A Bayesian Algorithm for Tangential Deconvolution of Weather Radar Images

Transcription

1 A Bayesan Algorthm for Tangental Deconvoluton of Weather Radar Images J. Lenonen 1, V. Chandrasekar 1,,3, D. Mosseev 3 1 Fnnsh Meteorologcal Insttute, P.O. Box 503, FI Helsnk, Fnland, juss.lenonen@fm.f Colorado State Unversty, Fort Collns, Colorado, USA 3Unversty of Helsnk, Helsnk, Fnland (Dated: 30 May 01) Juss Lenonen 1. Introducton Applcaton of deconvoluton to weather radar mages has remaned a challengng problem. Deconvoluton algorthms tend to amplfy measurement nose, whch makes straghtforward nverson of the convoluton process mpractcal. A number of algorthms have been publshed to mprove radar mage qualty n the radal drecton from the radar, but success n tangental deconvoluton has been lmted. Wth tradtonal mages such as photographs, deconvoluton has been studed extensvely. However, most mage deconvoluton algorthms cannot be appled on radar mages n a straghtforward manner as the algorthms are typcally formulated by assumng addtve nose, whch s not applcable for weather radars n whch the sngle-pulse statstcal nose s multplcatve (as we shall explan below). Stll, we can dentfy a few general prncples that have led to success n varous types of deconvoluton algorthms. Frstly, t has been understood that the statstcal model of the mage nose should be accurate n order to properly understand the effect of nose on the resultng mage. Secondly, the ll-posed nature of the problem should be dealt wth by applyng a regularzaton that weghts the result towards more realstc mages. Although the dfferences between radar mages and photograph-lke mages necesstate a somewhat dfferent approach for radar mage processng, the above mentoned prncples do not present any nsurmountable obstacles for radar deconvoluton. For radar mage nose modelng, a partcular model of mage nose, exponentally dstrbuted multplcatve nose for snglepulse powers, has rather strong theoretcal justfcatons supportng t; ndeed ths s more than can be sad for many photograph nose models. The regularzaton can be approached n several ways, and there s no such strong theoretcal support for any partcular model, but on the other hand, the dfference from photographs s perhaps smaller than n the case of nose. Thus, these approaches seem justfable n weather radar deconvoluton as well. A general and statstcally justfed method to combne the nose model and the regularzaton s offered by Bayesan probablstc treatment. We have developed a deconvoluton algorthm for weather radar mages that s based on rgorous Bayesan treatment of the propertes of the mage, and the process of deteroraton through convoluton and nose. Typcally for Bayesan formulaton, the a posteror probablty s wrtten n terms of condtonal and a pror probabltes. The condtonal probablty s formulated by requrng that the probablty dstrbuton of the multplcatve nose s the exponental dstrbuton; ths property s true for the sngle-pulse powers receved by the radar. The sngle-pulse powers are requred by the algorthm as nput, necesstatng the use of. The a pror probablty s based on the propertes of the spatal varablty of ranfall, and restrcts the gradent of the logarthm of pulse power, resultng n an effectve "requested sharpness" parameter for the algorthm. From the Bayesan probablty, a cost functon s derved; ths can be mnmzed usng a standard multvarate optmzaton algorthm. We tested the result wth radar RHI scans and observed a sgnfcant sharpenng of the mages.. Formulaton of the problem Consder an deal radar reflectvty feld, whch we shall call x. Ths s the mage that would be seen by a radar that has a perfect pencl beam and a neglgbly short pulse length. Manly due to the statstcal varaton of the radar return sgnal and the nonzero wdth of the beam from the radar antenna, a real radar wll see a degraded verson of ths mage, called y here. In ths secton, we shall descrbe the process that forms the mage y. When a weather radar scans, the underlyng true mage s spatally averaged (blurred) by the square of the antenna pattern. Ths results n a convoluton n the polar coordnates. Mathematcally, the expected value of the radar pulse power y s y ( ϕ)= x ( ϕ ϕ' ) g ( ϕ' ) dϕ' (1) where x s the true mage, g=f s the square of the antenna pattern f, and ϕ s the polar angle. The mage s also blurred by the radar pulse shape n the radal drecton, but we neglect ths effect n ths study. The convoluton can be dscretzed and wrtten n lnear algebrac form as y=gx ()

