Centroid Calculation Algorithm Using Weight Table to Increase Accuracy of Center Position Detection

Transcription

1 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. sts28, pp. Pj_1-Pj_8, 2012 Orgnal Paper Centrod Calculaton Algorthm Usng Weght Table to Increase Accurac of Center Poston Detecton B Yuta FUJII 1), Katsumasa IYATAKE 1), asak HARUNA 1), Kazuhde KODEKI 1), asateru NAGASE 1), Jro SUZUKI 1), Tatsuuk HANADA 2) and Shro YAAKAWA 2) 1) tsubsh Electrc Corporaton, Amagasak, Japan 2) Japan Aerospace Exploraton Agenc (JAXA), Tsukuba, Japan (Receved June 20th, 2011) In ths paper, a new algorthm that measures the centrod poston of a spot n the mage obtaned from an mage sensor wth sub-pxel accurac was presented. The lens forms a spot on the mage sensor, whch spreads over a few pxels on the mage. The algorthm s used to carr out centrod calculaton usng the brghtness values of the spread pxels, and t ncreases the accurac of the center detecton of the spot mage. In the calculaton, a weght table s used, whch has a crcular form wth a blurred border. The weght value s multpled b the correspondng brghtness value, whch further ncreases the accurac. An mage contanng a spot and random nose s produced n the smulaton. Centrod calculaton s carred out usng the produced mage, and the accurac of the centrod poston s evaluated. It s confrmed that the presented algorthm can detect the centrod poston wth an accurac of 0.02 = 1/50 pxel when the spot s defocused. It s 1.5 tmes more accurate comparng wth one of the tradtonal algorthms and the presented algorthm proves to be qute effectve. Ke Words: Free-Space Laser Communcaton, Optcal Antenna, Acquston, Centrod Estmaton, Trackng and Pontng 1. Introducton In the future, earth orbters, ncludng observaton satelltes and geostatonar satelltes, would be requred to transmt a large amount of data at a hgh speed rate. On the other hand, a conventonal rado frequenc (RF) sstem requres a larger communcaton devce for rapd transmsson, and the devces developed so far are approachng the speed-up lmtaton. To solve ths problem, laser communcaton between two satelltes, for example, a low Earth orbt (LEO) satellte and a geostatonar Earth orbt (GEO) satellte shown n Fgure 1 can meet the ncreased demand of hgh data volume. The advantages of laser communcaton nclude wde bandwdth, small antenna and swept volume, low weght and power, and secure communcaton. However, laser communcaton requres 10 or 100 tmes more precse pontng of the optcal lne than the RF sstem because of ts narrow optcal beamwdth. Therefore, one of the most mportant technologes of laser communcaton s to precsel pont the transmt beam to the narrow receve feld of vew n the recevng satellte, wth an accurac of around 10 μrad or less. Usng a spot mage s one of the approaches for ensurng precse pontng. The receved beam passes through a lens and forms a spot mage. B accuratel measurng the center poston of the spot, we obtan the accurate drecton of the beam. Ths accurate drecton s used to acheve precse pontng and fne trackng. Fgure 2 shows one of the concepts of the confguraton of the devces equpped wth optcal antenna and steerng mrrors for laser communcaton. The confguraton s supposed to be equpped on a LEO satellte. A GEO satellte transmts a laser beam, and the coarse pontng mechansm (CP) acqures t. Then, the beam enters the fne pontng mechansm (FP), Fg. 1. Laser communcaton between two satelltes, for example, a low Earth orbt (LEO) satellte and a geostatonar Earth orbt (GEO) satellte. whch conssts of a steerng mrror and a controller. In the FP, the movable mrror reflects the beam and the mrror rotates b a small angle to correct the dsturbed drecton of the beam generated b local dsturbances such as satellte oscllaton. Then, the beam passes through the beam spltter, whch combnes the transmtted beam and the receved beam. The beam passes through another beam spltter where t s dvded nto two. One of the two beams enters the acquston and trackng mage sensor, where the lens forms a spot mage of the beam on the mage sensor. B accuratel measurng the center poston of the spot, we obtan the beam drecton, Coprght 2012 b the Japan Socet for Aeronautcal and Space Scences and ISTS. All rghts reserved. Pj_1

