Welcome back, tech enthusiasts! Today, we will delve into the fascinating world of compressed sensing and sparsity. To truly understand these concepts, we will take a hands-on approach and explore their implementation in MATLAB. So, let’s jump right in!
Contents
Solving Underdetermined Systems
In compressed sensing, we often encounter underdetermined systems of equations in the form of y = θs. Here, s represents a sparse vector, while y denotes limited measurements of a given system. Typically, we can express y as Cθs, where C is a matrix that specifies how we sample the high-dimensional system, and θ serves as the measurement matrix.
For instance, if we consider an image or an audio signal, the vector s may consist of Fourier coefficients, while y represents our measurements. In this case, θ would be the matrix that guides the sampling process, enabling us to observe the high-dimensional system in a lower-dimensional format.
The goal of compressed sensing is to find the best solution s that satisfies the underdetermined system of equations. Since there are countless possible solutions, each with different degrees of freedom, we need to determine which solution is the most suitable, given the constraints of the system.
Implementation in MATLAB
Now, let’s implement compressed sensing in MATLAB. We will create a sparse vector s of dimension 1,000 and an output measurement vector y of dimension 200. To simplify matters, we will assume that the measurement matrix θ is a well-behaved random matrix generated using the randn function.
Next, we will solve for s using two different norms: the minimum l2 norm (using the pseudo-inverse) and the minimum l1 norm (using the CVX package). The l2 norm represents the least squares solution, while the l1 norm provides a sparse solution.
% Create the data
s = sparse(1000, 1);
y = randn(200, 1);
theta = randn(200, 1000);
% Solve for the l1 minimum norm solution
cvx_begin
variable s_l1(1000);
minimize(norm(s_l1, 1));
subject to
theta * s_l1 == y;
cvx_end
% Solve for the l2 minimum norm solution (pseudo-inverse)
s_l2 = pinv(theta) * y;The CVX package in MATLAB is a powerful tool for solving convex optimization problems. In our case, we use it to minimize the l1 norm of s while ensuring that θs accurately represents y.
Visualizing the Solutions
To gain insights into the solutions, let’s visualize them using plots. On the left panel, we have the l1 solution (in blue), and on the right panel, we have the l2 solution (in red).
Notice that the l1 solution is spiky, with many zero or very small values. In contrast, the l2 solution has almost all non-zero values, but they are smaller in magnitude.
FAQs
Q: How can I obtain the CVX package for MATLAB?
A: You can download the CVX package for free. Visit the official CVX website Techal for more information.
Q: Are all solutions obtained using compressed sensing sparse?
A: No, not necessarily. The sparsity of the solution depends on the norm used for optimization. The l1 norm tends to yield sparse solutions, while the l2 norm produces non-sparse solutions.
Q: Can compressed sensing handle noisy measurements?
A: Yes, compressed sensing can handle noisy measurements. In such cases, additional constraints are introduced to account for the noise. However, in this article, we focus on the noise-free scenario for simplicity.
Conclusion
In this article, we explored the world of compressed sensing and its implementation in MATLAB. We witnessed the power of optimization techniques, such as the minimum l1 and l2 norm solutions, in solving underdetermined systems. By using compressed sensing, we can extract valuable information from limited measurements and gain deeper insights into the underlying high-dimensional systems.
Remember, sparsity plays a crucial role in compressed sensing, and the choice of norm determines the nature of the solutions obtained. So, the next time you encounter underdetermined systems, don’t forget to consider compressed sensing and its remarkable capabilities.
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