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Unformatted text preview: (11] (11] pts} (all (El {fl
(si Figure 1: Image patch For the image patch in Figure 1 at the pixel at the center [1 that is
the pixel marked by; the hlack square] apply; the following ﬁlters
and round to the nearest integer t‘alue: 1 2 1
i. a 3 x 3 Gaussian ﬁlter 1115 2 =1 2
1 2 1 ii. a 3 x 3 Box ﬁlter (that is averaging in a 3 x 3 neighl’mrhmd]. lCompute the edge direction and strength [: that is the direction
and ahsolute value of the image gradientl at the center pixel using
the masks of the Sohel edge detector. 1 —1 £1 1 1 1 2 1
31: E —2 U 2 52 = E U I] I]
k —1 U 1 k —1 —2 —1 Apply; a median ﬁlter to the center pixel. Explain for what kind
of noise median ﬁltering 1will work hest. 1illThy is the Gaussian ﬁlter a good smoothing ﬁlter? [:How' can it
he implemented fast? How can we implement repeated Gaussian
ﬁltering in one operation?) 1iill"hat happens to the two edges at the boundaries of a dark line
on a white hackground it the image is smoothed with a Gaussian with kernel size larger than the width of the line?
Explain why Box ﬁltering (that is averaging] attenuates the noise. Explain the concept of aliasing and give an example. (2] (11] pts} lConsider the cube 1writh points P1. P2. P3. P4. P5.PG.PT. P8 and 3D coordinates in the world coordinate system as given in Figure 2. A. PilI:.D.1.'2.1: 33—:1; 1'11: F?[1.1.'2.1] P5IZI.1‘2.EI: PE:1.1.'2.E: Figure 2: calihrated camera with focal length f = 1 whose origin is at (ﬁll. —3]I and which has a rotation of —¢L'3“' around the Y—axis with respect to the world coordinate system takes an image of the cube. The image ccoor— dinates of the corners of the cube are labeled 331. p2. p3. pl. p5. 3313. pT. pd Remember a rotation of angle o: around the Y—axis can he expressed
cos (I: 1] sin or hv the rotation matrix R = [l 1 [l _. aud cosi—zlEro} =
—sino 1] cos n %u’§. and EDIE—45'") = —%u’§.
(a) Derive the projection matrix mapping homogeneous world coor—
dinates to homogeneous image coordinates. [hi] ICompute the homogeneous and the non—homogenous image coor—
dinates of points 335. 305. pT. 338. (c) Derive the non—homogenous coordinates of the 3 vanishing points,
corresponding to the 3 parallel lines. {d} ICompute the vanishing point of the line PIS—PT {e} How would the camera need to he positioned with respect to the
cuhe. such that 2 of the vanishing points are ideal [that is are at
inﬁnith 7.’ [3:] {5pm} Describe the Emmy edge detector. Exlain its three modules. Figure =1: Image patches at two instances in time 5:] (Epts) {a} lIC‘onsider the cube with a stripe pattern in front of a plane with
a stripe pattern shown in Figure 3. The scene is observed by
a moving camera which translates to the right [: parallel to the
X—aXlSl. Draw the optical ﬂow ﬁeld and draw the normal ﬂow
ﬁeld. [hi] At time t5. and time t1 the camera observes the image patches
shown in Figure :1 [i the image patch at to is the same as the one
in problem 1). lC‘n:+1'.n1:rute the normal ﬂow vector at the center of
the patch. Estimate the spatial derivative If and Iy by averaging
the derivatives computed with the Sohel operator at the two time
instances. lCompute the time derivative I; as difference between
the intensity; at the center pixels. ...
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