2 where G s a Toepltz matrx (or more generally, a lnear operator) whch produces a convoluton by f on a vector multpled by t. The actual receved power s randomly determned from an exponental dstrbuton wth the gven expected value. The latter effect s explaned as follows: snce the number of scatterers n a meteorologcal target s large, the central lmt theorem apples such that the receved real and magnary phasor voltages I and Q are normally dstrbuted. The power from the pulse s I +Q, and the sum of the squares of two normally dstrbuted random varables s well known to be exponentally dstrbuted. The actual realzed value y of the receved power from a sngle pulse s a random varable wth probablty densty functon (PDF) p ( y)= 1 y ( ϕ) exp ( y y ( ϕ ) ). (3) As ths PDF only depends on the rato of the realzed value and the expected value, the statstcal varaton of the sgnal can be seen as multplcatve nose, and thus the one-pulse mage formaton process can be wrtten as y=gx n (4) where n s a sampled an exponental dstrbuton wth a rate parameter of Algorthm We attempt to solve the nverse problem of restorng the deal mage x from the degraded mage y by wrtng the probablty of x, gven y and some assumptons about the nature of x, usng Bayes' theorem as. p (x y)= p (y x) p (x). (5) p (y ) A soluton that maxmzes ths condtonal probablty s called the maxmum a posteror (MAP) estmate. The condtonal probablty p(y x) can be formulated n terms of the nose. From (4), we see that Gx =n. (6) y Snce the PDF of n s known, we can wrte (for N dscrete samples). p(y x) exp( y. (7) The problem wth ths defnton s that the modal and expected values of the exponental dstrbuton are very dfferent (0 and 1, respectvely, n ths case), whch can cause problems when usng the MAP estmator and amng to fnd the expected value. Also, the PDF s dscontnuous at 0, whch s problematc partcularly for unbounded optmzaton methods. We can work around the problem by usng the condtonal dstrbuton problem by usng the condtonal probablty of the logarthms nstead. It can be shown (we omt the proof here) that p (ln y ln x) exp( ( ln y (Gx ) ) y ) (Gx ) (8) and ths expresson s adopted as our condtonal probablty. For the pror p(x), we need to assume somethng about the nature of x. It s a common assumpton that meteorologcal targets exhbt scale-free varablty statstcs. Ths means that the logarthm of the ran rate can be modeled as a Gaussan random walk wth a constant varablty. Snce the logarthm of the receved radar power s approxmately lnearly related to the logarthm of the ran rate, ths assumpton should apply to the receved power as well. Wth ths reasonng, we construct the pror n terms of the spatal rate of change n ln(x), wth p (ln x ) exp( D r ln (x ) σ r (Gx ) ) D ϕ ln (x ) ) σ (9) ϕ where D r and D ϕ are ''dfferencng'' matrces whch gve the dfference of adjacent values n the radal and tangental drectons, respectvely. Ths pror also allows us to convenently operate wth ln(x), as wth the condtonal part. The parameters σ r and σ ϕ should be as close to the real varablty of the measured target as possble. In practce, small values of σ r and σ ϕ lead to softer mages, whle large values produce sharper results but can also ntroduce artfacts. Thus, these can be seen as a ''requested sharpness'' parameters. Our optmal estmate s the x that maxmzes the probablty p(x y). Snce we can equally valdly maxmze the probablty of ln(p(x y)), the deconvoluton becomes a cost functon mnmzaton problem where the cost functon E s

