相位梯度自聚焦算法

【实例简介】
相位梯度自聚焦算法论文，原作者所写，毕业论文里的引文用的就是这篇
blurring function. Proper alignment of these low breaks down. This is because the low contrast contrast edges is crucial to maximizing the accuracy of point and edge responsc functions do not risc he phase-error estimation process. Circular shifting, sufficiently above the surrounding clutter to allow however, is only one part in the process of robust proper thresholding of s(x). We have found that a phase-error estimation and must be complemented by progressively decreasing window width works quite the operations that follow is selected to span the maximum possible blur width B. Windowing (which could be several hundred samples for extremely defocused imagery) and progressively reduced by 20%a The next important step is windowing the for each iteration circularly shifted imagery. Windowing has the effect of preserving the width of the dominant blur for C, Phase Gradient Estimation each range bin while discarding data that cannot contribute to the phase-error estimation. This allows After the image data is circularly shifted and the phase-error estimation to proceed using input data windowed, the phase gradicnt is estimated. Let us having the highest signal-to-noise rati denote the shifted and windowed image dala as gn(x) The question remains as to how to choose the and Gn(u)=G,(ulexpgIPE(u)+Bn(u)d be the proper window size w, i.e., how to distinguish between inverse Fourier transform. The scatterer-dependent that data associated with the dominant blur and the surrounding clutter. For images containing many strong It has been shown [10] that a linear unbiased). phase function for each range bin is denoted by Bn (u) non-clutter-like)scatterers such as buildings, vehicles, minimum variance (LUMv) estimate of the gradient metallic structures, etc., the signal-to-clutter ratio even of the phase error, pe(u), is given by, for severely blurred scenes is large enough to allow automatic window determination. To do automatic windowing we again exploit the fact that thc phasc ∑,|Gn()2 error we seek is redundant in the image. Thus, the brightest scatterer on cach range bin is subjected to an =e(4)+ Gn(u)/26 identical blurring function. We can therefore estimate Gn(u)l2 the width of his blur W, by registering the dominant Thus, this LuMv estimate yiclds the gradient of the blur on each range bin and averaging over the bins true phase error, plus an error term that has been Since the circular shifting operation already performed made as small as possible(at this step) by the circular the registration, we just need to incoherently sum shifting and windowing operations over the range bins to obtain a one-dimensional The PGia phase gradient estimation kernel is function whose width at some chosen level adequately defined as the second term in the above equation captures the support of the point spread function. This kernel is only one of a number of phase gradient Mathematically, this one-dimensional function is phase derivative, or phase difference estimation computed as formulas. For example, another kernel used to estimate s(x)=∑(x) )phase differences is given by where fn(r)is the circularly shifted image △=ag∑F(+△n Because of the registration, s(0)will be maximum over x, Furthermore because of the This kernel is found in the Knox -Thompson method redundancy of the blur function, s(r) will typically for stellar speckle interferometry and in the shear exhibit a plateau approximately w in width and be averaging technique for SAR phase-error correction significantly smaller outside this region. Thus, w can 5]. It has been shown that this kernel is a maximum be estimated from s(r) using a variety of methods likelihood estimate of the phase and does work well Wc havc choscn to mcasurc w by thresholding s(x) when used within the full PGa construct ll. Without at the point 10 dB down from its peak, S(0), then the processing steps of center shifting, windowing and increasing this width by 50%. The window width will iteration, this kernel as well as the PGA kernel will not decrease for subsequent iterations because the image is uniformly produce excellent results over a variety of becoming more focused, The progress of the automatic scene content and phase-error functions window can be monitored as a means for assessing convergence. At convergence, W is typically a few (i.e, D. iterative Phase Correction 5-6) pixels wide When the scene to be focused has low dynamic The estimated phase gradient, lumy(u) is integrated range and consists almast entirely of clutter-Iik to obtain e(u), and any bias and linear trend is objects, the automatic window determination method removed prior to correction Phase correction WAHL ET AL. PHASE GRADIENT AUTOFOCUS is imposed by complex multiplication of the TABLE I range-compressed phase-history domain data by Case Parameters, rms Errors and Associated Fig. Number exp[-joe (u)]. The estimation and correction proces is repeated iteratively. As the image becomes more Case focused, individual scatterers become more compact, Number C.s. Wind Iteration (rads) Fig the signal-to-clutter