1. Convolution properties: Which one of the following is not true, where h, k, and m are all convolution
kernels and H, K, and M are their corresponding filters?
a) h*k =k*h.
b) h* (k + m) = h*k +h*m.
c) (h + k) *m = h*m + k*m.
(HK) = h*k.
e) DFT(hk) = H*K.
f) None of the above, i.e., all of a-e are true.
2. Say which one of the following is not a property of cubic B-splines in 2D:
a) It uses the same basis cubics for each patch.
b) It provides continuity across patch boundaries of 3rd derivatives.
c) If at control point 4 patches away from a reference patch the control value changes, the spline in the reference
patch does not change.
d) Its effect can be computed using a successive subdivision algorithm.
e) It is applied separably, first in x and then in y or equivalently, first in y and then in x.
3. Edges and bars and blobs: Using directional derivatives of Gaussians as matched filters can provide useful
means of finding edges, bars, and circular blobs in 2D images. Which one of the following statements is false.
a) Eigenanalyis of the (D
1G) * I tensor at each pixel will yield positions of bar centers and the bar orientation
b) Using the gradient direction will yield the direction perpendicular to an edge of contrast in an image.
c) The width of Gaussian used to find edges should be somehow matched with the slope of the edge sought.
d) The Laplacian of the Gaussian, with an RMS width equal to the radius of the circular blob sought, can be
used to locate in an image centers of circular blobs of roughly constant intensity and of the specified radius.
e) None of the above, i.e., all the statements a-d are true.
4. Assume an image made by an ordinary light camera is polluted by zero-mean white noise. Thus the image
can be understood as the scene S blurred by a Gaussian G (call that blurred result the “signal”) with the white
noise added to the signal. Say which one of the following statements is true.
a) The RMS noise level at any frequency after taking the DFT of the image is dependent on the RMS width of
the impulse response function associated with the blurring due to the camera.
b) The actual noise level relative to the signal at each frequency can be computationally modified by convolving
the image with an appropriate kernel.
c) The RMS noise level relative to the signal at the low frequencies will be lower than at the high frequencies.
5. Consider the rect function and the brush function in 3D both as filters and kernels. Which one of the
following operations was not useful in the analysis of pixelization of an image that we covered?
a) Convolution with a brush in the space domain
b) Multiplication with a brush in the space domain
c) Multiplication with a rect in the frequency domain
d) Convolution with a narrow rect in the space domain
e) None of the above, i.e., all were useful.
6. Within a Gaussian aperture with RMS width , various levels of detail are possible. Which pair of the
following pairs of levels of detail is unreasonable:
a) First derivative and second derivative
b) Cosines with at frequencies with wavelengths 4 and 8.
c) Cosines with at frequencies with wavelengths 0.25 and 0.5.
d) None of the above, i.e., all are reasonable.
7. Compression: If you wish to store an image via the coefficients of only 10 basis images, which is the best
basis in terms of sum of squares errors over a particular set of training images, written as rows of a data matrix
a) The separable sine and cosine images for the lowest frequencies.
b) The separable cosine images for the lowest frequencies.
c) The eigenimages of A AT with the largest eigenvalues.
d) The eigenimages of AT A with the smallest eigenvalues.
e) The eigenimages of AT A with the largest eigenvalues.
8. Assume an image I is written as a linear combination of one of the following sets of basis images. For which
one set can the sum of the squares of the coefficients different from the others?:
a) The delta (e) images.
b) The unit eigenimages of the convolution with a Gaussian.
c) The unit eigenimages of the operator that is computed by multiplying the DFT of its input image by the
magnitude of the frequency and taking the DFT-1 of the result.
d) The Hadamard-Walsh images.
e) None of the above, i.e., that sum or squares is the same for all of basis sets a-d.
9. Images as vectors: Images can be understood as vectors, i.e., 1-dimensional arrays, with an appropriate zero
vector, additive inverse for each vector, and property that the sum of two vectors is a vector. In considering the
image this way, which one of the following statements is true.
a) The neighbor relations among an image’s pixels is especially well exposed when considering the image as a
b) Multiplying an image pixel by pixel by a scalar gives a different result than multiplying the vector by a
c) The Euclidean (Pythagorean) length squared of the vector corresponding to an image is not a useful image
d) The dot product between certain image pairs can produce a useful value.
e) None of the above.
