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EECS101 Exam f1
TOTAL: l?) a) Does the set of points ,S lie on a straight lirre in 3-D? Explain. Ynr' + ttr.e '*f"' s b) Will orthographic projection be a good approximation to perspective of points ^9? Explain, No, Tk* sef ,{ Pu'*l' 5 ir^'luJer a ,It{. i -r-n* # {rt,:.tvetthale{.V1 O {,u Nn c) Find the set of points in the image that u'iil result froin takirrg the perspective projection of every point in the set S. projection for the set rahf € 2 Question 1 (15 points) Consiier: the imaging systeni ciescribed in ciass having a pinhole at the origin with the z-axis horizontal (positive z to tire left), tiie g-axis vertical (positive g up) and the r-axis forward (positive r out). The irnage plane is located at z - 2. Consider a set S of points in 3-D defined by {'= 2 r - 3t Y:4t .L forallt<-7. Q* frhl s hm*3t iw f,{ # e".fi' { # ir\i" /dg i,i Y 5r 0 fi,'' {r: , f>/ Y'= J_I_= 2'$&* *6 I Y= lurS* Ltv * :t/\ li /./.qf/ttL I IMi@ l* I fe-i i r # p,t€ p a,u -t (x', yJ . (A, q d) Plot the set of points found iri pait c) iu iire 2-D x.r' irnage plane. (x,y,*) fc;,'+ d€{raed b/lan€ 3-D d;nr *f ; o "-i n *{",e t .) tlie irnage that "l,iil result from takiiig the oithographic projectione) Find the set of points in of every point in the set ,S. x'= x.=3{ ii(c, -i .' t 7' ,"': "' : "vi V/: y = t{ T*/ i r i^l y'i f < " I Vf, f o r *L-II jl(-l f) Plot the set of points found in part e) in the 2-D x-v in:rgc plane. (- 3j-q) f .,' *tl( bt-di
4Question 2 (I2 points) Consider an irnagirrg svstem using a iens liith focal length 3cm having an image plane 12cm behind the lens. Suppose the iens diametel is 1cm. The z axis is the optical axis and the lens is located at z : A. a) How far in front of the lens on the optical axis of the system must we place a point to get an image without blur? It*\* I r1Tl * *i* o a = ?*^ I + J-* nIL e; I : b.= d 7 o/_ L= *L + *L*{ci t2-7'- b-O,5 -l Iz'dt #l t1a aXf\J /- l+l -d- | F' 0i I rt ele ^r>tv f tlz ..) 5 t_- blur diameter of less than or equal to 0.5cm? 15) U What is the closest to the lens on tire opticai axis tirat r,ve can piace a point and still have a z- tL ***-F t_ 2+ inraq e 7: tz (g 4*=L+)., Z= 3.421 *n ., i L zr-- )-_--L+ I ,,*i, rr,o3*'" .u,,.-$'o"" Ff,o,r., and still hai ev a blur diameter of less than or erirral to 0.5r'rrr'.' iuaae) Z= l2 lens Z=o \, Ecrui Question 3 (f2 points) Consicler an image definecl by E(t:,Y): 10u(5 - 't) + 2}tt(5 - Y) over the continuous ranges 0 ( r < 10, U < y! l0 rvher-e u is lhe unit step function. a) Draw the image E(r.a) arrd lalrcl all irrrpoltnirt puirits Io @ b) Comput" # as a function of r and y. DE /xy) : -lofi|*r)4 \. r/,/ bx S "l Compute # u" a function of r arrd g. dtr{'s:,,,,) = *8# d(5 - y}:J* \r'"!/ /sy ( 7) d) Compute the squared gradient magnitude (SG\4) as a function of r and gr.\7 r "rll{'..':r 4p"iJ {'i -:,Ji .t4jt LJ56 u {x,yi 5 to o$( r - r.t)' o e) Suppose we apply a threshold T : Show where edges wiil be detectecl in Itl I 2S0 id(0)1'z to the SGIVI resuit where d(.) is a delta function. the irnage. J, iiune ttt e T* cted t* \! .