improves, the circular shifting 1 Y 1(c) more precisely removes the Doppler offsets, and the 234 YNY Y 11.2 algorithm is driven toward convergence. Removal 031(f of any linear trend in the phase-error estimate prevents image shifting and the bias rcmoval allows computation of the rms phase error removed at each reducing the window size by 1/3 at each iteration iteration as a means for monitoring convergence. Our Finally, Case 4 implements the full PGA algorithm. experience has shown that when the rms error drops to The original and the phase-corrupted images a few tenths of a radian, the image is well focused and are shown in Figs. 1(a)and(b), respectively. The will not improvc with additional iterations focused result of the algorithms described in cases 14 are shown in Figs. 1(c)(). By visually comparing the focused rcsults. it is clear that the best result is III. PROOF OF NECESSITY produced by the algorithm using all the PGA steps. The critical steps of the PGA algorithm consist subtracted from the estimated phase error for each w Additionally, an rms value of the original phase en center shifting, windowing and phase gradient case was calculated. The rms value of the applied estimation. These steps are repeated iteratively until pl phase error is 5.61 rads. These results are shown in the algorithm converges to the true phase error. This Table I section illustrates the necessity of all thesc steps by The rms errors and the visual comparisons of howing the effect of the PGA focusing algorithm when the focused results indicate that eliminating any of any of these critical steps are eliminated. By doing the steps of the PGA algorithm produces an inferior this experiment, we show by counter example that the quality focus. These experiments show that for at least quality of the resultant image is inferior to that which one SAR image, any algorithm claiming computational can be achieved by using the full PGa process. advantage over PGa by essentially eliminating one We approach the problem by focusing an image or more steps of PGa produces rcsults which are with only one of the critical steps eliminated from suboptimal. Although these results are specific to one the PGa algorithm. The focused result can then be image, it has been the authors experience that all the compared with the result using the full PGA algorithm. steps are required in order to achieve high quality There are three possibilities using this approach: PGa focus on the vast majority of SAr images without center shifting, PGA without windowing, and PGA without iterating.(Although phase gradient ∨. ROBUSTNESS OF PGA estimation is stated as one of the steps of PGA, it is the heart of this phase correction process and thus t is of course impossible to predict and test every cannot be eliminated as one of the possibilities in our conceivable phase-error function and scene content experiment. combination to demonstrate the infallibility of any a typical SAR image was selected for the example. autofocus algorithm. The authors, however, feel the The image is that of a rural scene with no apparent PGa algorithm is exceedingly robust over a wide ra cultural targets. The image was artificially corrupted by of applications. This confidence is born of analyss nge a tenth-order phase error using optimal estimation theory [10], as well as its The following four examples illustrate the effect application to a large number of fielded SAR systems of implementing a subset of the PGa algorithm on by the authors and others. In this section, we present this phase corrupted image. Case 1 implements the the results of applying PGa to a variety of scenes and PGA algorithm without iterating. In other words, one phase-error functions to demonstrate this robustness pass was done which included center shifting and As in the previous section, all of these examples were windowing at 1 3 the aperture size before the PGa generated using actual SAR imagery with synthetic kernel was uscd to estimate the phase errors. Case 2 phase-error functions applied deletes the windowing step from the PGa alg It has be that Pga is a discrete This case was implemented by leaving the window point-type algorithm generalized to average over many wide open. At each iteration, the brightest point on such points. Indeed, the analysis cited above models each range bin was circularly shifted to the center of a Sar image as a collection of such points for the the image before using the PGA kernel to estimate the sake uf mathematical tractability. However, the PGA phase error. In case 3, the circularly shifting process algorithm has demonstrated its ability to focus an is removed. The windowing was implemented by mage in the complete absence of dominant reflectors IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL 30, NO 3 JULY 1994 (a) Fig 3. SAR image of urban scene with dense set of large cross tion targets.(a) d with 10th- ordcr pal ial phas error.(b)Corrected using PGA Fig. 4. SAR image of low dynamic range scene. (a) Corrupted by largc magnitudc phase crror function with power law second -order statistics.