Indrasen Bhattacharya - Revision history

imported>Student221 at 00:37, 8 December 2020

2020-12-08T00:37:58Z

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	Case 3: Vertical astigmatism and horizontal coma (<math>\alpha^1_3</math>) are both present		Case 3: Vertical astigmatism and horizontal coma (<math>\alpha^1_3</math>) are both present

	The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.		The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section. Please note that the aberrations are specified by both the descriptive name and the (m, n) order of the Zernike coefficient in the comparison graphs below.

	[[File:Fig5_Indrasen.png\|900px]]		[[File:Fig5_Indrasen.png\|900px]]

imported>Student221: /* Analytical Forward Model */

2020-12-08T00:30:59Z

Analytical Forward Model

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	== Analytical Forward Model ==		== Analytical Forward Model ==

	An important aspect of the inverse problem solution is the development of a linearized model for the field point spread function. As a result, it will turn out that the field PSF is linear in the Zernike coefficients and the ~~<math>Insert formula here</math>~~intensity PSF is quadratic. The error in the field PSF is of third order in the Zernike coefficients. This linearization will later turn out to be greatly beneficial for solving the inverse problem. We omit the derivation here but specify the key assumptions and results.		An important aspect of the inverse problem solution is the development of a linearized model for the field point spread function. As a result, it will turn out that the field PSF is linear in the Zernike coefficients and the intensity PSF is quadratic. The error in the field PSF is of third order in the Zernike coefficients. This linearization will later turn out to be greatly beneficial for solving the inverse problem. We omit the derivation here but specify the key assumptions and results.

	If the aberration phase is small, we can linearize it as follows:		If the aberration phase is small, we can linearize it as follows:

imported>Student221 at 00:27, 8 December 2020

2020-12-08T00:27:05Z

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	== Appendix ==		== Appendix ==

	The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[Indrasen_final.zip]]		The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[File:Indrasen_final.zip]]

imported>Student221 at 00:26, 8 December 2020

2020-12-08T00:26:17Z

← Older revision		Revision as of 00:26, 8 December 2020
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	== Appendix ==		== Appendix ==

	The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[~~indrasen_final~~.zip]]		The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[Indrasen_final.zip]]

imported>Student221 at 00:22, 8 December 2020

2020-12-08T00:22:28Z

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	== Limitations of Method and Future Work ==		== Limitations of Method and Future Work ==

	There are some well known limitations of the approach as presented in this write-up:		There are some well known limitations of the approach as presented in this write-up (which have been addressed in later publications [3]-[5]):

	1. The approach is valid only for pure phase aberrations. There is an extension to phase and amplitude aberrations using complex Zernike coefficients that may be worth investigating.		1. The approach is valid only for pure phase aberrations. There is an extension to phase and amplitude aberrations using complex Zernike coefficients that may be worth investigating.
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	== Appendix ==		== Appendix ==

	The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[~~Code_okkeun~~.zip]]		The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[indrasen_final.zip]]

imported>Student221 at 00:19, 8 December 2020

2020-12-08T00:19:08Z

← Older revision		Revision as of 00:19, 8 December 2020
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	Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)		Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)

			Due to time and resource constraints, the simulation settings were rather coarse. For instance, the choice of the quantity M, the number of radial orders per azimuthal order could only be 2. Ideally, this number would be on the order of 10 in order to consider a sufficiently large number of radial orders. The Gram matrix method attempts a best fit of the data to the basis functions included in the projection, up to M. Hence, the results may be poor if M is rather low. Another constraint was the spatial discretization, which could not be too fine due to memory and run time constraints on my personal computer. It would be very interesting to perform a more detailed study exploring the impact of focus and spatial discretization, as well as M on the accuracy of the results.

	~~Unfortunately, due to time~~ and ~~resource constraints, the simulation settings were rather coarse. For instance, the choice~~ of the ~~quantity M, the number of radial orders per azimuthal order could only be 2. Ideally,~~ this ~~number would be on the order of 10 in order to consider a sufficiently large number of radial orders~~. The ~~Gram matrix method attempts a best fit of the data to the basis functions included in the projection, up to M~~. ~~Hence, the results may be poor if M~~ is ~~rather low. Another constraint was the spatial discretization, which could not be too fine due~~ to ~~memory~~ and ~~run time constraints on my personal computer. It would~~ be ~~very interesting to perform a more detailed study exploring the impact of focus and spatial discretization, as well as M on the accuracy of the results~~.		== Limitations of Method and Future Work ==

			There are some well known limitations of the approach as presented in this write-up:

			1. The approach is valid only for pure phase aberrations. There is an extension to phase and amplitude aberrations using complex Zernike coefficients that may be worth investigating.