* lqy'' ^lonj tl Question 4 (L2 points) Consicier atr altpiifii:ri i;i;iieclion siio rneasurernent C in electrons for a cooled CCD camera described bv C: (,9+l,',r 1'.\i,),.1 .;r'litlc S is tire signal in electrons, 1V.1 is a zero-mean amplifier noise soul'ce-' r,,ith a ,"'ar;'ianari: o1' circ elei:trou, ;\rp is the zero-mean phct,ou noise source, and A is the constalit anrplifier gilili. r\iisiluie tire noise sorLrces are independent of each other A a) Suppose S : 1000 electroris. Firrd tire lneal a.rid r.arii,rirci: of the rneasrirement C. M€ an{ r) = 1'v\(,,4" (".i A + I!^d *" furA j =* 5A; l': *,;,4 -"7 vtr{r) = v^, (S A + NaA * ,\Jf,A,} = A'' !: A'5 n2./ , s ''i:/\{,lt>) r , nZ _ ljutA _*L t0A: { b) Define the signal-to-noise Suppose S: 1000 electrons. 5\iff ' l,:r*A lat,ic, as tiic rnlrlii of il rli-,,irleri bv tirt: lVlriit is li;e srgnal-io'lcist iatii; lbl C'l I r, r,i -\ I / -jkgY- :Y 5 !. 6 ,4'-*'.#l,,^..' i 'j 1 '''t;""' t standard deviation of (.' ,f-'r ilfl .) Will the signal-to-noise ratio'be 1:Lrger fcr riaik i'egir-r;rs cr }.-r'ighi rrrgi.)ir-q irr arr image? Explail 5NR'5fr €- A\€tArJJ+/ n , ril*\J*',t. I 4I- .\J I \ +",, ianqf )r.-t V J lut l 5ir,ec 5hJ i? incrt'**{r"*ef wi+l' 5} *'l.u ;1;j Le lavjer 4r' blr;,.hT vr:3ictv^5". d) Suppose ,S : 1000 electroirs a,nti wi.' t.1kc different times. Let C' : 0.5(Cr -y Ct). F'rnLi source is independent of itself when mr::rsuLeil M(*arr, 4_t" rir-, r {r)) = /v,ta. {* ,# i2 uavi*qrrrrflo fti:r * Cz )) = "pv- inn{{'(C ,{A(Nur + Nr, + Npr + Nea )) t ttj * i ; ,,*l i '- 't'* ' yti; r;,I' t' ,r'{ ,r.: ttvir ni easi tii.rttii:nt s tiit. rrieiln :r,rid i.aliar ar ciiffcleni tiniirs. sA + A {No, ;* #, asAz (t * : 5i* r-J- f A' : a. L5 A'(ro' + 6; + 6; of C denoted C1 ir:e of O'. Assrrme and C2 at each nci:e *il,r, )* A(ruu ,, tlil)) = SA= looiln ' t,eo s A . *-d. I -I /i tI vf) II./ iIt(r\ttf J r JJ Question 5 (I2 points) Lel g(.i'.g) Le tlre I x iglay-lelci irnage lnlzalzellsl I ____ I ____ I ____ I ____ illttt l16lzslzalrsl I ____ I ____ I ____ I ____ itttriItzllslzglrsi I ____ r ____ I ____ I ____ |ttrttlrslr0lralr0l I ____ I ____ I ____ I __-__ I ffi ,, Find the gray-level histogranr iixi tI itoiO +o it'i t{ o $ O, Find a threshold T thatwill separate tlie briglrt objerct from the dark background. cble*t {ur 3 {xi I,i > T u,.r lnere T' f T f- gqq*y suvi*rt a 3elr 1',ll c,i ed,t l'"e) c) Use 7 to find abinary image b(r,g) where l represents ob.ject pixels.\_:# 3l o z{o d) Find the center of area (z,A) for thc cbjer-:t. Assu.rnc th:;t tire center of the lower left pixei is at (r,g) : (0,0). Indicate the location (:r,g) ln iire hinair, iurage. Y= il* 2 +z+3+ 3)15= iifr * ?'?" 3 Y llctot i) sl* # c o **-"-a**-*-"*'L 7 ? X ,jrr !l.f,\0 l_J{_ o 5 "l\t?g1*--:'t *. It({,;l )*{{.d,4" 8and 0 denote back-Question 6 (8 points) For the binarf images below, let 1 rienote object ground. a) Circle the 4-connected object regic-ins in the binarSi image beiorv. O Ol How many 4-connected object regions aie theie iu this image? B (9 .l For the binary image below, circle the S-connecteci objecl regions. O Ol How many S-connected objeci regions are there in this imagel 3 0 0 U 0 0 U 0 0 0 0 Ar:-1*7s o o o -nrr f 0-6 o o ,D fD o (*l- r Q o'0" 0 d'-1 1 1*l , iJo o o -ol -1.