3 y E= ln( (Gx ) ) y + D r ln (x ) + D ϕ ln ( x ) (Gx ) σ r σ ϕ, (10) whch can be mnmzed wth respect to x wth standard optmzaton methods. 4. Experments For testng the deconvoluton, we used test data from a case observed usng the Unversty of Helsnk Kumpula radar on March 19, 011. From the orgnal RHI scan, we selected a smaller part of 30 range gates from each of 1500 pulses, at roughly km dstance and an elevaton angle of The range was convenent for the beamwdth of 1.05 to ntroduce a suffcent level of blurrng to the meteorologcal features. Addtonally, there was a pont target (probably an arplane) n ths part of the scan, whch presented a good test for deconvoluton. We appled the MAP method usng varous parameters for σ r and σ ϕ. Lackng an accurate measured antenna pattern, t was assumed to be Gaussan, and varous standard devatons of f were tred. The best parameters for ths case were found wth some expermentaton to be σ r 3.1 and σ ϕ 0.36, usng a standard devaton of 1.05 for f. The results of applyng the MAP method are shown n Fgs The results of usng 64-sample ntegraton to produce a standard radar mage are also shown. The examples show that the deconvoluton method seems to add a sgnfcant vsual mprovement to the qualty of the radar mage. The meteorologcal targets on the center rght and the bottom of the mages are denosed and sharpened at the edges; the wdenng of objects ntroduced by the convoluton s also removed from the center rght target. The object n the center of the mage that was assumed to be a pont target has collapsed nto a much more compact and ntense target. It s encouragng to note that the method works even though the sgnal from the pont target (whch consttutes only a small part of the whole mage) does not follow the exponental multplcatve nose assumpton. Another nterestng fndng was that even though the convoluton only operates n one dmenson, the method works better for two-dmensonal mages than t does for one-dmensonal tme seres. Thus, processng the whole mage at once yelds better results than processng each range ndependently. An ntutve explanaton for ths s that n a -dmensonal mage, there are more neghborng pxels avalable from whch to recover a nose-corrupted pxel by usng the pror. Fg. 1 Sngle-pulse powers from a pont target and meteorologcal targets n an RHI scan of the UH radar. The coordnates are n klometers, and Cartesan wth the horzontal and vertcal axes defned as those at the radar ste (thus gnorng the curvature of the Earth).

4 Fg. The radar mage of Fg. 1 after 64-pulse averagng. Fg. 3 The radar mage of Fg. 1 after deconvoluton usng the MAP method. 5. Concluson The MAP deconvoluton method seems to be able to provde a resoluton enhancement that could be useful n usng radar to study targets where hgh resoluton s essental. In fact, snce t operates on one-pulse power data, t can be sad to combne deconvoluton, denosng and superresoluton methods n nvertng the mage degradaton process. The falure of the conventonal deconvoluton methods to tackle the problem hghlghts the need for talored algorthms for the weather radar deconvoluton problem. Fndng optmal methods for determnng the values of the free parameters of the pror, σ r and σ ϕ, s essental for makng the MAP method more wdely applcable. Ideally, they should be derved from a physcal bass, usng the horzontal and vertcal rates of varaton as the startng pont. It should also be possble to derve the parameters as a functon of the dstance between adjacent samples by dervng statstcal relatons from large quanttes of measured data. At the current state of the algorthms, however, the parameters have to be found expermentally. The computatonal requrements for the MAP method are also heavy n the current mplementaton. Enhancng the 1500x30 pont mage used here takes roughly mnutes on a desktop computer, so t would be desrable to fnd an optmzaton method that s talored for ths problem, rather than usng the general L-BFGS method. Other than that, the computaton can be effcently parallelzed, whch allows for code optmzaton that was not used n ths study. Acknowledgment We would lke to thank Dr. Matt Lesknen for provdng us test data for the deconvoluton method. References Brng V., Chandrasekar, V., 001: Polarmetrc Doppler weather radar: prncples and applcatons. Cambrdge Unversty Press. Rchardson W., 197: Bayesan-based teratve method of mage restoraton. J. Opt. Soc. Am., 6,

5 Shan Q., Jaya J., Agarwala A., 008: Hgh-qualty moton deblurrng from a sngle mage, ACM Trans. Graphcs 7.