Introduction to Digital Image Processing Example exam study problems by tyt31586

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									   Introduction to Digital Image Processing:
        Example exam study problems.


                           March 12, 2007


Study Topics:

1. Histogram Processing
2. Spatial Filtering
3. Linear Systems theory and properties.
4. Fourier Transform and it’s Properties.
5. Maximum likelihood Detection and Estimation.




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1. The diagram at the right contains several curves that could be used
   to transform the brightness values of a monochrome image by the
   operation B = T [A] where A and B are image arrays. Shown below
   are four pairs of histograms. Identify the transformation curve best
   associated with each pair and write the letter in the space in the
   center column.




                           Input image histogram       Transform    Output image histogram
                                                                                                  5
3. A 4×4 gray-scale original image passes through three spatial linear shift-invariant filters,
resulting in three filtered images

                                        Shift-Invariant
                                                                  ?   ?   ?   ?
                                        Linear Fiter 1:
                                                                  ?   ?   ?   ?
                                             0   1   0
                                         1                        ?   ?   ?   ?
                                             0   0   0
                                         2                        ?   ?   ?   ?
                                             0   1   0
                                                               Filtered Image 1


                                        Shift-Invariant
                12 10 8      6                                    7   0   0   2
                                        Linear Filter 2:
                10 8     6   4                                    6   0   0   1
                                             ?   ?   ?
                 8   6   4   2                                    5   0   0   0
                                             ?   ?   ?
                 6   4   2   0                                    4   0   0 -1
                                             ?   ?   ?
                original image                                 Filtered Image 2


                                        Shift-Invariant
                                                                  ?   ?   ?   ?
                                        Linear Filter 3:
                                                                  ?   ?   ?   ?
                                             0   1   0
                                         1                        ?   ?   ?   ?
                                           -1 2 -1
                                         2                        ?   ?   ?   ?
                                            0 1 0
                                                               Filtered Image 3


   (a) Compute Filtered Image 1 (Use zero-padding of the original image).

   (b) Compute Filtered Image 3 (Use zero-padding of the original image).

   (c) Based on relationship between Filtered Image 1, Filtered Image 2, and Filtered Image 3,
       determine the filter coefficients in Shift-Invariant Linear Filter 2.

ANSWER:
2. The Sobel operator computes the following quantity at each location (x, y) in an image array, A:

        Gx [j, k] = (A[j + 1, k + 1] + 2A[j + 1, k] + A[j + 1, k − 1]) − (A[j − 1, k + 1] + 2A[j − 1, k] + A[j − 1, k − 1])
        Gy [j, k] = (A[j − 1, k − 1] + 2A[j, k − 1] + A[j + 1, k − 1]) − (A[j − 1, k + 1] + 2A[j, k + 1] + A[j + 1, k + 1])
         G[j, k] = |Gx [j, k]| + |Gy [j, k]|

  The position of A[j, k] is column j and row k of the array.
  The operation is implemented as the convolution of the image array A with two masks, Mx and My followed by the
  magnitude operation.

   (a) Write a 3 × 3 array for each mask, Mx and My .
   (b) What mathematical operation on an image array is approximated by the Sobel operator? Show how the Sobel
       operator is related to the mathematical operation.
    Discrete Fourier Transformsand it’s properties Practice proving prop-
erties of the Discrete Fourier Transform such as:
   • Prove that the circular convolution is multiplication in Fourier domain.
   • Prove the shifting property.




                                      2
    Maximum Likelihood Detection: An object of uniform intensity is im-
aged in an noisy environment. You as an image processing expert are asked to
develop an algorithm to generate a binary image of the object. It is expected
that the object has an intensity of 100. The noise is modeled as additive Gaus-
sian noise of variance σ 2 = 20. Assume that the background has an average
intensity of 0.
    Basic outline developed in class: This is a basic hypothesis testing
problem. At each pixel you need to test the hypothesis if the pixel belongs to the
object or not. Hypothesis H1 is that the pixel belongs to the object. Hypothesis
H0 is that the pixel belongs to the background. Under the modeling assumption
you have to compare the likelihood of the data given the two hypothesis.
                                               1       x2
                           P (x|H0 ) = √           e− 2∗20
                                              2π20
                                           1     (x−100)2
                          P (x|H1 ) = √        e− 2∗20
                                          2π20
   Taking the logs and simplifying we get that Hypothesis H1 is true if

                             −x2 + (x − 100)2 < 0

This implies that

                        −2 ∗ 100 ∗ x + 100 ∗ 100 < 0
                                               x > 50

   Now the optimal algorithm becomes simply threshold the image at 50.




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