2 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. sts28 (2012) whch s necessar for the correcton of the mrror rotaton n the FP. The other beam enters the lght recever, whch reads the communcaton data. On the other hand, the transmtter produces a laser beam, whch enters the pont ahead mechansm (PA). In the PA, the movable mrror reflects the beam and the mrror rotates b a small angle to control the drecton of the beam. The beam then passes through the beam spltter that combnes the receved beam, before t passes through the FP and the CP. Fnall, the beam s transmtted toward the GEO satellte. The GEO satellte has bascall the same mechansm. In ths paper, a new centrod calculaton algorthm s presented, whch accuratel measures the center poston of the spot mage. The lens of the acquston and trackng mage sensor aggregates the receved laser beam to form a spot on the mage sensor, and the spot spreads over a few pxels on the mage. The algorthm s used to carr out centrod calculaton usng the brghtness values of the spread pxels, and t ncreases the accurac of the center detecton of the spot mage. In the calculaton, a weght table s used, whch has a crcular form wth a blurred border. Each weght value s set n the range from 0 to 1, and the weght value s multpled b the correspondng brghtness value. The weght table s desgned such that the weght values nsde the crcle are taken as 1, those outsde the crcle are taken as 0, and those along the crcumference of the crcle are taken as decmal numbers between 0 and 1. Ths weght table further ncreases the accurac. Fg. 2. One of the confguratons of the devces ncludng optcal antenna and steerng mrrors. Some of the other methods for measurng the center poston of the spot mage b centrod calculaton are known, such as the method reported b Sung-Hoon Bak et al. 1) and the method reported b Shnhak Lee 2). The method reported b Sung-Hoon Bak et al. squares each brghtness value n the centrod calculaton to reduce the nose mpact. In ths paper, the method reported b Sung-Hoon Bak et al. and the presented algorthm are compared n terms of the accurac of detecton of the center poston of the spot. Other applcatons that requre centrod calculaton var from the centrod measurement for the Shack-Hartmann wavefront sensor (SHWS) 3,4) to the centrod measurement of ndvdual fluorescent partcles and molecules for accurate localzaton and trackng n lght mcroscopes 5), noncontact three-dmensonal (3-D) pont acquston for the optcal trangulaton 6), star tracker poston measurement 7), and so on. 2. Centrod Calculaton Algorthm The new centrod calculaton algorthm presented n ths paper conssts of the followng steps: 1. Fnd the maxmum brghtness value p n the mage that has a spot. 2. Let p be p multpled b. ( p p ). 3. Set zero to the brghtness values that are lower than p n the mage. 4. After truncaton, carr out centrod calculaton usng Eq. (1). Let us call the obtaned poston the frst centrod poston. 5. ake a crcle wth radus r around the frst centrod poston. 6. Calculate the rato of how much the th pxel overlaps the crcle usng Eq. (2), and let the obtaned rato be w. 7. Truncate w between 0 and Carr out centrod calculaton usng Eq. (3), multpl the obtaned value b w, and obtan the fnal centrod poston. Step 4 s calculated usng Eq. (1), where p denotes the brghtness value of the th pxel, x and denote the locaton of the th pxel, and c x and c denote the frst centrod poston. cx c p x p p p Step 6 s calculated usng Eq. (2), where a denotes the dstance between the frst centrod poston and the th pxel, and W s a parameter that decdes the wdth of the crcle border. A set of w values s the weght table. (1) r a w 0.5 (2) W Step 8 s calculated usng Eq. (3), where w s calculated usng Eq. (2), p denotes the brghtness value of the th pxel, x and denote the locaton of the th pxel, and c and c denote the fnal centrod poston. x cx c w p x w p w p w p Let us call the presented algorthm [BC]-[BW] (bottom cuttng and blurred weghted method). The three ke ponts of ths algorthm are as follows: (3) Pj_2

