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Paul R. Shearer, Richard A. Frazin, Anna C. Gilbert, Alfred O. Hero III (University of Michigan)

Effective Stray Light Removal for EUV Images

Coronal images from all EUV telescopes (EIT, TRACE, EUVI, AIA) are contaminated with stray light caused by entrance aperture diffraction and mirror scattering. The contamination is worst in faint regions such as coronal holes, prominence cavities, and the off-limb corona. EUV stray light contamination is caused by a point spread function (PSF) which varies with wavelength band but not with time. In the time series of images for each band, each observation is the convolution of the bands PSF with the unknown true-emission image, plus Poisson-distributed photon noise:Fig. 1: Severe stray light contamination can be directly observed in an EUVI-B lunar transit from Feb. 25, 2007. Since the Moon (circled) is not an EUV emitter, all of its apparent emissions are in fact stray light. Since the apparent emissions of the off-limb region above and below the Moon are nearly identical to the apparent emissions from the Moon, the true emission from this region must be nearly zero. Quantitative study of this regions emission is therefore meaningless without correction.Fig. 2: Deconvolutions using PSFs output by SolarSofts euvi_psf.pro utility (far right: SolarSoft 171 PSF) are the only stray light corrections publicly available today. They predict high levels of stray light contamination in faint coronal regions, but do not remove all stray light. Left: A SolarSoft deconvolution predicts that a prominence cavitys apparent emissions are 40% stray light. Right: In a Feb. 25 lunar transit observation, the SolarSoft 171 PSF removes only 30-50% of the apparent lunar emissions. Since all apparent lunar emissions are in fact stray light, we conclude that the SolarSoft PSF removes less than half of the stray light in 171 images. When this light is fully removed by our method, faint regions such as the prominence cavity become much fainter than when deconvolved with SolarSoft PSFs (cf. Fig 5).

observeddeconvolvedEUVI 171 The EUV Stray Light Problem

Nonparametric Blind Deconvolution For EUVI-B, 171 The PSF and images must conform to the EUV stray light model of Eq. 1.The PSF h must be everywhere positive, normalized to 1, and must decay roughly monotonically with distance from its central core. This assumption appears to be reasonable in the 171 band, as mirror scattering is estimated to account for >97% of stray light in 171 and mirror scattering typically decays roughly monotonically.To formulate the rough monotonic decay condition precisely, we require that at any point p in the PSF domain I, h(p) should lie in a cone K(p) pointing towards p. Level sets of a PSF that conforms to this inward gradient model are given in Fig. 3 below. The pictured PSF conforms because at every point p, h(p) is in the red region K(p), an infinite cone of interior angle 90 with symmetry axis OP. This model admits a rich class of PSFs with roughly elliptical level sets and arbitrary decay profiles.The images (u1 uN) must be everywhere nonnegative, and zero within the disk of the Moon and the vignetted image corners; we denote these zero-constrained regions by Z = (Z1, , ZN). The Moons intensity is known to be zero, while comparison of intensities inside and outside the far off-limb Moon (Fig. 1) provide evidence that the true emissions are very close to zero in the far off-limb vignetted region.To remove stray light from all images in a band, we must simultaneously determine h and u from f; this notoriously difficult and underdetermined signal processing problem is known as blind deconvolution. Our groups ultimate goal is to remove stray light from all EUV images; here we present stray light corrections for EUVI-B 171 .Conclusion and Ongoing WorkUsing a novel nonparametric model and algorithm, we have obtained a single PSF for the EUVI-B 171 band which removes nearly all stray light from the Moon in every image of the Feb. 25, 2007 transit series (except for those in the extreme off-limb, where imposing proper boundary conditions becomes important). We currently seek to

refine our EUVI-B 171 PSF by imposing correct image boundary conditions and applying our algorithm to the full, unbinned images. Determining the proper boundary conditions is a challenging problem, as the emissions near and beyond the image-forming region may be significant. We currently seek to impose a decay condition similar to the inward gradient model on the off-limb.estimate PSFs in all EUVI-A/B bands, accounting for entrance aperture diffraction in the bands where it is significant. We may do this by relaxing our inward gradient model to allow for limited deviations from monotonicity.adapt our methods to estimate PSFs for the entire EUV telescope fleet.

