-quadratic functional is to be such that they
do not penalize large values of their argument as much as the quadratic.
The L2 penalty does.
A regularizer with such properties is the lp norm, that's shown here.
So p is between 1 and 2, and we take the ith
element of the vector z, we raise it to the p power.
And with some of the old reentries of the vector of the pixels of the image.
So if p is chosen between one and two as
shown here, these functions are still convex functions of their arguments.
And therefore, the optimization is rather straight forward relatively speaking.
This lp norm can be used both for the J1, and the J2 functionals here.
We're going to revisit actually these norms when we talk about sparsity
in the last week of classes, since these norms promote sparse solutions.
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