Application of Image Restoration Technique in Flow Scalar Imaging. Experiment

Transcription

1 Final Reprt Applicatin f Image Restratin Technique in Flw Scalar Imaging Experiment Guanghua Wang Abstract Center fr Aermechanics Research Department f Aerspace Engineering and Engineering Mechanics The University f Texas at Austin Austin, Texas Scalar imaging techniques are widely used in fluid mechanics, but the effects f imaging system blur n the measured scalar gradients are ften inadequately cnsidered. Depending n the flw cnditin and imaging system used, the blurring can cause unacceptable errrs in gradient related measurements, which are much larger than thse fr the scalar itself. Planar Laser-Induced Flurescence (PLIF) images f turbulent jet fluid cncentratin were crrected fr blur based n the Richardsn-Lucy Expectatin Maximizatin (R-L-EM) image restratin algrithm. This algrithm relies n the sht-nise limited nature f PLIF images and the measured Pint Spread Functin (PSF). The restred PLIF images shw much higher peak dissipatins and thinner finescale structures in the images, particularly when the structures are clustered.

2 . Intrductin The spatial reslutin f the ptical system is very imprtant fr flw imaging experiments and it depends n many factrs [], e.g. pixel size f the detectr array, depth f the cllectin ptics and magnificatin. In many scalar imaging experiments, the reslutin is usually quted in terms f the area that each pixel images in the flw. Fr an ideal ptical system r fr an ptical system used at high f# (f# = f/d, where f is the fcal length and D is the diameter f the lens) and magnificatin is clse t the design cnditin, reslutin is nearly diffractin limited. Hwever fr lw light level flw imaging experiments, such as Raman, Rayleigh and Planar Laser Induced Flurescence (PLIF) imaging, fast (lw f#) ptics are cmmnly used. Fr these experiments, pixel size is nt the nly factr that limits the reslutin. The imaging system blur shuld als be cnsidered, since the image is the cnvlutin f the Pint Spread Functin (PSF) with the irradiance distributin f the bject. The smallest bjects that can be reslved are related t the size and shape f the PSF. PSF als tends t prgressively blur increasingly smaller structures. This is essentially a result f the system s inability t transfer cntrast variatins in the bject t the image. Depending n the flw cnditin and imaging system used, the PSF culd be f the same rder as the characteristic length scale f the scalar structures in the flw field, which may cause unacceptable errr in the scalar measurement. T represent the TRUE scalar structure statistics, e.g. thickness, dissipatin and Prbability Density Functin (PDF), it is necessary t restre the flw scalar experimental images. This prject will intrduce image restratin technique in flw scalar imaging applicatins and fcus n hw scalar measurements are affected. PLIF images f turbulent jet fluid cncentratin were crrected fr blur based n the Richardsn-Lucy Expectatin Maximizatin (R-L-EM) image restratin algrithm. The sht-nise limited nature f PLIF images and the measured PSF are used t get reliable restratin. 2. Backgrund The general mdel [3-9] fr a linear degradatin caused by blurring and additive nise is i ( = h( ( + n( () f 8

3 where i ( is the blurred and nisy bserved image crrespnding t the bservatin f the true image (, h( is the blurring functin r PSF f the imaging system, is the cnvlutin peratr, n ( dentes the additive nise such as the electrnic r quantizatin nise invlved in btaining the image. In Furier dmain, the degradatin is I ( u, = H ( u, O( u, + N( u, (2) where I ( u,, H ( u,, O ( u, and N ( u, are the cntinuus Furier transfrms f i (, h (, ( and n ( respectively and u and v are the spatial frequencies. The purpse f restratin is t determine ( knwing i ( and h (. This inverse prblem has led t a large amunt f wrk. Main difficulties are cming frm the additive nise [3-9] and the PSF [7-9]. The mdeling f blurring can be divided in tw parts: blurring functin and nise mdeling. Sme ideal PSF mdels are Gaussian, ut-f-fcus and linear mtin blur [3]. In astrnmy, data extracted frm clear stars in bserved image is used t fit a synthetic PSF functin by weighted least square methd [7, 8]. The PSF measurement techniques are als discussed in []. The diversity f algrithms [3-9] develped nwadays reflects different ways f recvering a best estimate f the true image. Wiener and regularized filters are better fr knwn PSF and additive nise [3, 4]. Sme iterative restratin techniques [], e.g. Expectatin Maximizatin (EM) algrithms, wrk better fr knwn PSF and unknwn additive nise. Blind image restratin algrithms [5, 6] are mre prper fr unknwn PSF and additive nise. Flw imaging experiments have a lt in cmmn with astrnmy bservatins. They bth invlve lw light level imaging. In bth cases images are degraded by imaging ptical system and suffer frm signal dependent nise (Pissn nise), CCD camera read-ut nise and quantizatin nise. These physical similarities suggest that a better starting pint in applying image restratin techniques in flw scalar image restratin is t cnsider thse successful nes in astrnmy. 3. Richardsn-Lucy Expectatin Maximizatin (R-L-EM) algrithm The Richardsn-Lucy algrithm ([, 2]), als called the expectatin maximizatin (EM) methd, is an iterative technique used heavily fr the restratin f astrnmical images in the presence f Pissn 2 f 8