3 Y. FUJII et al.: Centrod Calculaton Algorthm Usng Weght Table to Increase Accurac of Center Poston Detecton The executon of step 3, n whch we set the brghtness values that are lower than p n the mage as zero, makes the algorthm exclude nose, whch the pxels outsde the spot generate. The executon of step 4, n whch we calculate the frst centrod poston, ncreases the accurac of the fnal centrod poston. The executon of step 8, n whch we carr out the centrod calculaton b multplng w, further ncreases the accurac of the centrod poston, especall when the center of the spot s located n and around the boundar between two pxels. durng the repetton, the poston of the spot s moved wthn 1 pxel n a sub-pxel and n a unforml random manner to obtan a number of centrod errors. These centrod errors have a dstrbuton smlar to the normal dstrbuton. Therefore, we can calculate the standard devaton (σ) of these errors and obtan the value of 3σ. We have evaluated the centrod accurac usng the magntude of 3σ n ths paper. Fgures 3 and 4 show examples of the forms of the weght tables obtaned n steps 6 and 7. The weght values w of each pxel are calculated over the mage and are shown n grascale mages, where black ndcates that the weght value s 0 and whte ndcates that the weght value s 1. Fgure 3 shows the weght tables when radus r s changed from 2.0 to 4.0 and Fgure 4 shows those when wdth W s changed from 0.5 to 2.5. When r s changed, the sze of the crcular form ncreases, and when W s changed, the border of the crcular form becomes more blurred. Fg. 5. Confguraton of the optcal elements, whch s used n the smulaton of ths paper. Fg. 3. Weght tables when radus r s changed from 2.0 to 4.0. Fg. 4. Weght tables when wdth W s changed from 0.5 to Smulaton Confguraton Fgure 5 shows the confguraton of the optcal elements, whch s used n the smulaton reported n ths paper. A parallel laser beam passes through the crcular aperture and enters the lens. We assume that the profle of the spot should be the Ar dsc pattern. The Ar dsc pattern s a descrpton of the best focused spot of lght that a perfect lens wth a crcular aperture can form, caused b the dffracton of lght 8). Based on ths assumpton, an mage contanng a spot dentcal to that shown n Fgure 6 s produced n the smulaton. Random nose s added to the mage. Then, the centrod poston of the produced spot mage s calculated wth sub-pxel accurac usng the centrod calculaton algorthm, and the centrod error between the calculated poston and the true poston s obtaned. Ths s a sngle ccle for the calculaton of centrod error. We repeat ths ccle several tmes to analze centrod error statstcall. In each ccle Fg. 6. Image contanng a spot. (The mage on the left s the orgnal mage produced n the smulaton b drawng n the range of [DN], whereas the mage on the rght s redrawn n the range of [DN]. Both are produced wth the condton n whch the focal length s 168 [mm], the lens poston s 168 [mm] and the energ njected to the mage sensor s 3.0 [pj].) 4. Smulaton Condtons There are man factors that nfluence the accurac of the centrod calculaton. These factors and ther relatonshps wth each other are shown n Fgure 7. The factors that have the most nfluence are nose characterstc, lght energ, and defocus dstance. The nose characterstc of the amount of random nose n the spot mage reduces the accurac of the centrod calculaton. The accurac also depends on the lght energ. When the lght energ decreases, the brghtness value of the spot n the mage decreases and the sze of the nose becomes relatvel large; Pj_3

4 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. sts28 (2012) thus, t reduces the accurac of the centrod calculaton. The defocus dstance sgnfcantl changes the profle of the spot, and the accurac of the centrod calculaton greatl depends on the profle of the spot. ndcated b A n the fgure. Usng the two patterns, we create the mages n the smulaton, whch are supposed to be obtaned from the mage sensor. Fgures 10, 11, and 12 show the mages produced b the two patterns. The brghtness values are expressed n the unt of DN, whch s the dgtal output value from the mage sensors. It s a nondmensonal value of pxel brghtness voltage [V] normalzed b ts sensor dnamc range [V]. We assume the use of an mage sensor wth 12 bts output dnamc range and t outputs [DN]. The sstem parameter condton shown n Table 1 s used n the smulaton. Let case 1 be the case where the non-defocus pattern n Fgure 9 s used, and case 2 be the case where the defocus pattern s used. The smulaton s performed n the two cases. In case 2, the length to dsplace the lens for defocus s found b determnng the approprate locaton that gves the most accurate centrod calculaton. The unt of e- n Table 1 means the number of electron. The nose to be added