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FDGI-0.5
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مشاهده بیشترتوضیحات
A package for frequency-domain ghost imaging
| ویژگی | مقدار |
|---|---|
| سیستم عامل | OS Independent |
| نام فایل | FDGI-0.5 |
| نام | FDGI |
| نسخه کتابخانه | 0.5 |
| نگهدارنده | [] |
| ایمیل نگهدارنده | [] |
| نویسنده | Jun Wang |
| ایمیل نویسنده | junwang9@stanford.edu |
| آدرس صفحه اصلی | https://github.com/congzlwag/spook |
| آدرس اینترنتی | https://pypi.org/project/FDGI/ |
| مجوز | - |
# Spooktroscopy: Frequency-Domain Ghost Imaging
## Target Problem
Solve <img src="https://render.githubusercontent.com/render/math?math=(A \otimes G)X = B"> under [regularizations](#Regularizations). A, G are matrices acting on the two indices of X, i.e.
G is optional, by default (`G=None`) it is identity, in which case this is the conventional Spooktroscopy, i.e. to solve <img src="https://render.githubusercontent.com/render/math?math=AX=B"> under regularizations.
The reason for G is to accommodate the combination with [pBASEX](https://github.com/e-champenois/CPBASEX), in which case this is solving the two linear inversions in one step.
### Regularizations
Common regularizations are the following three types, all of which optional, depending on what _a prior_ knowledge one wants to enforce on the problem solving.
1. Nonnegativity: To constrain <img src="https://render.githubusercontent.com/render/math?math=X\succeq 0">
2. Sparsity: To penalize on <img src="https://render.githubusercontent.com/render/math?math=\|X\|_1"> or <img src="https://render.githubusercontent.com/render/math?math=\|X\|_2^2">
3. Smoothness: To penalize on roughness of X , along the two indices, independently. For <img src="https://render.githubusercontent.com/render/math?math=\omega">-axis, which is the photon energy axis, the form is fixed <img src="https://render.githubusercontent.com/render/math?math=\|(L_{N_w}\otimes I)X\|_2^2"> where <img src="https://render.githubusercontent.com/render/math?math=L_{N_w}"> is the [laplacian](https://en.wikipedia.org/wiki/Laplace_operator). Roughness along the second axis of X is customizable through parameter `Bsmoother`, which by default is laplacian squared too.
Sparsity and Smoothness are enforced through penalties in the total obejctive function, and the penalties are weighted by hyperparameters `lsparse` and `lsmooth`. `lsmooth` is a 2-tuple that weight roughness penalty along the two axes of X respectively. The hyperparameters can be passed in during instantiation and also updated afterwards. It is recommended to call method `getXopt` with the hyperparameter(s) to be updated, because it will update, solve, and return the optimal X in one step. Calling `solve` with the hyperparameter(s) to be updated and then calling `getXopt()` without input is effectively the same, and the problem will be solved once as long as there is no update.
## Solvers
Different combinations of regularizations can lead to different forms of objective function. Solvers in package always formalize the specific problem into either a [Quadratic Programming](https://en.wikipedia.org/wiki/Quadratic_programming) or a linear equation. Examples can be found in [unit tests](#UnitTests)
| Nonnegativity | Sparsity | Smoothness | Solver | Notes |
| ------------- | ------------------- | ---------- | ------------------- | ------------------------------------------------------------ |
| True | L1 or False | Quadratic | `SpookPosL1` | This solver can serve tasks like in [Li _et al_](https://iopscience.iop.org/article/10.1088/1361-6455/abcdf1) |
| True | L2 squared | Quadratic | `SpookPosL2` | |
| False | L2 squared or False | Quadratic | `SpookLinSolve` | This solver is so far the work-horse for SpookVMI |
| False | L1 | Quadratic | `SpookL1` | |
### Quadratic Programming
For cases where it can be formalized into a [Quadratic Programming](https://en.wikipedia.org/wiki/Quadratic_programming) , [OSQP](https://osqp.org) does the job. Thus the root numerical method is alternating direction method of multipliers (ADMM). Looking into the [solver settings of OSQP](https://osqp.org/docs/interfaces/solver_settings.html) is always encouraged, but the default settings usually work fine for `spook` . If one needs to pass in settings, the OSQP solver is `SpookQPBase._prob` .
### Linear Equation
A rare case that it can be formalized into a linear equation is the third line in the table above: no nonnegativity constraint, and the sparsity is L2 norm squared. This is implemented in `SpookLinSolve` , which calls `numpy.linalg.solve` or `scipy.sparse.linalg.spsolve` .
## Normalization Convention
The entries in <img src="https://render.githubusercontent.com/render/math?math=A^TA, G^TG"> are preferred to be on the order of unity, because regularization-related quadratic form matrices have their entries around unity. The scale factors are set as
},s_g=\sqrt{\frac{1}{N_q}\mathrm{tr}(G^TG)})
where <img src="https://render.githubusercontent.com/render/math?math=N_w, N_q"> are the dimensions along w-axis and q-axis, respectively. <img src="https://render.githubusercontent.com/render/math?math=s_as_g"> is an accessible property of the solver `AGscale`. To normalize or not is controlled by parameter `pre_normalize` in instanciation.
**By default** `pre_normalize=True`, i.e. `self._AtA` =<img src="https://render.githubusercontent.com/render/math?math=A^TA/s_a^2">, `self._GtG` =<img src="https://render.githubusercontent.com/render/math?math=G^TG/s_g^2">, and `self._Bcontracted` = <img src="https://render.githubusercontent.com/render/math?math=(A^T \otimes G^T)B/(s_as_g)">. In this case, the direct solution `self.res` is scaled as <img src="https://render.githubusercontent.com/render/math?math=s_as_g X_\mathrm{opt}"> , but in `getXopt` the final result of <img src="https://render.githubusercontent.com/render/math?math=X_\mathrm{opt}"> is scaled back before returned.
The entries in B are not always accessible, because of the option to pass in precontracted results and in `mode='contracted'`. Therefore B is not normalized.
## Unit Tests
`unittest/testPosL1.py` is a good example to play with `SpookPosL1`.
`unittest/testL1L2.ipynb` include good examples to play with `SpookPosL1`, `SpookPosL2`, and `SpookL1`.
## Dependencies
See `requirements.txt`
## Acknowledgement
This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences Division (CSGB).
test
زبان مورد نیاز
| مقدار | نام |
|---|---|
| >=3.7 | Python |
نحوه نصب
نصب پکیج whl FDGI-0.5:
pip install FDGI-0.5.whl
نصب پکیج tar.gz FDGI-0.5:
pip install FDGI-0.5.tar.gz