(b) Corrected using PGA Convcrscly, dcns collections of scattcrcrs with ws intermingled sidelobe structure can be handled as well ) That is, if large magnitude scatterers are present, they need not be isolated In Figs. 3(a)and(b), we show the Fig. 1. SAR image uf fiel and trees.(a) Original, undegraded. results of applying pga to an urban sccnc with a dense (b) Corrupted with 1Oth-order polynomial phase crror. (c) Case 1 no iteration. (d) Case 2, no windowing(e)Case 3, no circular set of large cross section targets shifting.()Case 4, full PGA algorithm The above three examples possessed tenth-order polynomial phase-error functions(see Fig. 6) This type of phase error might be expected as a residual after the data has been motion compensated (although typically not this severe). Propagat induced phase errors such as that generated by a turbulent ionosphere or troposphere, however, can s havc much highcr frcqucncy components [11] lIs Case, It is imperative to use a nonparametric phase-error estimation technique such as PgA. due to the exponential modulation, large magnitude high frequcncy phasc crrors causc scvcrc spreading of thc Fig. 2. SAR image of rural scene with no cultural, metallic, or impulse response specular targets. (a) Corrupted with 10th-order polynomial phase The next example is particularly difficult from error.(b) Corrected using PGA the standpoint of phase correction. A low dynamic range scene has been corrupted by a large magnitude with sidelobes above the surrounding clutter. Fig. 2 phase-error function with power law second-order is such an example. The underlying image is one of statistics(see Fig. 7). Thus, use can be made of a rural scene with no cultural, metallic, or specular neither a dominant scatterer nor smoothing by fitting targets whasoever. Were it not for some tree shadows, a low-order polynomial to the error function. The this scene would be almost featureless, As can be efficacy of PGA, however, can be seen in Figs. 4(a) seen from Figs. 2(a)and(b), PGA has no difficulty and (b) removing the substantial phase-error function from this The last example is of largely academic interest Practical SAR systems are designed with adequate WAIL ET AL: PHASE GRADIENT AUTOFOCUS 831 (b) APERTURE POSITION Fig. 7. Power law phase error function uscd to produce corrupted n in Fig. 4(a) (c) Fig. 5. SAR image on urban scene. (a) Corrupted by white phase crror function.() Corrected using PGA(c Original undegraded Image 6~0Q40 12 APERTURE POSIT。N Fig. 8. White phase error function used to produce corupted 34 A户 ER TURE P。sIT1。N Fig. G. Tenth-order polynomial phase error function used to the estimated phase-error function turns out to b produce corrupted images shown in Figs. 1(b), 2(a), and 3(a) function with the same principal values of phase as the original error function and slopes less than pi radians pcr samplc. Thus, whilc it is not the same function as motion compensation/phase stabilization so the phase that impressed on the image, after the phase wrapping errors al adjacent azimuth samples will not exceed pi due to e/f, it has the same effect on the image. To radians. However, a question arises as to what would put it another way, given any aliased, random phase happen if a white phase error function were applied function, it is possible to find an unaliased phase to an image. This function would clearly exceed pi unction with the same principal values. 'The Pga radians at many adjacent azimuth samples producing algorithm converges to this function. Fig. 5(a) depicts a spread of the IPR across all azimuth samples. That an urban scene that has been corrupted by a white is, neglecting range walk(which cannot be corrected phase-error function(scc Fig 8 ) As cxpcctcd, thc IPR by autofocus), what are the consequences of a random, has been blurred the entire width of the image. The white error function? It has been suggested that PGA, PGA restored image is shown in Fig. 5(b). This image with its differential-based estimation kernel, could not appears to be about as well focused as the origina handle such a situation. This is not true. In this case, (Fig. 5(c)) IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL 30, NO, 3 JULY 1994 V COMPUTATIONAL CONSIDERATIONS sample"image"so constructed becomes the input image to the PGA algorithm, amounting to a 400: 1 Upon initial inspection, thc PGa algorithm appcars data reduction at the front end to be burdensome from a computational standpoint. This construction may also be accomplished from It is iterative, requires Fourier transforms between the image and range-compressed domains for each compression)in several ways. A set of range bin.e the range-compressed domain(prior to cross-range iteration, and such processing must be performed over be selected based on a rank ordering of total energy many range bins to take advantage of the redunda in each bin, or by compression of a fraction of the in the data that is the key to the success of the range-compressed data to obtain a low cross-range algorithm resolution image. The appropriate cross-range samples However, in this section we show that a variety may then