10. Eigenimages of shift-invariant, linear operators: Which of the following is not a possible property or use
of 2D eigenimages of shift-invariant, linear operators:
a) The eigenimage’s coefficients for a given operator and image can be given by two real coefficients.
b) The eigenimage’s coefficients can be given by one real coefficient and one angle.
c) Each eigenimage for a given operator is formed from two 2D images.
d) The eigenimage for a given operator is formed from two 1D eigenimages.
e) Decomposition of an image in terms of a shift-invariant, linear operator is useful for applying nonlinear,
11. SVD: Which one of the following is a true statement about a not-necessarily square matrix A?
a) The eigenvectors of A AT are the same as the eigenvectors of AT A.
b) AT A has the same number of non-zero eigenvalues as A AT
c) AT A can have negative eigenvalues.
d) You cannot derive the eigenvectors of A AT
from the eigenvectors of AT A.
e) None of the above.
12. Derivative tensors: The fourth derivative tensor at each pixel in a 3D image has how many entries?
a) 3. b) 4. c) 9. d) 64. e) 81.
13. Directional derivatives: Which one of the following second directional derivatives of an image at each
pixel cannot be used to compute D2
I) v. b) (v
I) u. c) u
I)T v). d) u
I v). e) None of the above.
14. Taylor series: Which one of the following statements in not true about a Taylor series representation of an
a) It is linear combination of basis images.
b) The first few terms of a Taylor series are useful as a global approximation to the image.
c) For a 2D image the first 7 terms involve partial derivatives of order 0-2.
d) Any Taylor series must involve derivatives at a common image point.
e) All the derivatives of the same order in a Taylor series are divided by the factorial of the same integer.
15. Basis images: Which of the following are not basis images of interest for image computing?
a) The delta-images (also called e images).
b) Eigenimages of a linear operator on images.
c) Eigenimages related to a matrix listing example images.
d) Cosine images at many frequencies.
e) None of the above.
16. Linear operators: Which of the following is not a linear operator on an image?
b) Diffusion with constant conductance.
c) Sharpening with a “volcano-filter.
d) Setting to zero every pixel with a negative value.
e) None of the above, i.e., all of the above are linear operators on an image.
17. Invariance to geometric transformations: An operator on images that is magnification-invariant
(equivariant) has the property that
a) If the image is magnified by some factor, the result of the operator is unchanged.
b) If the image is magnified by some factor and the result of the operator is magnified by the same factor, the
result is the same.
c) If the image is magnified by some factor and the result of the operator is magnified by the reciprocal of that
factor, the result is the same.
d) If the image is magnified by some factor and the result of the operator is magnified by the negative of that
factor, the result is the same.
e) None of the above.
18. For each of the following questions about aliasing, say whether the statement is true or false.
a) The aliasing pollution across the folding frequency depends on the imaging impulse response function’s
DFT value at the folding frequency. T F
b) Aliasing can be improved by Gaussian convolution after pixelization. T F
c) The folding frequency is the reciprocal of the distance between adjacent pixels. T F
19. Specify true or false for each of the following properties as to whether that property’s necessity is why you
want to use the Gaussian as aperture:
a) The convolution of two Gaussians is a Gaussian. T F
b) The DFT of a Gaussian is a Gaussian. T F
c) The Gaussian is isotropic. T F
d) Convolution with a Gaussian makes relative maxima lower and relative minima raise. T F
e) Convolution with a Gaussian can be accomplished via the diffusion differential equation. T F