n the smulaton s defned b Eq. (4), where p represents the brghtness value of the th pxel and both n a and n b are the parameters that decde the nose characterstc. The obtaned n represents the 3σ of the random nose that has normal dstrbuton. The parameters n a and n b are chosen as n Table 1. n n p n (4) a b Fg. 7. Factors and ther relatonshps wth each other. (Parameter dependenc graph) An mportant case that we have to take nto consderaton whle producng a spot mage b smulaton s a case where almost all of the energ of a spot goes nto a sngle pxel and the spot does not spread to an other pxel around the center pxel, as shown n Fgure 8, dependng on the pxel ptch wdth of the mage sensor and the sharpness of the spot. In ths case, t s dffcult to ncrease the accurac of the centrod poston b centrod calculaton because the calculaton uses the spread pxels. In order to avod such a case, an ntentonall desgned defocus s necessar, where the lens s delberatel dsplaced slghtl from the focal poston. In ths paper, we also consder the case where defocus s necessar and compare the accurac of the centrod postons n the two cases,.e., one n whch the defocus s used and the other where t s not used. In ths paper, we use a sngle wavelength laser, whch s used n laser communcatons sstem as well. When a sngle wavelength laser s collected b a lens, the profle of the spot should deall be the Ar dsc pattern 8). The non-defocus lne n Fgure 9 shows the Ar dsc pattern under the condtons n Table 1 wth case 1. In contrast, when a sngle wavelength laser s defocused, the pattern changes lke that of the defocus lne shown n Fgure 9. It has a bump n the mddle of the spot, Fg. 8. Image n whch almost all the energ of a spot enters a sngle pxel. (The mage on the left s the orgnal mage produced n the smulaton b drawng n the range of [DN], whereas the mage on the rght s redrawn n the range of [DN]. Both are produced wth the condton n whch the focal length s 10 [mm], the lens poston s 10 [mm] and the energ njected to the mage sensor s 3.0 [pj].) 5. Other Algorthms to Compare The centrod calculaton algorthm [BC]-[BW] was presented n Secton 2. In order to compare the effectveness of [BC]-[BW], we perform smulatons usng other algorthms of [BC], [BW], and [EW] n Secton 6. In ths secton, we brefl explan those algorthms. [BC] and [BW] comes from [BC]-[BW], whch conssts of two stages of centrod calculaton. We separate [BC]-[BW] nto sngle stages named [BC] and [BW] and perform smulatons usng those two algorthms as well. One of the other effectve methods of centrod calculaton s the method reported b Sung-Hoon Bak et al. 1) It squares each brghtness value to reduce the nose mpact. Let us call the method [EW] (exponental weghted method) n ths paper. Pj_4

5 Y. FUJII et al.: Centrod Calculaton Algorthm Usng Weght Table to Increase Accurac of Center Poston Detecton Lght Energ Fluence [J/m^2] 9.E+05 8.E+05 7.E+05 6.E+05 5.E+05 4.E+05 3.E+05 2.E+05 1.E+05 0.E Pxel Locaton [px] Non Defocus Defocus Fg. 9. Profle of the spot. (The non-defocus lne s the Ar dsc pattern and the defocus lne s the pattern when the lens s defocused b 0.3 [mm]. The non-defocus lne s computed wth the smulaton condton n whch the focal length s 84 [mm], the lens poston s 84 [mm] and the energ njected to the mage sensor s 3.0 [pj], whereas the defocus lne s done wth the same condton except the lens poston s 84.3 [mm].) Brghtness Value [DN] Pxel Locaton [px] Non Defocus Defocus Fg. 10. Dstrbuton of the brghtness values of the mage produced b the two patterns n Fgure 9. (The defocus lne s n the case where the lens s defocused b 0.3 [mm].) Fg. 11. Spot mage n the non-defocus case. (The mage on the left s the orgnal mage produced n the smulaton b drawng n the range of [DN], whereas the mage on the rght s redrawn n the range of [DN]. Both are produced wth the non-defocus condton n Fgure 9.) A Fg. 12. Spot mage n the defocus case. (The mage on the left s the orgnal mage produced n the smulaton b drawng n the range of [DN], whereas the mage on the rght s redrawn n the range of [DN]. Both are produced wth the defocus condton n Fgure 9.) Table 1. Sstem parameters used n the smulaton wavelength of lght 1064 [nm] energ njected to sensor 2.0 [pj] focal length 84 [mm] lens poston (Case1) 84 [mm] lens poston (Case2) 84.3 [mm] aperture radus 10 [mm] pxel ptch wdth 8 [um] saturaton number of electron [e-] quantum effcenc 0.09 [-] range of brghtness [DN] nose parameter n a [-] nose parameter n 15 [DN] b 5.1. Algorthm [BC] The centrod calculaton algorthm [BC] conssts of the followng steps: 1. Fnd the maxmum brghtness value p n the mage that has a spot. 2. Let p be p multpled b. ( p p ). 