Classes | |
| class | CartesianFrame |
| class | EllipseFrame |
| class | EllipticalParametricComponent |
| class | ParametricComponent |
Functions | |
| np.ndarray | gaussian2d (np.ndarray params, EllipseFrame ellipse) |
| np.ndarray | grad_gaussian2 (np.ndarray input_grad, np.ndarray params, np.ndarray morph, np.ndarray spectrum, EllipseFrame ellipse) |
| np.ndarray | circular_gaussian (Sequence[int] center, CartesianFrame frame, float sigma) |
| np.ndarray | grad_circular_gaussian (np.ndarray input_grad, np.ndarray params, np.ndarray morph, np.ndarray spectrum, CartesianFrame frame, float sigma) |
| integrated_gaussian (np.ndarray params, CartesianFrame frame) | |
| np.ndarray | grad_integrated_gaussian (np.ndarray input_grad, np.ndarray params, np.ndarray morph, np.ndarray spectrum, CartesianFrame frame) |
| np.ndarray | bounded_prox (np.ndarray params, np.ndarray proxmin, np.ndarray proxmax) |
| np.ndarray | sersic (np.ndarray params, EllipseFrame ellipse) |
| np.ndarray | grad_sersic (np.ndarray input_grad, np.ndarray params, np.ndarray morph, np.ndarray spectrum, EllipseFrame ellipse) |
Variables | |
| int | MIN_RADIUS = 1e-20 |
| SQRT_PI_2 = np.sqrt(np.pi / 2) | |
| SERSIC_B1 = gamma.ppf(0.5, 2) | |
Function Documentation
◆ bounded_prox()
| np.ndarray lsst.scarlet.lite.models.parametric.bounded_prox | ( | np.ndarray | params, |
| np.ndarray | proxmin, | ||
| np.ndarray | proxmax ) |
A bounded proximal operator
This function updates `params` in place.
Parameters
----------
params:
The array of parameters to constrain.
proxmin:
The array of minimum values for each parameter.
proxmax:
The array of maximum values for each parameter.
Returns
-------
result:
The updated parameters.
Definition at line 535 of file parametric.py.
535def bounded_prox(params: np.ndarray, proxmin: np.ndarray, proxmax: np.ndarray) -> np.ndarray:
536 """A bounded proximal operator
537
538 This function updates `params` in place.
539
540 Parameters
541 ----------
542 params:
543 The array of parameters to constrain.
544 proxmin:
545 The array of minimum values for each parameter.
546 proxmax:
547 The array of maximum values for each parameter.
548
549 Returns
550 -------
551 result:
552 The updated parameters.
553 """
554 cuts = params < proxmin
555 params[cuts] = proxmin[cuts]
556 cuts = params > proxmax
557 params[cuts] = proxmax[cuts]
558 return params
559
560
◆ circular_gaussian()
| np.ndarray lsst.scarlet.lite.models.parametric.circular_gaussian | ( | Sequence[int] | center, |
| CartesianFrame | frame, | ||
| float | sigma ) |
Model of a circularly symmetric Gaussian
Parameters
----------
center:
The center of the Gaussian.
frame:
The frame in which to generate the image of the circular Gaussian
sigma:
The standard deviation.
Returns
-------
result:
The image of the circular Gaussian.
Definition at line 388 of file parametric.py.
388def circular_gaussian(center: Sequence[int], frame: CartesianFrame, sigma: float) -> np.ndarray:
389 """Model of a circularly symmetric Gaussian
390
391 Parameters
392 ----------
393 center:
394 The center of the Gaussian.
395 frame:
396 The frame in which to generate the image of the circular Gaussian
397 sigma:
398 The standard deviation.
399
400 Returns
401 -------
402 result:
403 The image of the circular Gaussian.
404 """
405 y0, x0 = center[:2]
406 two_sigma = 2 * sigma
407 r2 = ((frame.x_grid - x0) / two_sigma) ** 2 + ((frame.y_grid - y0) / two_sigma) ** 2
408 return np.exp(-r2)
409
410
◆ gaussian2d()
| np.ndarray lsst.scarlet.lite.models.parametric.gaussian2d | ( | np.ndarray | params, |
| EllipseFrame | ellipse ) |
Model of a 2D elliptical gaussian
Parameters
----------
params:
The parameters of the function.