be obtained by compression of only those of processing expediencies may be employed to selected range bins dramatically improve throughput. In fact, it Thc third computation-saving technique is shown that the processing burden is essentially fixed embeddcd in the PGa algorithm itsclf. Since, at every independent of image size, and as such represents a iteration, many crOss-range samples are discarded by proportionately cvcr decreasing computational load the windowing operation, it makes sense to perform as imagc sizc grows. Thus, in thc authors'cxpcricncc, FFTs on vectors of length equal to the number of PGA represents a tiny fraction of image formation non-zero samples retained by the window. This has a time for large images dramatic effect on the later iterations as the window In what follows, we consider only the phase-error sizes become quite small. Of course, the estimated stimation portion of the problem. The final correction phase-error vector must be up-sampled to the full of the data is generic to the autofocus problem in cross-range dimension to be applied to the range SAR, and as such, is a fixed computational burden compressed data at each iteration. also, the final independent of PCA. Further, we consider here only correction vector obtained from this 500 x 500 sample the case of large problem sets. For small SAR images, image must again be up-sampled to the original image the economies to be described here would have little impact. In order to fix these ideas, let us exaggerate If we accept the hypothesis that an input image to the point and consider by way of example a very large PGa has an uppcr bound of say 500 range samples by SAR image of 10 K x 10 K samples 500 crOss-range samples(exact numbers dcpcnding on There are three fundamental steps that may be system design), then it is clear that the computational taken to reduce the computational burden of PGa burden of PGa does not grow with actual inage size They are: 1)limiting the estimation to a reduced above these numbers. On the other hand, virtually number of range bins, 2)limiting the estimation to a every other processing step in SAR image formation reduced number of cross-range bins, and 3)utilizing (motion compensation, polar formatting, range and variable-length fast Fourier transforms (FFTS) cross-range compression, and geometric correction) Consider our hypothetical 10K X 10 K sample does grow linearly (or faster) with image size. It is thus image, and suppose for the moment that the data easily seen that, while PGa represents a substantial have been fully compressed to the image domain. It is fraction of image formation for small images, it unlikely that all 10 krange bins are required to make becomes inconsequential for large images. an excellent estimate of the phase-error function In fact, our experience has demonstrated that estimation VL SUMMARY over 500 well chosen range bins is adequate in virtually every circumstance. Likewise, even the most severely The PGA algorithm for phase-error correction degraded or poorly motion-compensated image is in sar was introduced in 1988. Since that time likely to suffer less than 500 samples of cross-range it has bccn provcn to bc a robust technique that impulse response smearing. While this is certainly can provide excellent rcsults ovcr a widc variety of subject to various system dcsign paramcters, the vast both scene content and structure of the degrading majorily of SAR systems will nevertheless have an phase-error Tunction. Indeed, PGA is capable of bound on Ipr si defocused SAR to their designed Consider, then, constructing an image consisting of quality specifications in nearly all instances. The 500 range samples by 500 cross-range samples taken uccess of the technique relies upon four fundamental from the original 10 K x 10 K image as follows. Locate signal processing steps. These include center shifting, the 500 highest magnitude samples in the original windowing, phase gradient estimation, and iterative image. Place these 500 samples in the central column correction. The question has been raised, however, of the 500 x 500 image. Finally, fill in the samples as to whether or not these four elements are in fact to the left and right of this central column with the crucial to the robustness of the algorithm. Several corresponding adjacent samples from the original techniques now exist that utilize a subset of the image, circularly buffering as required. This 500 x 500 PGa algorithmic steps We demonstrate here that WAHL ET AL: PHASE GRADIENT AUTOFOCUS 83 omission of any of these basic clements from the PGa [5] Ficnup, J (1989) construct has deleterious effects on the final product Phase error correction by shear averaging a result, we suggest that no shortcuts to PGa are Technical Digest Series, 15, Signal recovery and synthesis II appropriate, since use of the complete Pga technique (June1989),134-137 Altia, E H, and Steinberg, B D.