3. Set zero to the brghtness values that are lower than p n the mage. 4. After truncaton, carr out centrod calculaton usng Eq. (1), and obtan the fnal centrod poston Algorthm [BW] The centrod calculaton algorthm [BW] conssts of the followng steps: 1. Fnd the maxmum brghtness value p n the mage that has a spot. 2. ake a crcle wth radus r around the pxel whch s found n Step Calculate the rato of how much the th pxel overlaps the crcle usng Eq. (2), and let the obtaned rato be w. 4. Truncate w between 0 and Carr out centrod calculaton usng Eq. (3), multpl the obtaned value b w, and obtan the fnal centrod poston Algorthm [EW] The centrod calculaton algorthm [EW] conssts of the followng steps: 1. Fnd the maxmum brghtness value p n the mage that has a spot. Pj_5

6 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. sts28 (2012) 2. Let p be p multpled b. ( p p ). 3. Set zero to the brghtness values that are lower than p n the mage. 4. After truncaton, carr out centrod calculaton usng Eq. (5), and obtan the fnal centrod poston. [BC] Centrod Error (3σ) [px] Step 4 s calculated usng Eq. (5), where p denotes the brghtness value of the th pxel, s a parameter that enhances the brghtness values near the center of the spot, x and denote the locaton of the th pxel, and c x and c denote the frst centrod poston. c x c p p x 6. Smulaton and Results p p We perform smulatons usng the presented centrod calculaton algorthm [BC]-[BW]. In order to compare the effectveness of [BC]-[BW], other algorthms of [BC], [BW], and [EW] ntroduced n Secton 5 are used n the smulaton as well. Each algorthm contans several parameters and those shown n Table 2 are used n the smulaton. The are determned b teratve smulaton and the optmum values that gve the most accurate centrod calculaton are used. The results of centrod calculaton are shown n Fgures 13 and 14. In the fgures, the vertcal axes lst the names of the methods of centrod calculaton and the horzontal axes lst the centrod errors n the 3σ, whch are expressed n the unt of pxel. Fgure 13 s the result of case 1 and Fgure 14 s the result of case 2. Table 2. Algorthm parameters used n the smulaton. Case Algorthm r W Case 1 [BC] [BW] [BC]-[BW] [EW] Case 2 [BC] [BW] [BC]-[BW] [EW] Fgure 13 shows that [BW] s the most accurate method wth an accurac of 0.07 [px] = 1/15 [px] for the condton of case 1, and Fgure 14 shows that [BC]-[BW] s the most accurate method wth an accurac of 0.02 [px] = 1/50 [px] for the condton of case 2. It means that the presented algorthm [BC]-[BW] can calculate the most accurate centrod poston (5) [BW] [BC]-[BW] [EW] Fg. 13. Results of centrod calculaton smulaton for case 1. [BC] [BW] [BC]-[BW] [EW] Centrod Error (3σ) [px] Fg. 14. Results of centrod calculaton smulaton for case 2. when the spot s defocused. It was found b dong several smulatons that the accurac of the centrod calculaton s lkel to ncrease when the form of the weght table s smlar to the profle of the spot. In ths case, the profle of case 2 has a characterstc bump n the mddle of the spot ndcated b A n Fgure 9. eanwhle the weght table of [BC]-[BW] has the blurred border of the crcular form as seen n Fgures 3 and 4. That blurred border of the weght table fts the poston where the bump exsts, and t ncreases the accurac of the centrod calculaton. In addton, we can calculate the centrod poston wth hgher accurac b combnng the two stages as n [BC]-[BW] than b usng onl a sngle stage lke [BC] or [BW]. In [BW], the center poston of the brghtest pxel s used to make a crcle n Step 2 of [BW]. eanwhle n [BC]-[BW], the frst centrod poston whch s calculated b [BC] n the frst stage s used to make a crcle n Step 5 of the second stage of [BC]-[BW]. Thus [BC]-[BW] s able to make a crcle at a more precse poston, and t ncreases the accurac of the centrod calculaton hgher. Table 3 shows the processng performance of each algorthm. The mddle column n the table shows the tme necessar for a sngle process. The cost tme was measured b carrng out each algorthm whch was mplemented n C++ one mllon tmes on Wndows PC (Intel Core T 2 Duo CPU 2.1GHz) and the tme was calculated b dvdng the measured tme b one mllon. Although [BC]-[BW] takes about 35 tmes longer than [BC] and about 10 tmes longer than [EW], t fnshes the centrod calculaton less than 150 [ns], whch means t s short enough for real-tme processng. Pj_6