In this case there are none outside of the ellipticity
ellipse:
The ellipse parameters to scale the radius in all directions.
Returns
-------
result:
The 2D guassian for the given ellipse parameters
Definition at line 335 of file parametric.py.
335def gaussian2d(params: np.ndarray, ellipse: EllipseFrame) -> np.ndarray:
336 """Model of a 2D elliptical gaussian
337
338 Parameters
339 ----------
340 params:
341 The parameters of the function.
342 In this case there are none outside of the ellipticity
343 ellipse:
344 The ellipse parameters to scale the radius in all directions.
345
346 Returns
347 -------
348 result:
349 The 2D guassian for the given ellipse parameters
350 """
351 return np.exp(-ellipse.r2_grid)
352
353
◆ grad_circular_gaussian()
| np.ndarray lsst.scarlet.lite.models.parametric.grad_circular_gaussian | ( | np.ndarray | input_grad, |
| np.ndarray | params, | ||
| np.ndarray | morph, | ||
| np.ndarray | spectrum, | ||
| CartesianFrame | frame, | ||
| float | sigma ) |
Gradient of the the component model wrt the Gaussian
morphology parameters
Parameters
----------
input_grad:
Gradient of the likelihood wrt the component model
params:
The parameters of the morphology.
morph:
The model of the morphology.
spectrum:
The model of the spectrum.
frame:
The frame in which to generate the image of the circular Gaussian.
sigma:
The standard deviation.
Definition at line 411 of file parametric.py.
418) -> np.ndarray:
419 """Gradient of the the component model wrt the Gaussian
420 morphology parameters
421
422 Parameters
423 ----------
424 input_grad:
425 Gradient of the likelihood wrt the component model
426 params:
427 The parameters of the morphology.
428 morph:
429 The model of the morphology.
430 spectrum:
431 The model of the spectrum.
432 frame:
433 The frame in which to generate the image of the circular Gaussian.
434 sigma:
435 The standard deviation.
436 """
437 # Calculate the gradient of the likelihod
438 # wrt the Gaussian e^-r**2
439
440 _grad = -morph * np.einsum("i,i...", spectrum, input_grad)
441
442 y0, x0 = params[:2]
443 d_y0 = -2 * np.sum((frame.y_grid - y0) * _grad)
444 d_x0 = -2 * np.sum((frame.x_grid - x0) * _grad)
445 return np.array([d_y0, d_x0], dtype=params.dtype)
446
447
◆ grad_gaussian2()
| np.ndarray lsst.scarlet.lite.models.parametric.grad_gaussian2 | ( | np.ndarray | input_grad, |
| np.ndarray | params, | ||
| np.ndarray | morph, | ||
| np.ndarray | spectrum, | ||
| EllipseFrame | ellipse ) |
Gradient of the the component model wrt the Gaussian
morphology parameters
Parameters
----------
input_grad:
Gradient of the likelihood wrt the component model
params:
The parameters of the morphology.
morph:
The model of the morphology.
spectrum:
The model of the spectrum.
ellipse:
The ellipse parameters to scale the radius in all directions.
Definition at line 354 of file parametric.py.
360) -> np.ndarray:
361 """Gradient of the the component model wrt the Gaussian
362 morphology parameters
363
364 Parameters
365 ----------
366 input_grad:
367 Gradient of the likelihood wrt the component model
368 params:
369 The parameters of the morphology.
370 morph:
371 The model of the morphology.
372 spectrum:
373 The model of the spectrum.
374 ellipse:
375 The ellipse parameters to scale the radius in all directions.
376 """
377 # Calculate the gradient of the likelihod
378 # wrt the Gaussian e^-r**2
379 _grad = -morph * np.einsum("i,i...", spectrum, input_grad)
380 d_y0 = ellipse.grad_y0(_grad, True)
381 d_x0 = ellipse.grad_x0(_grad, True)
382 d_sigma_y = ellipse.grad_major(_grad, True)
383 d_sigma_x = ellipse.grad_minor(_grad, True)
384 d_theta = ellipse.grad_theta(_grad, True)
385 return np.array([d_y0, d_x0, d_sigma_y, d_sigma_x, d_theta], dtype=params.dtype)
386
387
◆ grad_integrated_gaussian()
| np.ndarray lsst.scarlet.lite.models.parametric.grad_integrated_gaussian | ( | np.ndarray | input_grad, |
| np.ndarray | params, | ||
| np.ndarray | morph, | ||
| np.ndarray | spectrum, | ||
| CartesianFrame | frame ) |
Gradient of the component model wrt the Gaussian
morphology parameters
Parameters
----------
input_grad:
Gradient of the likelihood wrt the component model parameters.
