Description
1. Recovering a blurred signal. This continues the deblurring problem that you
started in Period 15 Activity. The original problem statement is repeated here for
convenience.
Many sensing and imaging systems produce signals that may be blurred or distorted,
such as an out-of-focus camera, and have noise in them. In such situations, algorithms
are needed to deblur the data to obtain a more accurate estimate of the true signal.
A famous example is the Hubble telescope, which was launched with imperfections
in the mirror. These imperfections blurred the image. The blur could be identified
by measuring how much a star (a point) was distorted. Eventually a shuttle mission
installed corrective optics, but prior to that algorithms were used to deblur the images.
Suppose our true signal is a vector x ∈ R
n with ith element xi and the blurring
produces a new signal b ∈ R
n according to the model:
bi =
1
k
xi + xi−1 + · · · + xi−k+1
+ wi
for i = 1, . . . , n
In other words, each bi
is the average of the previous k values of xi
, plus some extra
noise wi
. Here we treat xj as zero when j < 1.
We rewrite these n scalar equations in matrix-vector form
b = Ax + w
where the n-by-1 vector b is the noisy, blurred, measured signal, the n-by-1 vector x is
the true signal, and the n-by-1 vector w contains the noise samples. In this formulation
the n-by-n matrix A is a model for the blurring operation.
a) Write code that generates A ∈ R
n×n matrix as a function of n and k.
b) Suppose the true x is given in the file xsignal.csv. Generate b by using k = 30.
To generate w, make each wi normally distributed with standard deviation σ.
For example, you can do this in Pyhton via: w=sigma*np.random.randn(n,1).
Plot x and b using σ = 0.01 and σ = 0.1.
c) Reconstruct x in the three following ways, and for each one plot the true x and
its reconstruction.
(i) Ordinary least squares
(ii) Truncated SVD; only keep the largest m singular values of A and try different
values of m.
(iii) Regularized (Tikhonov) least squares; try different values of the parameter
λ.
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d) Experiment with different degrees of averaging (i.e., different values of k). and
with different noise levels (σ in the code). How do the blurring and noise level
affect the value of the regularization parameters that produce the best estimates?
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