Examination in Image Processing

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1 Umeå University, TFE Ulrik Söderström Examination in Image Processing Time for examination: Please try to extend the answers as much as possible. Do not answer in a single sentence.

. Histogram equalization Below there is a histogram of an image. Explain how histogram equalization functions and how the histogram will look like after equalization.

It will still retain its original shape but it will be spread out. 2. Filtering Give an explanation to how filtering works in the spatial domain and in the frequency domain.

2 . Histogram equalization Below there is a histogram of an image. Explain how histogram equalization functions and how the histogram will look like after equalization. Histogram equalization stretches the histogram so that it is spread from 0 to max. This increases the contrast in the image. The lowest value is 0 and the highest is the max value; in this case 255. It will still retain its original shape but it will be spread out. 2. Filtering Give an explanation to how filtering works in the spatial domain and in the frequency domain. Provide examples of occasions when one or the other could be a better choice. In the spatial domain a filter kernel is convoluted with the image. The result is based on the pixelwise multiplication of the values in the filter kernel and the image pixels. It is also possible to use filter kernels where the result isn t based on convolution but instead depends on the content of the pixels below the filter kernel. In the frequency domain the frequency content of the image is multiplied with a frequency filter. To get the result you usually transform an image with a Fourier transform, perform filtering with multiplication and the do an inverse Fourier transformation. Instead of operating on a local pixel patch the frequency filter can remove frequencies from the entire image. The spatial domain filtering is suitable when you have a small filter kernel and frequency domain filtering is suitable when you need to remove certain frequencies or frequency bands.

3 3. Laplace filter The image below has 256 grey levels (-27 to 28). Apply a 3x3 Laplace filter to the image and calculate the magnitude. The image is already zero-padded. Show what kind of filter you choose to use. (3p) The Laplace filter used: The result: Fourier transform It is all about rotations. When the image is rotated in the spatial domain it corresponds to a rotation of the frequency spectra with equal angle. The images display a white box and the corresponding Fourier representation for each box is the spectra which is rotated in the same way. -C 2-A 3-B

4 4. Fourier transform Four plots of magnitude of FFT2 are shown on the left side of the page. These are Fourier transform spectras of the images to the right. Discuss which FFT-plot that corresponds to which image. Give a motivation for your choice! (3p) A 2 B 3 C 4

5 5. Signatures Create the signature for the shape below. You can use the chart under the figure. r θ r(θ) θ

6 6. Chain coding In the figure below there is a shape. Find the chain code for the shape with d 8 -metric. Make the code invariant to starting point and rotation. Starting point Chain code invariant to starting point: Chain code invariant to rotation: Here are some important areas. For all the other edges it is possible to go two directions but here you have to follow the boundary.

7 7. Structure elements All morphological operations work with hit and fit of structure elements. Below there are two structure elements and a figure. Explain where the two structure elements will fit the image. Especially note the difference between the two elements. You can mark the positions with A and B. Structure elements A B Figure 0 0 A A A 0 0 A A B B 0 0 B Domains Below there is an image of a filter kernel in the spatial domain. Give an approximation of how the kernel for such a filter will look like in the frequency domain. Explain why it will look like this. This is a high-pass filter so it will be similar to the image to the right. In the central part there are low frequencies and they are removed. Further out toward the edges there are high frequencies and they are retained. Black=0, White=. -4

9. Segmentation In segmentation there are two main methods; one where you look for discontinuities and one where you look

Explain how these methods work and give an example of a kind of segmentation that functions according to the given methods.

Discontinuities: The image is segmented based on the discontinuities between different areas.

8 9. Segmentation In segmentation there are two main methods; one where you look for discontinuities and one where you look for similarities. Explain how these methods work and give an example of a kind of segmentation that functions according to the given methods. Similarities: The image is segmented based on the similarities of the pixels. Example: Thresolding. Discontinuities: The image is segmented based on the discontinuities between different areas. Example: Boundary detection or edge detection finds the discontinuities. 0. Noise models Below there are three histogram for images. All the images are distorted by noise. Which kind of noise are the three images degraded by? A B C A= Gaussian noise B= Uniform noise C= Exponential noise

9 . Coding efficiency and Huffman coding You have a source with 6 symbols {a, a 2, a 3, a 4, a 5, a 6 }. The probability for each symbol is z=[0,05 0,5 0,55 0, 0, 0,05]. a) Calculate the entropy of the source. (p) H(z)=-(0,05*log 2 (0,05) + 0,5*log 2 (0,5) + 0,55*log 2 (0,55) + 0,*log 2 (0,) + 0,*log 2 (0,) + 0,05*log 2 (0,05)) =,98 b) Create a Huffman code for the source. (p) Code = [ ] c) Calculate the average word length of the source. (p) 0.4*+0.25*2+0.5*3+0.*4+0.05*5+0.05*5 = 2.00 d) Calculate the coding efficiency for the Huffman code. (p)

10 2. Morphology In the image below there are three objects (black) on a white background. Explain what will happen with the image if you perform the stated morphological operations with a 3x3 square structure element. a) Erosion (p) b) Dilation (p) c) Opening (p) pixel d) Closening (p) Choose a structure element. In this example I have used a square 3x3 SE. Since the size of a pixel is it is not possible to create these figures at this resolution but I just want an approximate answer. a) b) c) d)

Boundary descriptors. Representation REPRESENTATION & DESCRIPTION. Descriptors. Moore boundary tracking

Boundary descriptors. Representation REPRESENTATION & DESCRIPTION. Descriptors. Moore boundary tracking Representation REPRESENTATION & DESCRIPTION After image segmentation the resulting collection of regions is usually represented and described in a form suitable for higher level processing. Most important