To find the PSF and image series that best fit these conditions, we solve the optimization problem given in Eq. 2. For simplicity we replace the Poisson likelihood with a Gaussian likelihood, giving us a constrained nonlinear least squares problem. The large dimensionality, complex constraint structure, and highly ill-conditioned quadratic terms make even this simplified problem intractable by general-purpose optimization methods. To solve it, we created a novel algorithm based on recent advances in dual decomposition and proximal splitting, and applied it to 8x downsampled images (i.e. 2048x2048 EUVI-B images became 256x256). Synthetic EUVI-B Blind DeconvolutionsFig. 4: A synthetic blind deconvolution experiment which demonstrates that we can accurately recover EUVI-like PSFs and images from noisy, blurry observations by solving Eq. 2. In our experiment, we generated a synthetic EUVI-B 171 PSF and a series of 8 true-emission images. The PSF had a power-law decay profile simulating mirror scattering; each true-emission image was a real EUVI lunar transit image in which we set the Moon and off-limb to zero. We convolved the images and PSF together and added Poisson noise to generate synthetic EUVI observations. Col. 1: 1 of the 8 synthetic observations and its true-emission zero set (red). Col. 2: the true-emission image and our deconvolved image are visually indistinguishable. Col. 3: the relative deviation of the deconvolved image from the true emission (bottom) is at most 5%, comparable to the signal-to-noise ratio (top). Col. 4: Our recovered PSF closely resembles the true PSF despite our not imposing any prior formula.Fig. 6: EUVI-B 171 features before deconvolution with our PSF (top) and after (bottom). To create these images, we upsampled our 256x256 PSF by 8x, and deconvolved the 2048x2048 images with the upsampled PSF. Col. 1: a coronal hole dims by up to 70% after deconvolution. Col. 2: a prominence cavity dims by up to 70%, which is 30% more than was predicted by the SolarSoft PSF (cf. Fig. 2). Col. 3: an active region brightens by 30%. Col. 4: stray light is removed to a tolerance of less than 5% over almost all of the Moon. The negative counts on the lower lunar boundary do not exceed 15% of the apparent emission, and appear to be an artifact of upsampling the 256x256 PSF to 2048x2048.

obsdeconvFig. 5: We solved Eq. 2 using 8 observations from the Feb. 25 transit series to estimate the EUVI 171 PSF (far right). For independent verification of the PSFs accuracy, we deconvolved several transit series images not in the training set of 8, including the 3 shown here. We remove at least 90% of the Moons stray light in every image except those where the Moon is very close to the image boundary (not pictured). The sensitivity of deconvolution to boundary conditions is a common problem and a subject of our current work. Unlike the SolarSoft PSF, our PSF is highly anisotropic. The dark triangles on the sides are boundary effects.Removal of Stray Light from EUVI-B Data

observeddeconvolved% stray light

Eq. 2: Problem Formulation.Fig. 3: The Inward Gradient PSF Model.Eq. 1: The EUV Stray Light Model. log10 PSF

log10 PSFTo remove stray light from EUVI-B 171 images, we seek a physically reasonable PSF that removes all stray light from the Moon and produces physically reasonable deconvolved images. We give a precise meaning to reasonable by specifying a mathematical model for the images and PSF. Note that our model is nonparametric; we do not assume the PSF is given by a formula with just a few free parameters. Nonparametric modeling greatly facilitates discovery of unanticipated PSF structure.