(1989) will essentially always deliver nearly diffraction-limited IEEE Transactions on Antenna Propagation, 37(1989) imagery. We then show that the computational load of PGA is really not a very large fraction of the complete [7 Press, W(1991) image formation process when mid to large size images Recovery of sar images with one dimension of unknown ha are involved finally the digital sar data sets used JASON report JSR-91-175. here to demonstrate our assertions are offered to [8] Barbarossa,S, and Farina, A( 1992) other researchers at no cost, in order to stimulate Detection and imaging of moving objects with SAR, further research in this area. The use of a common Part 2; Joint time-frequency analysis by Wigner-Ville set of realistic SAR images should facilitate useful distribution (WD). comparative performance studies JEE Proceedings, PL. F, 139, 1(Feb. 1992) [9 Barbarossa, S(1990) New autofocusing technique for SAR images based on the REFERENCES Wigner-Ville distribution. Electronics Letters, 26, 18(Aug 1990) [10 Eichel, P,, and Jakowatz, C. Motion compensation for SAr Thase-gradient algorithm as an optimal estimator of the TEEE Transactions on Aerospace and Electronic Systems phase derivative. (May1975) Optics Letters, 14, 20(Oct. 1989) 2] Brown, W.D., and Ghiglia, D. C(1988) [11] Jakowatz, C,Jr, and Wahl, D.E(1993) Some methods for reducing propagation -induced phase ging systems, Part I, Fo Eigenvector method for maximum-likelihood estimation of Phase errors in synthetic-aperture-radar imagery Journal of the Optical Society of America, 5( June 1988) Journal of thc Optical Saciety of America, 10(Dec. 1993) [3] Ghiglia, D. C, and Brown, W D(1988) [12] Jakowatz, C, Jr, Eichel, P, and Ghiglia, D.(1989) Some methods for reducing propagation-induced phase Autofocus of SAR imagery degraded by errors in coherent imaging systems, Part I, Numerical ionospheric -induced phase error res sFE,1101(1989 Journal of the optical society of america, 5 June 1988) [4 Eichel, P, Ghiglia, D. Jakowatz, C, Jr( 1989) Speckle processing method for synthetic-aperture-cadar phasc correction Send postage prepaid mailing envelope with a QIC(1/4 "cartridge) tape to the authors at the above address IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL 30 NO. 3 JULY 1994 Daniel E. Wahl received his B.S. and M.S. degrees in electrical engineering from Colorado State University, Ft Collins, in 1981 and 1983, respectively and the Ph.D. degree from the University of New Mexico in electrical computer engineering in 1993 Since 1983 he has been a senior member of the technical staff at sandi National Laboratories, Albuquerque, NM. His current interests include digital signal and image processing, synthctic-apcrturc radar signal processing, and SONAR array signal processing Paul i. Eichel (S77-A79M85) received the B E. degree from Vanderbilt University, Nashville, TN in 1978, the M.S. degree from Stanford University, Palo Alto, CA in 1979 and the Ph. D. degree from the university of michigan, Ann Arbor, in 1985, all in electrical engineering From June 1978 through August 1981 he was a member of the technical staff at Bell Laboratories, Holmdel, NJ. Sincc July 1985, Dr. Eichcl has been with the Sandia National Laboratories in Albuquerque, NM where he is a distinguished member of the technical staff. his activilies and interests include thevretical and experimental aspects of modulation and channel coding, digital signal processing, and synthetic aperture radar. Dr. Eichel is a member of Tau Beta Pi, the IEEE Information Theory Group, and the IEEE Acoustics, Speech, and Signal Processing Society Dennis C. Ghiglia received the bachelors degree in electrical engineering at California State Polytechnic University, Pomona, in 1969 and the M.S. and Ph. D. degrees in electrical engineering from Arizona State University, Tempe, in 191 and 1976, respectively He is a distinguished member of the technical staff at Sandia National Laboratories, Albuquerque, NM, where he has been employed since June, 1978 He has been responsible for the development of the Image Processing Facility for the centralized support of a wide range of image processing activities. Before joining Sandia, he was cmploycd by Goodycar Acrospace near Phoenix, AZ, where he did signal processing, image processing, and systems anal lysis fo synthetic-aperture radars. His current research interests lie in the area of image restoration, synthetic-aperture radar signal processing, and phase problems in signal and image pr g Dr Ghiglia is a member of the Optical Society of America Charles v, Jakowatz, Jr (S"75--M'90)was born in 1951 and received the B.S., M.S., and Ph. D. degrees, all from the School of Electrical Engineering at Purdue University, W. Lafayette, IN, in 1972, 1973, and 1976, respectively He is currently Manager of the Signal Processing Research Department at the Sandia National Laboratories in Albuquerque, NM, where he has been employed since 1976. During his tenure at the national laboratory, his chief research interests have been in digital signal processing, computed imaging, as well as detection and estimation theory. During the past six years, he has worked in r&d for synthetic aperture radars WAHL ET AL. PHASE GRADIENT AUTOFOCUS 835 【实例截图】
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