7 Y. FUJII et al.: Centrod Calculaton Algorthm Usng Weght Table to Increase Accurac of Center Poston Detecton Table 3. Algorthm parameters used n the smulaton. Algorthm Processng tme [ns] Rato [-] [BC] [BW] [BC]-[BW] [EW] Parameter Optmzaton The presented algorthm [BC]-[BW] has three parameters, namel the rato, the radus r, and the wdth W. In ths secton, we examne the nfluence of the parameters b varng each parameter wthn a partcular range. Fgures 15, 16, and 17 are obtaned usng [BC]-[BW] under the sstem condtons of case 2 as shown n Table 1 and the algorthm parameters of case 2 as shown n Table 2. Centrod Error (3σ) [px] Centrod Error (3σ) [px] Fg Fg Parameter α [-] Results obtaned when the parameter changes Parameter r [px] Results obtaned when the parameter r changes. In Fgure 15, the rato = 0.09 mnmzes the centrod error but ts varaton becomes small,.e., n the range from 0.02 to [px], and the nfluence of the parameter s consdered to be weak. Instead, b usng two stages for calculatng the frst and then the fnal centrod poston brngs an effect. In Fgure 16, the radus r = 3 [px] mnmzes the centrod error and ts varaton s n the range from 0.02 to over 0.07 [px]. Its nfluence s large and the radus r should be well adjusted durng desgn. In Fgure 17, the wdth W = Centrod Error (3σ) [px] Parameter W [px] Fg. 17. Results obtaned when the parameter W changes. 2 [px] mnmzes the centrod error but t generates almost the same centrod errors n the range from 1 to 3 [px], and ts nfluence s consdered to be weak. 8. Concluson In ths paper, a new algorthm [BC]-[BW] was presented, whch measures the centrod poston of the spot n the mage obtaned from an mage sensor wth sub-pxel accurac. The lens forms a spot on the mage sensor and t spreads over a few pxels on the mage. [BC]-[BW] performs the centrod calculaton usng the brghtness values of these spread pxels and t ncreases the accurac of the center detecton of the spot mage. In the calculaton, a weght table s used, whch has a crcular form wth a blurred border. The weght value s multpled b the correspondng brghtness value and ths ncreases the accurac further. An mage contanng a spot and random nose s produced n the smulaton. Centrod calculaton s performed usng the produced mage and the accurac of the centrod poston s evaluated. It s confrmed that [BC]-[BW] can detect the centrod poston wth an accurac of 0.02 [px] = 1/50 [px] when the spot s defocused. It s 1.5 tmes more accurate than [EW], whch s one of the tradtonal algorthms and [BC]-[BW] proves to be qute effectve. References 1) Shnhak, L.: Pontng Accurac Improvement usng odel-based Nose Reducton ethod, Free-Space Laser Communcaton Technologes, SPIE, 4635 (2002), pp ) Sung, H. B., Seung, K. P., Cheol, J. K. and Bungheon, C.: A Center Detecton Algorthm for Shack Hartmann Wavefront Sensor, Optcs & Laser Technolog, ScenceDrect, 39 (2007), pp ) A, L. X. and Ca, W..: An Improved Centrod Detecton ethod Based on Hgher oment for Shack-Hartmann Wavefront Sensor, Optoelectronc Imagng and ultmeda Technolog, SPIE, 7850 (2010). 4) Scott, A. S., Bron,. W. and chael, C. R.: axmum a Pror Estmaton of Wavefront Slopes Usng a Hartmann Wavefront Sensor, Dgtal Image Recover and Snthess, SPIE, 2827 (1996), pp ) Russell, E. T., Danel, R. L. and Watt, W. W.: Precse Nanometer Localzaton Analss for Indvdual Fluorescent Probes, Pj_7

8 Trans. JSASS Aerospace Tech. Japan Vol. 10, No. sts28 (2012) Bophscal Journal, 82 (2002), pp ) De, N. F., Comper, F., Gonzo, L., Gottard,., Stoppa, D., Smon, A. and Beraldn, J. A.: A COS Sensor Optmzed for Laser Spot-Poston Detecton, Sensors Journal, IEEE, 5 (2005), pp ) Hancock, B., Strbl, R., Cunnngham, T., Pan, B., Wrgle, C. and Rngold, P.: COS Actve Pxel Sensor Specfc Performance Effects on Star Tracker/Imager Poston Accurac, the Internatonal Socet for Optcal Engneerng, SPIE, 4284 (2001), pp ) Herschel, J. F. W.: Lght, Transactons Treatses on Phscal Astronom, Lght and Sound Contrbuted to The Encclopaeda etropoltana, Rchard Grffn & Co., 1828, pp.491. Pj_8