-.-J, .r/o o /rl o o o o f- r"r\o qqo \,1_lr0 ou IQuestion 7 (9 points) Consider the potcntiir,l iveli or cach CCD array of a three-cirip color camera that corresponds to a single pixel location (:r. g). Assurne tliat ihe quantum effficiencir is q()) : 0.9, the incoming photon fiux has iigirt at onl1,'i1vg rvavelengths n'ith one million photors per second at wavelength 445nm arid threc irriilion piiotorrs per scconcl at wavelength 532nnr. Assume that one-third of the incident light is sent in tire rlireclir,-n of each CCD array and th:r,t the integration time is one milliseconcl. Assurne tirele is no noi-"e. The red, green) and blue fl1tcr spectral transmission functions al-e given irv /n()) : u() - 6oonn) - u(A - 700nrn) /c(f) : u,() - 500niri) - u(.\ - 600nrn) f n(^) - "(l - .100rmr) - u().- 500rrrn) where u is the unit step function. a) PIot /a()), /c()), and /6(.\) betrveen 400nrrr aird 700nrrr. (^l I r*--t _l _ r_* It64(rt $o* trnn 7psxo, fo,ru,- {6u*^^' "\ ,h I r-**]ltI _ _ L_--- .. 4ou^t faan^ n nd the number of electrons S;. L(qAiQ (;')t'{'iid}3 )u tts {,\) \^ -:JK 6 0,,, Sc, Ss collecied ai pixel location = () V1o f i',:i'a"i I the camera. tu?' €E {,eJ (", g,) bv , Q"n:''"r*'d I iCI* ) *-(o,4 )(s * 1o* * derl'r,ea*; S B = *f nd,i) &#i$fA)J I* -# (o,a)Qon) = 3oo electr"wspj S ", What color will the light at tliis pixel appcar? Jveen #y h{"n -3rQuestion defined by 8 (10 points) Considr:r a 100 x 100 rii;git:ri 10 irrage G(.r,A) with gray levei vaiues 6kry1 qq qt .-x "rYJ y 0.1.....99 of r anrl gr for each pixel in G(r,:tt'). + E a1 A6;bel xelsp So e ix +at o -L 4 I o s ,1 c 'l fr,4)' rf the ndary l? ro, a) ; l0Itt G(, ito our 9 O b) Find the output of the Sobr'l ff\;/ You may ignore boundary pixels. fi 2 rllo c o I l-t -z "iJI pu ,b the out 'ignore Lnd 1 may e, L{ "/ ,/", . /r""'Z i00 r:0,1,...,99 operrr{or rS a litilt i iorr Ft a rl \-z o L {Ir c rJ )tr S-b-{ r # &x o tz opeiator as a furicti.rn of':r and y for eacir pixel in G(r.y). Jr Srbej = *8 t7\ V) ** s\t#,{f {U,y) rly c) Show the location of the largest conrrecteci region in G(.r. g).for n'irich the pixel gray levels in the region have a variance of zero. Assurne S-connectecl ireighbors. F{ow rnany pirels ale in i}rls region? What is the mean of the pixel .",aiucs in this region? €ua{y (x,>l Lavyr* Y*1Jro- i5" cir*feef J;nj#{"sf " lao yixefs ftl€au" 5 10fi o tv \t-l Y G) a) Fi You IZ 1 L} .{tJ ,/t 18 11rc0 4fi tca tot lou rol IS2 1d\\,Jl l1 Question 9 (5 points) Why is silicon a goocl r4ateriai to usc for buildiirg cameras? Tln* ene{qy t{^^t l,oiri, b,,, Js "for,rfllurr 'in s,lrlu is 'f in^ ilarj/ J Jb'lh* u*uoJy in u,r;b{r, i,**f f;r-'rt*r^S" Tlreor{iu* '' i -l "'i i r**i'*: fru* ef*ri"l.rr*f rr.rhic.l, co,.rn\ h r ; Cr e r'r 1 fr l,r',-i.'Lr i *.'r .i lr ,'' i b" ,,{{"rtrA t, /n(Q{ur€ *,ku $r^ar^t of i^rs/vt*J I yhf - Question 10 (5 points) Describe an applicaiion fcr n'hicir reglon growing would be more appropriate than region splitting. Kr3,r^ 3vavtina i r /va',t "(trro{n i'*{e" #^o^ Y€f in^ sP i 4 "na if wq *^^t +, Sucress€rt| s€jrnfuf s/$4ffiii trylons.