Low-Level Vision (Artificial)
1. Below is shown a convolution mask, H. Calculate the result of convolving this mask with (a) image I1, and (b) image I2. In both cases only calculate results for locations where the mask fits entirely inside the image.
10 000 000 H= 1 1 , I1=0 1 0, I2=1 1 0
2. Calculate H ∗ I, padding the image with zeros where necessary to produce a result the same size as I, when:
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0 0 0 H=0 0 1,
3. Calculate h ∗ hT (giving the result as a 3-by-3 pixel image), where h = [1, 0.5, 0.1]. Show that this is equal to hT × h.
Hence, calculate I ∗ H, where:
1 0.5 0.1 H=0.5 0.25 0.05,
0.1 0.05 0.01
111 I=1 1 1
0.25 1 0.8 I=0.75 1 1
4. List the categories of image features that can produce intensity-level discontinuities in an image.
5. Convolution masks can be used to provide a finite difference approximation to first and second order directional
derivatives. Write down the masks that approximate the following directional derivatives: (a) − δ , (b) − δ , (c) − δ2 , (d) δx δy δx2
−δ2,(e)−δ2 −δ2. δy2 δx2 δy2
6. Convolve the mask −1 1 with itself to produce a 3-by-1 pixel result.
7. (a) Write down a mathematical expression describing the effect of convolving an image I with a Laplacian mask (i.e. −1 −1 −1
L = −1 8 −1 ). (b) Hence, write down a mathematical expression describing the effect of convolving an image −1 −1 −1
−1 −1 −1 I withthefollowingmask: L′ = −1 9 −1
8. For edge detection, a Laplacian mask is usually “combined” with a Gaussian mask to create a Laplacian of Gaussian (or LoG) mask. (a) How are these masks “combined”? (b) Why is this advantageous for edge detection? (c) What other mathematical function can be used to approximate a LoG mask?
9. Use the following formula for a 2D Gaussian to calculate a 3-by-3 pixel numerical approximation to a Gaussian with standard deviation of 0.46 pixels, rounding values to two decimal places.
1 (x2+y2) G(x,y)=2πσ2exp − 2σ2
10. To perform multiscale feature analysis, it would be possible to either (1) keep the image size fixed and vary the size of the mask, or (2) keep the mask size fixed and vary the size of the image. (a) Why is the latter preferred? (b) Give and explicit example of the advantage of method (2) assuming that we have a 100 by 100 pixel image and a 3 by 3 pixel mask and we want to detect features at this scale and at double this scale.
11. Downsample the following image by a factor of 2.
0.3 0.2 0.1 0.5 0.5 0.4 0.4 0.1 0.3 0.6 0.2 1 0.6 0.3 0.8 0.2 0.5 0
I= 0.3 0.3 0 0.5 0.6 0.9 0 . 6 0 . 4 0 . 9 1 0 . 7 0 . 9
0.7 0.5 0.7 0.5 0.4 0.8
12. What is aliasing and how is this avoided when down-sampling images to create an image pyramid? 13. Briefly describe what is meant by (a) a Gaussian image pyramid, and (b) a Laplacian image pyramid.
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