Graphics Reference
In-Depth Information
of a high-dimensional candidate descriptor against a large library (e.g., for wide-
baseline matching) should take advantage of an approximate nearest-neighbor
search algorithm for computational efficiency (e.g., Beis and Lowe's best-bin-first
search [
36
]).
4.8
HOMEWORK PROBLEMS
4.1
Show that the Harris matrix for any positive set of weights must be positive
semidefinite. That is, show that
b
Hb
2
.
≥
0 for any
b
∈ R
4.2 Consider the
N
N
patch in Figure
4.24
a, where the slanted line passes
through the center of the patch at an angle of
×
θ
◦
from the positive
x
axis,
and the intensity is 1 above the line and 0 below the line. Estimate the
eigenvectors and eigenvalues of the Harris matrix for the pixel in the center
of the patch (assuming
w
(
x
,
y
)
is an ideal box filter encompassing the whole
patch).
B
A
(a)
(b)
Figure 4.24.
(a) A binary
N
×
N
patch. (b) A binary image, with two potential feature locations.
4.3 Consider the image in Figure
4.24
b. Will the Harris measure using an ideal
box filter give a higher response at point
A
(centered on a corner) or at point
B
(further inside a corner)? Think carefully about the gradients in the dotted
regions.
4.4 Write theHarrismeasure
C
inEquation (
4.4
) as a function of the eigenvalues
of
H
.
4.5 Show that if one eigenvalue of the Harris matrix is 0 and the other is very
large, the Harris measure
C
is negative.
4.6 Explain why the Gaussian derivative filters in Equation (
4.6
) act as gradient
operators on the original image.
4.7 Show that minimizing Equation (
4.11
) leads to Equation (
4.12
).
4.8 Determine the generalization of Equation (
4.12
) that corresponds to the
affine deformation model of Equation (
4.14
).
4.9
Sketch a simple example of an image location that would fail the Harris
corner test at a small scale but pass it at a larger scale.