params:
The parameters of the morphology.
morph:
The model of the morphology.
spectrum:
The model of the spectrum.
frame:
The frame in which to generate the image of the circular Gaussian.
Returns
-------
result:
The gradient of the component morphology.
Definition at line 478 of file parametric.py.
484) -> np.ndarray:
485 """Gradient of the component model wrt the Gaussian
486 morphology parameters
487
488 Parameters
489 ----------
490 input_grad:
491 Gradient of the likelihood wrt the component model parameters.
492 params:
493 The parameters of the morphology.
494 morph:
495 The model of the morphology.
496 spectrum:
497 The model of the spectrum.
498 frame:
499 The frame in which to generate the image of the circular Gaussian.
500
501 Returns
502 -------
503 result:
504 The gradient of the component morphology.
505 """
506 # Calculate the gradient of the likelihood
507 # wrt the Gaussian e^-r**2
508 _grad = np.einsum("i,i...", spectrum, input_grad)
509
510 # Extract the parameters
511 y0, x0, sigma = params
512 # define useful constants
513 x = frame.x_grid - x0
514 y = frame.y_grid - y0
515 c = 0.5 / sigma**2
516 sqrt_c = np.sqrt(c)
517 # Add a small constant to the radius to prevent a divergence at r==0
518 r = np.sqrt(x**2 + y**2) + MIN_RADIUS
519 # Shift half a pixel in each direction for the integration
520 r1 = r - 0.5
521 r2 = r + 0.5
522 # Calculate the gradient of the ERF wrt. each shifted radius
523 d_model1 = np.exp(-c * r1**2)
524 d_model2 = np.exp(-c * r2**2)
525 # Calculate the gradients of the parameters
526 d_x0 = np.sum(-x / r * (d_model2 - d_model1) * _grad)
527 d_y0 = np.sum(-y / r * (d_model2 - d_model1) * _grad)
528 d_sigma1 = -(r1 * d_model1 / sigma - SQRT_PI_2 * erf(r1 * sqrt_c))
529 d_sigma2 = -(r2 * d_model2 / sigma - SQRT_PI_2 * erf(r2 * sqrt_c))
530 d_sigma = np.sum((d_sigma2 - d_sigma1) * _grad)
531
532 return np.array([d_y0, d_x0, d_sigma])
533
534
◆ grad_sersic()
| np.ndarray lsst.scarlet.lite.models.parametric.grad_sersic | ( | np.ndarray | input_grad, |
| np.ndarray | params, | ||
| np.ndarray | morph, | ||
| np.ndarray | spectrum, | ||
| EllipseFrame | ellipse ) |
Gradient of the component model wrt the morphology parameters
Parameters
----------
input_grad:
Gradient of the likelihood wrt the component model
params:
The parameters of the morphology.
morph:
The model of the morphology.
spectrum:
The model of the spectrum.
ellipse:
The ellipse parameters to scale the radius in all directions.
Definition at line 589 of file parametric.py.
595) -> np.ndarray:
596 """Gradient of the component model wrt the morphology parameters
597
598 Parameters
599 ----------
600 input_grad:
601 Gradient of the likelihood wrt the component model
602 params:
603 The parameters of the morphology.
604 morph:
605 The model of the morphology.
606 spectrum:
607 The model of the spectrum.
608 ellipse:
609 The ellipse parameters to scale the radius in all directions.
610 """
611 n = params[5]
612 bn = gamma.ppf(0.5, 2 * n)
613 if n == 1:
614 # Use a simplified model for faster calculation
615 d_exp = -SERSIC_B1 * morph
616 else:
617 r = ellipse.r_grid
618 d_exp = -bn / n * morph * r ** (1 / n - 1)
619
620 _grad = np.einsum("i,i...", spectrum, input_grad)
621 d_n = np.sum(_grad * bn * morph * ellipse.r_grid ** (1 / n) * np.log10(ellipse.r_grid) / n**2)
622 _grad = _grad * d_exp
623 d_y0 = ellipse.grad_y0(_grad, False)
624 d_x0 = ellipse.grad_x0(_grad, False)
625 d_sigma_y = ellipse.grad_major(_grad, False)
626 d_sigma_x = ellipse.grad_minor(_grad, False)
627 d_theta = ellipse.grad_theta(_grad, False)
628 return np.array([d_y0, d_x0, d_sigma_y, d_sigma_x, d_theta, d_n], dtype=params.dtype)
629
630
◆ integrated_gaussian()
| lsst.scarlet.lite.models.parametric.integrated_gaussian | ( | np.ndarray | params, |
| CartesianFrame | frame ) |
Model of a circularly symmetric Gaussian integrated over pixels
This differs from `circularGaussian` because the gaussian function
is integrated over each pixel to replicate the pixelated image
version of a Gaussian function.
Parameters
----------
params:
The center of the Gaussian.
frame:
The frame in which to generate the image of the circular Gaussian
Returns
-------
result:
The image of the circular Gaussian.
Definition at line 448 of file parametric.py.
448def integrated_gaussian(params: np.ndarray, frame: CartesianFrame):
449 """Model of a circularly symmetric Gaussian integrated over pixels
450
451 This differs from `circularGaussian` because the gaussian function
452 is integrated over each pixel to replicate the pixelated image
453 version of a Gaussian function.
454
455 Parameters
456 ----------
457 params:
458 The center of the Gaussian.
459 frame:
460 The frame in which to generate the image of the circular Gaussian
461
462 Returns
463 -------
464 result:
465 The image of the circular Gaussian.
466 """
467 # Unpack the parameters and define constants
468 y0, x0, sigma = params
469 r = np.sqrt((frame.x_grid - x0) ** 2 + (frame.y_grid - y0) ** 2)
470 sqrt_c = 1 / np.sqrt(2) / sigma
471 # Integrate from half a pixel left and right
472 lhs = erf((r - 0.5) * sqrt_c)
473 rhs = erf((r + 0.5) * sqrt_c)
474 z = 0.5 * np.sqrt(np.pi) / sqrt_c * (rhs - lhs)
475 return z
476
477
◆ sersic()
| np.ndarray lsst.scarlet.lite.models.parametric.sersic | ( | np.ndarray | params, |
| EllipseFrame | ellipse ) |
Generate a Sersic Model.
Parameters
----------
params:
The parameters of the function.
In this case the only parameter is the sersic index ``n``.
ellipse:
The ellipse parameters to scale the radius in all directions.
Returns
-------
result:
The model for the given ellipse parameters
Definition at line 561 of file parametric.py.
561def sersic(params: np.ndarray, ellipse: EllipseFrame) -> np.ndarray:
562 """Generate a Sersic Model.
563
564 Parameters
565 ----------
566 params:
567 The parameters of the function.
568 In this case the only parameter is the sersic index ``n``.
569 ellipse:
570 The ellipse parameters to scale the radius in all directions.
571
572 Returns
573 -------
574 result:
575 The model for the given ellipse parameters
576 """
577 (n,) = params
578
579 r = ellipse.r_grid
580
581 if n == 1:
582 result = np.exp(-SERSIC_B1 * r)
583 else:
584 bn = gamma.ppf(0.5, 2 * n)
585 result = np.exp(-bn * (r ** (1 / n) - 1))
586 return result
587
588
Variable Documentation
◆ MIN_RADIUS
| int lsst.scarlet.lite.models.parametric.MIN_RADIUS = 1e-20 |
Definition at line 51 of file parametric.py.
◆ SERSIC_B1
| lsst.scarlet.lite.models.parametric.SERSIC_B1 = gamma.ppf(0.5, 2) |
Definition at line 57 of file parametric.py.
◆ SQRT_PI_2
| lsst.scarlet.lite.models.parametric.SQRT_PI_2 = np.sqrt(np.pi / 2) |
Definition at line 54 of file parametric.py.
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