4 nise [7-9]. It attempts t maximize the likelihd f the restred image by using the Expectatin Maximizatin (EM) algrithm. The EM apprach cnstructs the cnditinal prbability density [7, ] p ( i) = p( i ) p( ) p( i) (3) where p (i) and p () are the prbabilities f the bserved image i and the true image respectively. Here p ( i ) is the prbability distributin f bserved image i if were the true image. The Maximum Likelihd (ML) slutin maximizes the density p ( i ) ver ML = arg max p( i ) (4) where argmax means the value that maximizes the functin. Fr true image with Pissn nise p( i ) = Π e ( h )! ( h ) y i The maximum can be cmputed by setting the derivative f the lgarithm t zer i (5) ln p ( i ) = (6) The EM algrithm cnsists f tw steps []: expectatin step (E-step) and maximizatin step (Mstep). These tw steps are iterated until cnvergence. In practice, they are usually cmbined tgether t reduce the strage f results frm E-step. Assuming the PSF is nrmalized t unity, a typical iteratin is where h ( = h( ( k + ) i( ( = ( h ( h( ( ( ) and k ( is the estimate f the true image ( after k iteratins. (7) The image h( ( is referred t as the re-blurred image [3]. Figure is the flw chart f the basic iteratin prcedures f the R-L-EM algrithm [4] and it cnverges t the ML slutin fr Pissn statistics in the data. Cnstraints, e.g. nn-negativity (estimate f the image must be psitive), finite supprt (the bject belngs t a given spatial dmain), band-limited (the Furier transfrm f the bject belngs t a given frequency dmain) and lcal and glbal cnservatin f flux at each iteratin, can be incrprated in the basic iterative scheme. Every iteratin f the EM algrithm increases the likelihd functin until a pint f (lcal) maximum is reached. One way t suppress the nise amplificatin with increasing iteratins is t use the 3 f 8

5 ( k+ ) () ( k = i( ( = ( h ( h( ( ( ) k+ ( fllwing iterative scheme ( k ) { ( } ( = f (8) ( k + ) + where f is the prjectin peratr that enfrces the ( set f cnstraints n k + ) ( and sme frms f f are given in [7-9]. In rder t enfrce cnstraints, the prjectin and back-prjectin peratins can be perfrmed using the Fast Furier k = k + Transfrm (FFT). Each iteratin needs ne FFT in the prjectin step and ne in the back-prjectin step depending n hw many cnstraints are Fig. Flw chart f the R-L-EM algrithm [4] applied. Other cmputatins are vectr peratins that have little impact n CPU time. Thus the R-L-EM algrithm is essentially an algrithm with tw FFTs per iteratin. The stpping rule is ften based n the statistics f residual nise [4] Relative Errr = ( k + ) ( ( ( ε (9) where ε is a small number. There are many mdificatins and imprvements t vercme different drawbacks in the riginal R-L-EM algrithm, e.g. nise handling [7-9] and iteratin acceleratin (autmatic acceleratin [3]). 4. Simulatin Results PLIF is a well-develped technique and has been used extensively in studying nn-reacting flws []. The acetne PLIF images used here are frm Cnditin 2 in [2] fr a high Re jet flw and ne image is shwn in Figure 2. Here i( is the bserved PLIF image crrespnding t the jet fluid cncentratin field and is essentially sht-nise limited (Pissn nise) due t the high differential crss sectin ( -24 cm 2 /sr) and the large number f mlecules in the cld jet flw. Scalar dissipatin rate is an imprtant quantity in turbulent scalar flw thery and mdeling, which is defined as χ ( = D i( i( () 4 f 8

6 2.39E- - Nikn 5mm lens with f/2.8.9 SRF(x) Measured SRF(x) Curve fit LSF(x) LSF(x) SRF (x) E-2-8 Fig. 2 Acetne PLIF image [2] x (mm) Fig. 3 Measured SRF and LSF fr a Nikn lens f/2.8 [2].2 - Ttal Flux Rati Relative Errr Number f Iteratins (a) Relative errr vs. number f iteratins E-4 3.5E-4.3E-5.3E-5 (d) Restred dissipatin field 5 6 Dissipatin - Restred Dissipatin - Observed 5 Relative Errr (%).4 4 Dissipatin Number f Iteratins (b) Ttal flux rati at each iteratin (c) Scalar dissipatin field f Figure Pixel 8 9 (e) Crss-cut prfile f (d) at x= pixel Peak Index 4 5 (f) Relative errrs fr the peak dissipatin f (e) Fig. 4 Restratin results f the R-L-EM algrithm fr the acetne PLIF image in Figure 2 5 f 8

