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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.