20. For each of the following statements, say whether it is true or false.
a) The DFT provides good localization. T F
b) The DFT provides no information as to localization. T F
c) The DFT after multiplications by Gaussians centered at various image locations can provide useful
localization. T F
d) After localization, results with large spatial scale are removed from an image, localizations with smaller
spatial scales on the results can be useful. T F
21. Say whether the following statement is true or false. Let m(x) be an NN image, and let M() be the
result of applying the amplitude/phase FFT (used in your assignment) to m. If in 2D you want to convolve (not
multiply) M() by h() in the frequency domain to produce a result at =(½,½), where h is so narrow that you
want to do this computation as a weighted sum in the frequency domain, then the result will be affected by
M(-½+1/N,-½-1/N). T F
22. Assume you want to compute the Laplacian of a 3D image I, i.e., with a result at each voxel center. For
each of the following recipes say whether that would produce the desired result:
a) In the space domain convolve the image with a 3 3 kernel with the value zero in the corners, 5.0 in the
center, and -1.0 in the other four kernel locations. T F
b) Take the DFT of I, multiply it by -(2)
), and take the DFT-1 of the result. T F
c) Take the DFT of I, multiply it by -(2)
)], and take the DFT-1
of the result. T F
d) Take the DFT of I, multiply each x-row by exp[-½(4)
], multiply each y-column of the result by
], multiply each (z-depth array by exp[-½(4)
]; store that result as M; then
compute three multiplications of M: multiply first by -(2)
, second byy
, and finally by -(2)
sum the three just computed results and take the DFT-1 of the result. T F
e) Take the DFT of I, multiply each x-row by exp[-½(4)
], multiply each y-column of the result by
]; store that result as M; then multiply first by -(2)
, also byy
, then sum the two just
computed results and take the DFT-1 of the result. T F
23. FFT: For each of the following statements, say whether it is true or false
a) The FFT of an image gives a different result than the DFT of that image. T F
b) The FFT of an image gives the coefficients of the basis sinusoidal images for that image. T F
c) The FFT algorithm takes qualitatively more computing than the FFT-1
algorithm. T F
24. Let h(x) be a convolution kernel. For each of the statements below, say whether it is is possibly false or
always true when h is convolved in the space domain with an image I to produce a result image J?:
a) For each pixel y, I(y) is replaced by the product of I(y) and h(-x) centered at y. T Possibly F
b) For each output pixel x, J(x) has the value of the weighted sum over y of h(-y) centered at 0 times I(y)
translated so that x becomes 0. T Possibly F
c) For each output pixel x, J(x) has the value of weighted sum over y of h(-y) centered at 0 times I(y).
T Possibly F
d) In computing a Gaussian-smoothed image for an image I, I can be thought of as a convolution kernel
applied to the image formed by the Gaussian. T Possibly F
e) For output pixels next to the right edge of the image, the image pixels at the left image could be unused to
compute that output. T Possibly F
25. Answer each of the following true/false questions about matrix decompositions separately. In each question
R and S are appropriate rotation matrices and D is an appropriate (not necessarily square) diagonal matrix:
a) If a matrix is square and symmetric, it can be written as RT D R. T F
b) If a matrix is non-square with more rows than columns, it can be written as RT D S. T F
c) If a matrix is non-square with more columns than rows, it can be written as RT D S. T F
d) If a matrix is square but non-symmetric, it can be written as RT D R. T F
26. Eigenimages: True or false: A linear operator on an input image, where the operator is not shift-invariant,
has no eigenimages.
27. Which of the following is true about the Gaussian in 3 dimensions? For each, say whether the statement is true (T)
or false (F).
a. It is the best weighting function to use as the aperture for 3d images
b. It is separable across each of the three dimensions
c. That Gaussian centered at one image point is orthogonal to that Gaussian centered at a different point
d. Application of the Gaussian aperture can only be accomplished in the frequency domain
e. Application of the Gaussian aperture can be done using a partial differential equation.
f. When applied in the frequency domain the wrap-around problem can be completely avoided if the Gaussian kernel is
One-word Answer Questions
27. Answer each of the following questions with one word (it may be hyphenated). Any answer that has more
than one word will receive no credit.
A. An operator on a 2D image that is applied first in one dimension on the rows and on the result on the
columns is said to be ______________.
B. An operator on a 3D image that has no bias on orientation is said to be ____________.
C. The pollution of image components with frequencies less in magnitude than the folding frequency by
image components with frequencies greater in magnitude than the folding frequency is called
D. A patchwise image where each patch is a polynomial in each image dimension is called a
E. The general way to compute the coefficient for image I of basis image from a set of orthonormal set
of basis images is by combining I and using the operation called _______________
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