Image and PSF Model Assumptions

This poster template is from http://www.swarthmore.edu/NatSci/cpurrin1/posteradvice.htm. It is free, free, free for non-commercial use. But if you really like it, Im always thrilled to get postcards from wherever you happen to be presenting your poster. Have fun. Sincerely, Colin Purrington, Department of Biology, Swarthmore College, Swarthmore, PA 19081, USA. Email: cpurrin1@swarthmore.edu1

The effectiveness of a deconvolution can be judged by how well it conforms to these conditions. The transitcondition is particularly useful because it strongly constrains the PSF: for example, in the EUVI-B Moon transit,deconvolution with the PSF must remove all the stray light from the Moon. Using this constraint, I have found thatsome of the EUVI PSFs available in the SolarSoft IDL routine euvi psf.pro are extremely ineffective; for example,the 171 A PSF removes less than half of the stray light from the EUVI-B Moon transit.

Unfortunately, naive nonparametric blind deconvolution is an ill-posed, underdetermined problem. This meansthat, to deblur an image f , it is generally not enough to find a PSF h and image u such that hu f . There are generallymany PSF-image pairs (h,u) such that hu f up to noise, and most of these pairs are physically unreasonable. Theprocess of avoiding such unreasonable solutions by imposing prior knowledge on the solution is called regularization;regularization is critical to success for any nonparametric deconvolution method.

The Bayesian maximum a posteriori (MAP) approach provides a flexible, statistically-justified framework forregularizing blind deconvolution problems [?]. In the MAP approach, we seek the PSF-image pair (h,u) which ismost probable, given our data f and prior knowledge about htrue and utrue. Mathematically, this is equivalent tominimizing a penalized likelihood function

F(h,u) = L(h,u)+R(h,u),

where L(h,u) and R(h,u) are commonly called the likelihood and regularizer respectively. The likelihood measureshow much hu differs from the data f , and is determined by the distribution of the image noise (e.g. a 2 metric forGaussian noise). The regularizer penalizes unreasonable PSFs and images by taking relatively large positive valueswhen h or u display unreasonable structure; this helps ensure that unreasonable (h,u) pairs will not minimize F(h,u)even if they make L(h,u) small. To extend the MAP approach to the EUV time series problem - many images blurredby a common PSF - we introduce image variables (u1, . . . ,uN) and include a separate likelihood and regularizer termfor each image:

F(h,u1, . . . ,uN) =N

i=1

L(h,ui)+R(h,ui).

minimizeh,u

N

t=1

12hut ft2

subject to h 0, pI

h(p) = 1

h(p) Kp for all p Iu 0, u(Z) = 0

2

Poster equations for SDO workshop

Paul Shearer

April 28, 2011

1 PSF and Image CorrectionStray light effects in an EUV image (blur for short) may be modeled mathematically by convolving the image witha PSF. Let u and h represent a solar image and instrument PSF respectively. In the continuous case, u = u(x,y) andh = h(x,y) are real-valued functions on the domain I R2 where the image is formed. The convolution of h with u,denoted hu, is defined by

ft = hut +nt , t = 1, . . . ,N(hut)(x,y) =

Iut(x x,y y)h(x,y)dxdy.

h : band PSFu= (u1, . . . ,uN) : blur free imagesf = ( f1, . . . , fN) : EUVI observationsn= (n1, . . . ,nN) : noise

where u is extended to be zero at points outside of the image domain I. In the discrete case, the image domain I Z2is a discrete array of pixels; the PSF and image are functions on this discrete array rather than continuous functions;and the integral is replaced by a summation over I.

In each band, the EUV data consists of a time series of images blurred by a common PSF with additive photonnoise. Suppose we have N EUV images, f1, . . . , fN , in a given wavelength. Let htrue be the true instrument PSF, andutrue1 , . . . ,u

trueN be the series of true, blur-free EUV images. Each observed image fi is formed by convolving h

true andutruei together, then adding shifted Poisson noise i:

fi = htrue utruei +i, i= 1, . . . ,N.The problem of estimating the blur-free images from the observations is called deconvolution. When htrue is not knownand must also be estimated, the problem is called blind deconvolution.

