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Now that you mention it....

Thank you so much for your support, much appreciated!

Thanks - appreciated!

The reason is that I have been working on a textbook! I fit that into my spare time instead of videos. When completed, I hope to get back to some video making!

I hope to add videos soon, but could be a while...

Thanks for the comment. A GAN for denoising seems quite extreme. GANs can be unstable to train, and would compare a generated image to a reference training set. For denoising, I would think that a simpler approach suffices. Or, for state of the art, I would now consider diffusion models rather than GANs.

Thanks so much for the feedback. I hope this video will help for the Cartesian case: th-cam.com/video/TZzX-M1mVJ4/w-d-xo.html

Many thanks for the feedback! We have new methods being developed that could help with the potential crosstalk issue.

Thanks for the comment - yes, indeed it can be done with any image

A whole electron (there can't be a partial fraction of an electron) is detected. The wavefunction describes all of the information about the electron, including information about its position. Initially there is a superposition of many possible position states, and at detection the wavefunction, describing the electron, collapses to just one of those position states- the position where the electron is detected.

Thanks for the feedback! In this code I used the phantom that is available in Matlab, and then used the radon function to create the sinogram data from that phantom. Hence I did not need to import any data for the sinogram.

Thanks for the feedback!

Thanks for the feedback! There is a newer version of the video here: th-cam.com/video/G0V6ulOIlJc/w-d-xo.html

Thanks for the feedback. Convolution is only really defined in one way (if we overlook edge effects) - and that would mean that the kernel is not flipped in any way at all when placed into columns of the matrix, just shifted. Hopefully that would be true for your approach too?

@@AndrewJReader when the flipped vector is multiplied by the shifted matrix rows, the rows are not flipped but are just shifted. The resultants both ways end up being equivalent, possibly hence the associative property of convolution's intrinsic symmetry. I guess linear algebra is complicated in this way mostly because of all the moving parts, but it sure does do the job efficiently

Thanks so much for your feedback!

Many thanks for the feedback!

Great to hear, thanks so much for the feedback!

Glad this helped, so simple on the one hand, yet with massive consequences!

I have provided some example Python code here: th-cam.com/play/PL557uxcMh3xwUsqofih09ZPqHMdhe-rui.html&si=FyNe0xAO62HU_Of3 In terms of Matlab, I have only done one video: th-cam.com/video/r5lzacT3HkE/w-d-xo.html

For the libraries / packages that I had installed on my system when I did this video, there was no problem at all. So for me at that time there was no error report. Other installations / environments / OS can of course vary. So it did exist for my setup. I used that line to move the window to the top left corner. Hope that helps, and thanks for the feedback.

Many thanks for the feedback! In fact yes, I am working on a book now, but it might be a while yet for it to be released. I think the content I have presented here, and its level, is distinct from that in other courses / books.

Thanks for the feedback! If you use radon and iradon from skimage.transform, the gradients won't propagate for training due to the functions using numpy arrays, rather than torch tensors. So just converting between torch tensors and numpy arrays before/after use of radon or iradon would not be sufficient. Hope that makes sense (I assume your current version is not working, when you say that the loss remains constant for all epochs?).

@@AndrewJReader Exactly. Although it's not a huge issue, since I can use the approach with the system matrix. Was just curious whether I can embed radon/iradon into this calculation, to avoid a file with gigabytes in size.

Yes you can use a radon / iradon function, it's just that it needs to work on torch tensors all the way through. Some people have already done this, for example take a look here: torch-radon.readthedocs.io/ (note that I have not used anything from that package)

Thanks so much for the feedback, appreciated! Yes, the physics of what happens is remarkable!

Many thanks for the feedback, appreciated!

Thank you so much for the feedback, means a lot. Likewise, very best wishes for your next steps.

Thanks for the feedback and question. This method for denoising is applicable whatever the type or distribution of noise present in an image. But of course, more severe noise levels will always mean it is harder to reliably reduce the noise while preserving the signal.

Really good to know! Thanks for the comment!

Many thanks for the feedback!

Thanks for the feedback!

Many thanks! You can do a self supervised approach - a very simple example being to artificially create a noisier version of the image that you have, and train the CNN to denoise that back to the version of the image that had not been made noisier. Then use that trained CNN on the original image. But there are other methods that are more sophisticated of course.

Interesting question, it depends on the sampling of the image. In fact I would simply suggest that you could resize a highly (over)sampled version of the image (just regrid at lower resolution, assigning a value to the lower resolution image pixels according to the sum of corrsponding pixel values in the high res image). Hope that makes sense, no real need for DIP (which can do inpainting, and for a high res half tone image it would possibly just fill the gaps in accordance with the neighbouring values - and if at core the image is binary valued, then the result could be binary as well). It all depends on many things though. I would start with a simple non-deep learning approach for your problem.

@@AndrewJReader that’s gonna keep me fairly busy

Thank you so much for the feedback, really appreciated.

Thank you so much Parisa, your feedback really means a lot!

Really appreciate the feedback

Thanks so much for your feedback, really appreciated. I think histoimages have a definite future for practical and fast input to reconstruction networks. MLAP vs TOF-backprojection needs to be considered. Thanks also for recommending my channel to your peers!

Many thanks for the feedback! Yes indeed, one would need to calculate an integral (a product of 2 functions and then sum) for each and every frequency of interest. However, there is only a finite number of frequencies in the case of discrete functions (the discrete Fourier transform is used), so it is feasible. As it can however be slow, this is where the fast Fourier transform (FFT) comes in - it dramatically speeds up the calculation of the discrete Fourier transform. For continuous functions, analytic integration can be done, to give the solution in the form of another function.

Hi Jacob, thanks for the question. Yes, there is a probability of Compton scatter (inelastic) or even a tiny probably of Thomson scatter (elastic) of the 511 keV photons with the electrons in the attenuating medium (i.e. the brain, skull, etc.). However, a good fraction of these high-energy photons do escape without any interaction at all. (Photoelectric absorption can also occur with relatively low probability).

@@AndrewJReader Aha, thanks! What is modern physics without probabilities ;-)

Great to hear, thanks so much for the feedback!

Great question - yes, you can. One approach would be to use the deep image prior representation (or even just a pixel grid in fact, if not early stopping or regularisation sought) with a forward model only and use an optimiser such as those used in AI, such as Adam, but it would be slow

@@AndrewJReader i mean unfiltered backprojection without iradon

But backprojection is iradon (when used with no filter). The transpose of the radon transform can be found via iradon(no filter). If you mean to avoid using iradon as a function, then you can write your own version, such as, for example, what I did in this video: th-cam.com/video/BXXLoVyAT0Q/w-d-xo.htmlsi=D6PoRXZRiT3Jm9z3 (see about 18 mins into the video)

Really glad the videos have been helpful

Great to hear, thanks for the feedback

Thanks also!