7 .7.6 Observed Thickness Restred Thickness (a).4.2 Observed Peak Dissipatin Restred Peak Dissipatin (b) PDF.3 PDF Dissipatin Layer Thickness (Pixel) Peak Dissipatin Rate Fig. 5 PDF fr (a) Dissipatin layer thickness, (b) Peak dissipatin rate where D is the mass diffusivity and is set t ne here t simplify the analysis. The 2% dissipatin layer thickness is defined as twice the distance frm the peak dissipatin t where the dissipatin falling t 2% f the peak value. It is bvius that any change in peak dissipatin will als affect the layer thickness. The prblem is t restre the true cncentratin frm the bserved i and study the influences f image restratin n the scalar dissipatin rate and dissipatin layer thickness. The PSF is measured by the scanning knife edge technique in [2]. The Step Respnse Functin (SRF) is actually measured and differentiated t get the Line Spread Functin (LSF). Figure 3 shws the measured SRF and LSF fr a Nikn 5mm lens f/2.8 with a back-illuminated CCD camera (Crycam S5 series). The PSF used here is cnstructed frm the measured LSF by assuming an istrpic PSF. The iteratin stps when the relative errr is less than.2 r the maximum iteratin number f 5 is reached. The relative errr nrmally drps rapidly in the first few iteratins and slws with the increasing restratin. Figure 4 (a) shws the path f the relative errr frm a 382-iteratin R-L-EM restratin n a PLIF image in Figure 2. The cnservatin f ttal flux is illustrated in Figure 4(b). The ttal flux rati is defined as Ttal Flux Rati = ( k + ) y y ( i( () Maintaining the cnservatin f ttal flux is imprtant fr flw imaging applicatin since the restratin shuld nt alter the ttal number f phtns detected. The peak dissipatin in the restred field is higher and the dissipatin layer is thiner as cmparing bserved dissipatin field in Figure 4(c) and the crrespnding restred ne in 4(d). The peak dissipatin f the dissipatin layer centerline is significantly reduced by the imaging system blur as shwn in Figure 6 f 8

8 4(e), which are crss-cut prfiles at x = pixel frm Figure 4(c) and 4(d). The relative errrs between the bserved and restred peak dissipatin rate are shwn in Figure 4(f) and typically % relative errrs are bserved. The maximum relative errr ccurs when the dissipatin layers are thin and clustered tgether, which culd be f the rder f 6%, e.g. peak index 5 in Figure 4(e) and 4(f). The PDF f the dissipatin layer thickness and peak dissipatin rate, as shwn in Figure 5, are als generated by cunting 3 PLIF images. The restred layer thickness PDF shifts left cmparing with that f the bserved PDF, which means that the restred dissipatin layers are usually thinner than thse bserved layers. The restred peak dissipatin PDF shifts right with respect t that f the bserved PDF, which tells that the restred peak dissipatin rates are usually higher than thse bserved peak dissipatin rates. This clearly shws that imaging system blur reduces the peak dissipatin rate and brads the dissipatin layer. By the R-L-EM image restratin algrithm, t sme extend, the reslutin is imprved (we can measure thinner layers) and peak dissipatin rate measurement accuracy is imprved. 5. Cnclusins and future wrk Planar Laser-Induced Flurescence (PLIF) images f turbulent jet fluid cncentratin were crrected fr blur based n the Richardsn-Lucy Expectatin Maximizatin (R-L-EM) image restratin algrithm. This algrithm is mst cgnizant f the physical cnstraints n the PLIF images (sht-nise limited) and the measured PSF by the scanning knife-edge technique. The restred PLIF images shw much higher peak dissipatin and thinner fine-scale structures in the images, particularly when the structures are clustered. This is further verified by the right-shifting f the peak dissipatin PDF and the left-shifting f the dissipatin layer thickness PDF. These results illustrate the ptential f the R-L-EM algrithm t imprve the flw scalar imaging reslutin and gradient related measurement accuracy. Fr mst flw scalar imaging experiments, sequence shrt expsure f scalar images is usually recrded and multi-channel restratin technique can be used t get mre reliable restratins [5], especially when the PSF is prly knwn r unknwn. Multi-scale restratin techniques are als develped t restre astrnmy bservatin images, e.g. wavelet-lucy algrithm [7], which has a better nise handling capability. 7 f 8

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