Welcome back! In the world of sparsity and compressed sensing, a fascinating field of applied mathematics and statistics, we have been exploring the possibilities of reconstructing high-dimensional signals from a limited number of measurements. However, today we’re going to turn this paradigm upside down. Instead of relying on random measurements, we’ll ask the question: can we achieve better results by strategically placing our sensors? In other words, can we tailor our measurements to exploit prior information about the system we are sensing?
To give you a quick recap, compressed sensing involves a high-dimensional signal X that can be sparse in a transform basis. By measuring a down-sampled version of the signal, we can reconstruct it using an optimization problem that promotes sparsity. This approach works well when we have random measurements and don’t know the nature of the system we are sensing.
But what if we do know the system we are dealing with? For example, we might be measuring a specific flow field or capturing images of human faces. In these cases, using a universal basis like Fourier or wavelets might not be the most efficient approach. Instead, we can create a tailored library, a reduced basis specific to our system, using techniques like the singular value decomposition.
Once we have our tailored basis, we can ask two important questions: How do random sensors perform in this context? Can we optimize sensor placement in the measurement matrix to match our tailored basis?
It turns out that the performance of random sensors is not as effective in tailored systems. The key to achieving better results lies in minimizing the condition number of the matrix that relates the measurement matrix and the tailored basis. By doing so, we can find optimal sensor locations that are well-suited to our specific system.
This optimization process can be done using the QR factorization of the tailored basis matrix. The pivot locations obtained from the QR factorization correspond to the optimal sensor locations for measurement. This approach is computationally efficient and can be easily implemented in various programming languages like MATLAB and Python.
Let’s take a look at some examples. In the case of flow fields, by using QR-based sensors optimized for the tailored basis, we can achieve a much better signal reconstruction compared to random sensing. The same applies to capturing images of human faces. With optimized sensor placement, we can obtain a more faithful reconstruction of the original image.
This tailored sensing approach has wide applications in various fields such as insect flights, manufacturing, control systems, and ocean dynamics prediction. It allows us to gather more information about specific systems using fewer sensors, leading to efficient and accurate measurements.
In summary, the key idea is to match our sensing approach to the nature of the system we are measuring. By using a tailored basis and optimizing sensor placement, we can significantly improve signal reconstruction. This approach harnesses the power of high-dimensional geometry, sparsity, and optimization techniques.
If you’re interested in exploring this concept further, you can find all the code and additional resources on our website Techal. Stay tuned for more exciting topics on sensing, sparsity, and data-driven techniques.