Blind deconvolution is an underdetermined problem in general; therefore, the PSF and images must be constrainedwith strong background knowledge to secure an accurate estimate. Fortunately, we have a wealth of information whichcan be summarized in three conditions:

1. Positivity Condition: The PSF and each image in the time series must be everywhere positive: htrue 0 andutruei 0 for all images i.

2. Zero Transit Condition: Whenever a non-EUV emitter such as the Moon or Earth transits in front of the Sun,its true brightness must be zero: (utruei )Zi = 0 where Zi is the support set of the transit object (e.g. the apparentlunar disk) in image i. A Moon transit occurred for EUVI-B on February 25, 2007, and both Moon and Earthtransits are expected for AIA (Karel Schrijver 2010, personal communication).

3. PSF Structure Condition: In EUVI, the PSF arises from wire mesh diffraction and mirror scattering effects [?],so each PSF h can be written as h , where is the mirror scattering component and is the wire meshdiffraction component. is radially symmetric and decreasing, while is the Fourier transform of the opticaltransmission function (OTF) of the periodic wire mesh. A roughly similar model should exist for AIA and theAIA science team has offered to help us determine it.

1

Poster equations for SDO workshop

Paul Shearer

April 28, 2011

1 PSF and Image CorrectionStray light effects in an EUV image (blur for short) may be modeled mathematically by convolving the image witha PSF. Let u and h represent a solar image and instrument PSF respectively. In the continuous case, u = u(x,y) andh = h(x,y) are real-valued functions on the domain I R2 where the image is formed. The convolution of h with u,denoted hu, is defined by

ft = hut +nt , t = 1, . . . ,N(hut)(x,y) =

Iut(x x,y y)h(x,y)dxdy.

h : band PSFu= (u1, . . . ,uN) : blur free imagesf = ( f1, . . . , fN) : EUVI observationsn= (n1, . . . ,nN) : noise

where u is extended to be zero at points outside of the image domain I. In the discrete case, the image domain I Z2is a discrete array of pixels; the PSF and image are functions on this discrete array rather than continuous functions;and the integral is replaced by a summation over I.

In each band, the EUV data consists of a time series of images blurred by a common PSF with additive photonnoise. Suppose we have N EUV images, f1, . . . , fN , in a given wavelength. Let htrue be the true instrument PSF, andutrue1 , . . . ,u

trueN be the series of true, blur-free EUV images. Each observed image fi is formed by convolving h

true andutruei together, then adding shifted Poisson noise i:

fi = htrue utruei +i, i= 1, . . . ,N.The problem of estimating the blur-free images from the observations is called deconvolution. When htrue is not knownand must also be estimated, the problem is called blind deconvolution.

Blind deconvolution is an underdetermined problem in general; therefore, the PSF and images must be constrainedwith strong background knowledge to secure an accurate estimate. Fortunately, we have a wealth of information whichcan be summarized in three conditions:

1. Positivity Condition: The PSF and each image in the time series must be everywhere positive: htrue 0 andutruei 0 for all images i.

2. Zero Transit Condition: Whenever a non-EUV emitter such as the Moon or Earth transits in front of the Sun,its true brightness must be zero: (utruei )Zi = 0 where Zi is the support set of the transit object (e.g. the apparentlunar disk) in image i. A Moon transit occurred for EUVI-B on February 25, 2007, and both Moon and Earthtransits are expected for AIA (Karel Schrijver 2010, personal communication).

3. PSF Structure Condition: In EUVI, the PSF arises from wire mesh diffraction and mirror scattering effects [?],so each PSF h can be written as h , where is the mirror scattering component and is the wire meshdiffraction component. is radially symmetric and decreasing, while is the Fourier transform of the opticaltransmission function (OTF) of the periodic wire mesh. A roughly similar model should exist for AIA and theAIA science team has offered to help us determine it.

1