Graphics Reference
In-Depth Information
4.10 Use Equation (
4.21
) and a real image of your choice to duplicate the result in
Figure
4.4
. That is, create a zoomed version of the original image, and deter-
mine the characteristic scale of the same blob-like point in both images.
You should verify that the ratio of characteristic scales is approximately the
same as the zoom factor.
4.11 Verify with a simple real image that the Laplacian-of-Gaussian detector
responds strongly to edges, while the Hessian-Laplace detector does not.
4.12 Speculate about the formof a simple detector based on box filters (similar to
the Fast Hessian detector in Figure
4.7
) that approximates the normalized
Laplacian in Equation (
4.21
).
4.13 Prove that the Gaussian function in Equation (
4.7
) satisfies the diffusion
equation in Equation (
4.26
).
4.14 Show why the refined keypoint location in Equation (
4.31
) follows from the
quadratic fit in Equation (
4.30
).
4.15 If
C
is the matrix square root of a positive semidefinite matrix
H
, show how
the eigenvalues/eigenvectors of
C
and
H
are related.
4.16 The FAST Corner detector, which requires three of the four pixels labeled 1,
5, 9 and 13 in Figure
4.13
to be brighter or darker than the center pixel, can
miss some good corner candidates. For example, show that a strong corner
can exist if only two of the four pixels are significantly brighter or darker
than the center pixel.
4.17 Explain the shape of the curve in Figure
4.15
a — notably the low plateau,
sharp increase, and subsequent slow increase.
4.18 Describe how the computation of MSERs is related to the watershed
algorithm from image processing.
4.19 Explain why MSER detection is invariant to an affine change in intensity.
4.20
Show how the normalized cross-correlation between two real vectors can
be computedusing theDiscrete Fourier Transform. (Fast Fourier Transform
algorithms make this approach computationally efficient when searching
for a template patch across a large image region.)
4.21
In Lowe's definition of the SIFT descriptor, the gradient at each sample
location contributes to the surrounding spatial and orientation bins using
trilinear interpolation. This means the descriptor will change smoothly as
its center and orientation are varied. For example, the point with gradient
indicated in Figure
4.25
a will contribute to the eight labeled histogram bins
in Figure
4.25
b, with most of the weight in Bin 3. If the point lies two-
thirds of the way along the line segment connecting the center of the lower
left bin and the upper right bin, and the angle of the gradient from the
positive vertical axis is
π/
16, compute the weights
w
1
,
...
w
8
that represent
the contribution of the point to each orientation bin.