	~~== Limitations~~ of ~~Method~~ and ~~Future Work ==~~		2. The radiometric correction factor has been neglected in the current formulation.

			3. A vectorial formulation is required for higher numerical aperture systems.

			4. The effect of finite source extent has been neglected, although it is fairly straightforward to incorporate.

			Ideally, a little more investigation into the accuracy, numerical stability and a few more test cases would have been great - but hopefully this is a good starting point.

	== References ==		== References ==
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	== Appendix ==		== Appendix ==

	The computational implementation for this project can be found on this ~~iPython notebook~~: [[Code_okkeun.zip]]		The computational implementation (jupyter notebook) and results for this project can be found in this zipped folder: [[Code_okkeun.zip]]
	~~The final results for the 3 cases tested are in this spreadsheet:~~

imported>Student221 at 22:59, 7 December 2020

2020-12-07T22:59:30Z

← Older revision		Revision as of 22:59, 7 December 2020
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	The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.		The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.

	[[File:Fig5_Indrasen.png\|~~300px~~]]		[[File:Fig5_Indrasen.png\|900px]]

	Fig. 5: Results for Case 1 (sanity check with no aberrations)		Fig. 5: Results for Case 1 (sanity check with no aberrations)

	[[File:Fig6_Indrasen.png\|~~300px~~]]		[[File:Fig6_Indrasen.png\|900px]]

	Fig. 6: Results for Case 2 (only one form of aberration)		Fig. 6: Results for Case 2 (only one form of aberration)

	[[File:Fig7_Indrasen.png\|~~300px~~]]		[[File:Fig7_Indrasen.png\|900px]]

	Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)		Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)

imported>Student221 at 22:59, 7 December 2020

2020-12-07T22:59:06Z

← Older revision		Revision as of 22:59, 7 December 2020
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	The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.		The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.

	[[File:Fig5_Indrasen.png\|~~px 300~~]]		[[File:Fig5_Indrasen.png\|300px]]

	Fig. 5: Results for Case 1 (sanity check with no aberrations)		Fig. 5: Results for Case 1 (sanity check with no aberrations)

	[[File:Fig6_Indrasen.png\|~~px 300~~]]		[[File:Fig6_Indrasen.png\|300px]]

	Fig. 6: Results for Case 2 (only one form of aberration)		Fig. 6: Results for Case 2 (only one form of aberration)

	[[File:Fig7_Indrasen.png\|~~px 300~~]]		[[File:Fig7_Indrasen.png\|300px]]

	Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)		Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)

imported>Student221 at 22:58, 7 December 2020

2020-12-07T22:58:28Z

← Older revision		Revision as of 22:58, 7 December 2020
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	The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.		The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.

	[[File:Fig5_Indrasen.png]]		[[File:Fig5_Indrasen.png\|px 300]]

	Fig. 5: Results for Case 1 (sanity check with no aberrations)		Fig. 5: Results for Case 1 (sanity check with no aberrations)

	[[File:Fig6_Indrasen.png]]		[[File:Fig6_Indrasen.png\|px 300]]

	Fig. 6: Results for Case 2 (only one form of aberration)		Fig. 6: Results for Case 2 (only one form of aberration)

	[[File:Fig7_Indrasen.png]]		[[File:Fig7_Indrasen.png\|px 300]]

	Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)		Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)

imported>Student221 at 22:57, 7 December 2020

2020-12-07T22:57:48Z

← Older revision		Revision as of 22:57, 7 December 2020
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	The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.		The results below (with retrieved aberration in shown as red) demonstrate that there is a reasonable qualitative correspondence in the retrieved aberrations and ground truth. However, the quantitative correspondence was found to be lacking. Some of the reasons for this are explained at the end of this section.

	[[File:~~Fig5_indrasen~~.png]]		[[File:Fig5_Indrasen.png]]

	Fig. 5: Results for Case 1 (sanity check with no aberrations)		Fig. 5: Results for Case 1 (sanity check with no aberrations)

	[[File:~~Fig6_indrasen~~.png]]		[[File:Fig6_Indrasen.png]]

	Fig. 6: Results for Case 2 (only one form of aberration)		Fig. 6: Results for Case 2 (only one form of aberration)

	[[File:~~Fig7_indrasen~~.png]]		[[File:Fig7_Indrasen.png]]

	Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)		Fig. 7: Results for Case 3 (two dominant aberrations: same as the case shown in the PSF progression)