ECE 533 Final Exam
Assigned: Monday, April 30th, 2018.
Due: Tuesday, May 1st, 2018.
You must turn it in online or under my door
No collaboration is allowed.
You must direct all questions to the instructor ONLY. All problems are weighted equally.
Problem #1 (Sampling a continuous function). Please note that this problem essen- tially extends the first question in the 2015 final. Furthermore, consult the undergraduate final on sampling the Gaussian.
For this problem, we want to consider the simple ellipsoidal Gaussian case when f(x,y)=exp−(x2 +ry2)/σ2.
1(a) Consider limiting the ellipsoidal Gaussian using:
fbounded(x, y) = f(x, y) · rect x · rect y . (1)
AB
Derive a simple relationship between A and B so that we have f (A/2, 0) = f (0, B/2). Sketch
the 2D continuous-function for r = 2. 1(b)Deriveexpressionsforthetwo-dimensionalcontinuousFourierTransformsforF(Ω1,Ω2)
and Fbounded(Ω1, Ω2). You do not need to simplify the expression for Fbounded(Ω1, Ω2).
1(c) Suppose that you keep lots of samples. In this case, A and B are very large. For this case, show that the expression for Fbounded(Ω1,Ω2) approximates F(Ω1,Ω2).
Hint: Consider the 2D continuous-space Fourier Transform of f (x, y) = 1. 1(c) We next sample the truncated image fbounded. Sketch the Discrete-Space Fourier Trans-
form (DSFT) for the ideal case when r = 2 and
Fbounded(Ω1, Ω2) ≈ F (Ω1, Ω2).
1(d) To approximate the DSFT using FFT2, we will first append a large number of zeros, and then take the FFT2. In your sketch of 1(c), indicate how the FFT2 samples the 2D frequency plane in the ideal case.
Problem #2 (Directionality and the 2D FFT)
2(a) List all of the 2D FFT basis function for a 2 × 2 FFT.
2(b) List all of the 2d FFT frequencies for the 2 × 2 FFT.
2(c) Express the following image in terms of the 2D FFT basis functions:
1 2 I=12.
2(d) Express the following image in terms of the 2D FFT basis functions: 1 1
I=22.
2(e) Explain the results in 2(c) and 2(d) in terms of the directionality of the image.
Problem #3. Filterbank design.
In this example, we want to design a filterbank using the Discrete-Space Fourier Transform (DSFT). For this example, the filter coefficients formulas are taken from ”Least Squares FIR Filter Design Using Frequency Domain Piecewise Polynomial Approximations”. The basic idea is to use separability and the method of jumps.
1
(a)
(b)
Figure 1: Frequency-domain magnitude specifications for zero-phase and separable designs.
(a) F1(ω1), F2(ω1). (b) F3(ω2).
3(a) Derive the piecewise linear approximations for F1(ω1), F2(ω1), F3(ω2). We want to de- sign two separable filters using: G(ω1,ω2) = F1(ω1)F3(ω2) and H(ω1,ω2) = F2(ω1)F3(ω2). Sketch G(ω1, ω2) and H(ω1, ω2).
For each one, you will need to specify the equation for each interval as given here … if −∆≤ω1 ≤0
F1(ω1) = … if …
For your answer, make sure to consider one-period of the interval ( −π ≤ ω1 < π and
−π≤ω2 <π).
3(b) For each one of the three functions construct a table of jumps that accounts for the
discontinuity in the first derivative.
3(c) Use equations (3) and (4) to compute the digital filter coefficients. Note that the final filters would be implemented using separable convolutions.
Problem #4. Basic image analysis.
We want to design a simple system that analyzes the content of different objects. The basic approach is to use adaptive thresholding to detect objects, use connected component analysis to identify the resulting components, and then extract features within each component.
4(a) Based on the description, provide a block diagram that can be used to implement the system and measure the area of each resulting component.
4(b) Describe how you can use the system in 4(a) to remove objects that are too small. For full credit, you need to describe how you will use independent training and testing sets for setting free parameters and validating performance.
4(c) Describe how you can use the system in 4(b) to detect objects with image intensity that falls within a range of values. For full credit, you need to describe how you will use independent training and testing sets for setting free parameters and validating performance.
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Problem #5. Basic image filtering.
Consider the simple averaging filter:
h = 1/9 1/9
Define the simple image:
Define the noise image:
1 1 1 I=1 1 1 1 1 1
1/9 1/9 1/9 1/9
1/9 1/9 .
1/9 1
1. 1
1111
−1 −1 −1 2 N=−3 3 1 −2.
1 −1 1 1 −1 1 −1 1
Define the input image y = I + N.
5(a) Compute the convolution result g = h ∗ y which only includes the output points where the filter operates over the original samples of I ONLY. In other words, only compute the output values where the filter is fully contained within I without the need to extend the image. For this problem, you can use Python or Matlab to compute the outputs.
5(b) Compute there reduction (or increase?) in the mean-squared error over the points computed in 5(a).
Note: If A contains the pixels computed in 5(a), then define the mse using:
mse(g′ − I′) = 1 (g′(i, j) − I′(i, j))2
where g′ represents the result in 5(a) and I′ represents the original samples that correspond to g′, |A| represents the number of pixels that were computed in 5(a). Note that original MSE is given by:
1 i=3 j=3
mse(y − I) = N2(i, j).
|A|
(i,j )∈A
16 i=0 j=0
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