Q.1) Is there any limitation on the length of the signal? For example, compressed sensing works when the data set is large, but it may not work for a small signal length. Q.2) Can it be mathematically proven that taking random samples at a much lower rate than the Shanon-Nyquist will always work? Q.3) How to determine the dominant frequency using this method as the amplitudes of the frequencies may alter? For example, let's take the DMD of a set of images and try to find out the dominant frequency using this compressed sensing algorithm as often there will be a limitation on the FPS of a camera.
You are very welcome Dipanjan. For the L1 Norm explanation, check videos 8,9 and 10, from Proffesor Gilbert Strang, on Matrix Methods for DSP and ML ocw.mit.edu/courses/mathematics/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/
I tend to agree. I'm having a hard time believing that random is necessary vs random appears to work better statistically. I'd guess that a certain fraction of samples need to have 2*niquist sampling rate but it can be limited because of sparsity. That said, it's time to read the paper describing the method.
In computer graphics you could overcome aliasing by randomly sampling multiple points within a pixel, but that still could produce some noise when the distribution of the random samples ended up being too regular. A more structured approach was to create a sub-grid within the pixel, and then perturb each of the samples within a poisson disc. Has any work been done to understand the ideal structure of the "random" samples in terms of their ability to capture the range of possible frequencies, thus leading to possibly more "structured" random sampling?
Hi Derek, Did you get an answer to your question about designing the sampling matrix to promote the reconstruction process? Also, do you have any ideas on how to assess an already designed sampling matrix?
Awesome, a million thanks! Is the recovery of the spectrum exact, or is there any degree of uncertainty as you vary the sampling? What if for some type of limitation "the very precise clock" couldn't be uniform?
The thing is, if you know which frequencies your signal can contain, you only need 2 samples per each frequency, for your "FFT" set. If you know your signal can only contain 3 known frequencies, you only need 6 samples to get the complex amplitude of each component.
Hi Steve, I really like your explanation videos on CS. I was wondering what are your thoughts on multi-level CS wavelet scheme since natural images are sparse in levels rather than in x-lets (e.g. wavelets), and robust null space property (similar concept to RIP)? Would you be interested in doing such a video? Thanks for making these educational videos available for public!
Thank you Prof! really beautiful lessons! I have now some doubts: 1)Is it possible to apply CS to a complex signal?; 2) How we can improve the frequency resolution of the 's' vector? Thanks
Hi professor, in your previous video you mentioned the number of needed samples for compressed sensing p = K1*k*log(n/k) where k is the number of non zero coefficient in the sparse vector s. In this example, since it has only two frequency content, can we say it is k=2 sparse? If we go more aggressive randomly sample at the average of 64 or 32 will also work? (Assumed K1 = 4)
I find your way of teaching highly intuitive
LOVE the live coding! It's helpful to see the mistakes that anyone could make & also instills more confidence seeing that no one is perfect 👍🏼
Will watch after my exam :)
Q.1) Is there any limitation on the length of the signal?
For example, compressed sensing works when the data set is large, but it may not work for a small signal length.
Q.2) Can it be mathematically proven that taking random samples at a much lower rate than the Shanon-Nyquist will always work?
Q.3) How to determine the dominant frequency using this method as the amplitudes of the frequencies may alter?
For example, let's take the DMD of a set of images and try to find out the dominant frequency using this compressed sensing algorithm as often there will be a limitation on the FPS of a camera.
For question 2, as I know, this happen because of the properties of the L1 Norm, for the why and how is a current and very active area of research
@@elyepes19 Thanks
You are very welcome Dipanjan. For the L1 Norm explanation, check videos 8,9 and 10, from Proffesor Gilbert Strang, on Matrix Methods for DSP and ML
ocw.mit.edu/courses/mathematics/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/
why sample randomly? would it not be better sample in exponentialy smaller steps?
I tend to agree. I'm having a hard time believing that random is necessary vs random appears to work better statistically. I'd guess that a certain fraction of samples need to have 2*niquist sampling rate but it can be limited because of sparsity. That said, it's time to read the paper describing the method.
In computer graphics you could overcome aliasing by randomly sampling multiple points within a pixel, but that still could produce some noise when the distribution of the random samples ended up being too regular. A more structured approach was to create a sub-grid within the pixel, and then perturb each of the samples within a poisson disc. Has any work been done to understand the ideal structure of the "random" samples in terms of their ability to capture the range of possible frequencies, thus leading to possibly more "structured" random sampling?
Hi Derek,
Did you get an answer to your question about designing the sampling matrix to promote the reconstruction process? Also, do you have any ideas on how to assess an already designed sampling matrix?
Thank you, I will be following you for ever. How do you record your videos. What tools are you using?
Awesome, a million thanks! Is the recovery of the spectrum exact, or is there any degree of uncertainty as you vary the sampling?
What if for some type of limitation "the very precise clock" couldn't be uniform?
The thing is, if you know which frequencies your signal can contain, you only need 2 samples per each frequency, for your "FFT" set. If you know your signal can only contain 3 known frequencies, you only need 6 samples to get the complex amplitude of each component.
Hi Steve, I really like your explanation videos on CS. I was wondering what are your thoughts on multi-level CS wavelet scheme since natural images are sparse in levels rather than in x-lets (e.g. wavelets), and robust null space property (similar concept to RIP)? Would you be interested in doing such a video? Thanks for making these educational videos available for public!
Can you try your example with a chirp with a bandwidth greater than samplingrate/2 and proof that you can beat Nyquist?
Thank you Prof! really beautiful lessons! I have now some doubts: 1)Is it possible to apply CS to a complex signal?; 2) How we can improve the frequency resolution of the 's' vector? Thanks
Are you thinking about quantum-mechanical wavefunctions?
Hi professor, in your previous video you mentioned the number of needed samples for compressed sensing p = K1*k*log(n/k) where k is the number of non zero coefficient in the sparse vector s. In this example, since it has only two frequency content, can we say it is k=2 sparse? If we go more aggressive randomly sample at the average of 64 or 32 will also work? (Assumed K1 = 4)
your video is god damn helpful; why not put them on bilibili, Chinese student will love you
sir, could you please recommend me any channel based on nural science as I am doing a project on it please sir
Did anyone run the code ?
wowh nice👍🇩🇪
im an audio engineer and for some reason i ended up here lol