# EE 569: Homework #2 solved

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Problem 1: Edge Detection (50 %)
(a) Sobel Edge Detector (10%)
Implement the Sobel edge detector and apply to Gallery.raw and Dogs.raw images as shown in Fig. 1 (a)
and (b). Note that you need to convert RGB images to grayscale image first. The relationship between
RGB and grayscale images are provided in the Appendix. Include the following in your results:
(1) Normalize the x-gradient and the y-gradient values to 0-255 respectively and show the results;
(2) Calculate and show the normalized gradient magnitude map;
(3) Tune the thresholds (in terms of percentage) to obtain an edge map with the best visual
performance. The final edge map should be a binary image whose pixel values are either 0 (edge)
or 255 (background).
(b) Canny Edge Detector (10%)
The Canny edge detector is an edge detection technique utilizing image’s intensity gradients and nonmaximum suppression with double thresholding. In this part, apply the Canny edge detector [1] to both
Gallery.jpg and Dogs.jpg images from the Berkeley Segmentation Dataset and Benchmarks 500 (BSDS
500) [2]. You are allowed to use any online source code such as the Canny edge detector in the MATLAB
image processing toolbox or the OpenCV (i.e. Open Source Computer Vision Library).
(1) Explain the Non-maximum suppression in Canny edge detector in your own words.
(2) How are high and low threshold values used in Canny edge detector?
(3) Generate edge maps by trying different low and high thresholds and discuss your results.

Figure 1: Gallery and Dogs images
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 2 of 8
(c) Structured Edge (15%)
Apply the Structured Edge (SE) detector [3] to extract edge segments from a color image with online
source codes (released toolbox in MATLAB: https://github.com/pdollar/edges). Exemplary edge maps
generated by the SE method for the Helicopter image from BSDS 500 are shown in Figure 2. You can
apply the SE detector to the RGB image directly without converting it into a grayscale image. Also, the
SE detector will generate a probability edge map. To obtain a binary edge map, you need to binarize the
probability edge map with a threshold.
(1) Please digest the SE detection algorithm. Summarize it with a flow chart and explain it in your
own words (no more than 1 page, including both the flow chart and your explanation).
(2) The Random Forest (RF) classifier is used in the SE detector. The RF classifier consists of
multiple decision trees and integrate the results of these decision trees into one final probability
function. Explain the process of decision tree construction and the principle of the RF classifier.
(3) Apply the SE detector to Gallery.jpg and Dogs.jpg images. State the chosen parameters clearly
and justify your selection. Compare and comment on the visual results of the Canny detector and
the SE detector.
Helicopter Probability edge map (inversed) Binary edge map (with p>0.2)
Figure 2: The Helicopter image and its probability and binary edge maps obtained by the
Structured Edge detector
(d) Performance Evaluation (15%)
Perform quantitative comparison between different edge maps obtained by different edge detectors. The
ultimate goal of edge detection is to enable the machine to generate contours of priority to human being.
For this reason, we need the edge map provided by human (called the ground truth) to evaluate the quality
of a machine-generated edge map. However, different people may have different opinions about the
importance of different edges in an image. To handle the opinion diversity, it is typical to take the mean
of a certain performance measure with respect to each ground truth, e.g. the mean precision, the mean
recall, etc. Figure 3 shows the 5 ground truth edge maps for the Helicopter image. To evaluate the
performance of a generated edge map, we need to identify the error. Every pixel in a generated edge map
will be belong to either of the following four classes:
(i) True positive: Edge pixels in the edge map coincide with edge pixels in the ground truth. These
are edge pixels the algorithm successfully identifies.
(ii) True negative: Non-edge pixels in the edge map coincide with non-edge pixels in the ground
truth. These are non-edge pixels the algorithm successfully identifies.
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 3 of 8
(iii) False positive: Edge pixels in the edge map correspond to the non-edge pixels in the ground truth.
These are fake edge pixels the algorithm wrongly identifies.
(iv) False negative: Non-edge pixels in the edge map correspond to the true edge pixels in the ground
truth. These are edge pixels the algorithm misses.
Clearly, pixels in (i) and (ii) are correct ones while those in (iii) and (iv) are error pixels of two different
types. The performance of an edge detection algorithm can be measured using the F measure, which is a
function of the precision and the recall.
(1)
One can make the precision higher by decreasing the threshold in deriving the binary edge map. However,
this will result in a lower recall. Generally, we need to consider both precision and recall at the same time
and a metric called the F measure is developed for this purpose. A higher F measure implies a better edge
detector.

Ground Truth 1 Ground Truth 2 Ground Truth 3

Ground Truth 4 Ground Truth 5
Figure 3: Five ground truth edge maps for the Helicopter image
For the ground truth edge maps of Gallery and Dogs images, evaluate the quality of edge maps obtained
in Parts (a)-(c) with the following steps:
Precision 😛 = #True Positive
#True Positive + #False Positive
Recall : R = #True Positive
#True Positive + #False Negative
F = 2 ⋅ P⋅ R
P + R
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 4 of 8
(1) Calculate the precision and recall for each ground truth (provided in .mat format) separately using
the function provided by the SE software package and, then, compute the mean precision and the
mean recall. Finally, calculate the F measure for each generated edge map based on the mean
precision and the mean recall. Please use a table to show the precision and recall for each ground
truth, their means and the final F measure. Comment on the performance of different edge
detectors (i.e. their pros and cons.)
(2) The F measure is image dependent. Which image is easier to a get high F measure – Gallery or
(3) Discuss the rationale behind the F measure definition. Is it possible to get a high F measure if
precision is significantly higher than recall, or vice versa? If the sum of precision and recall is a
constant, show that the F measure reaches the maximum when precision is equal to recall.
Problem 2: Digital Half-toning (50%)
(a) Dithering (15%)
Figure 4 Light House is a grayscale image. Implement the following methods to convert it to half-toned
images. In the following discussion, �(�,�) and �(�,�) denote the pixel at position (�,�) in the input and
output images respectively. Show and compare the results obtained by the following algorithms in your
report.
Figure 4: Light House [4]
1. Fixed thresholding
Choose one value, T, as the threshold to divide the 256 intensity levels into two ranges. An intuitive choice
of T would be 128. For each pixel, map it to 0 if it is smaller than T, otherwise, map it to 255, i.e.
�(�,�) = ) 0 if 0 ≤ �(�,�) < � 255 if � ≤ �(�,�) < 256 2. Random thresholding In order to break the monotones in the result from fixed thresholding, we may use a random threshold. The algorithm can be described as: EE 569 Digital Image Processing: Homework #2 Professor C.-C. Jay Kuo Page 5 of 8 • For each pixel (�,�), generate a random number in the range 0 ∼ 255, so called ����(�,�) • Compare the pixel value with ����(�,�). If the pixel value is greater, then map it to 255; otherwise, map it to 0, i.e. �(�,�) = ) 0 �� 0 ≤ �(�,�) < ����(�,�) 255 �� ����(�,�) ≤ �(�,�) < 256 A choice of random threshold could be from uniformly distributed random variables. Check your coding language documentation for proper random variable generator. 3. Dithering Matrix Dithering parameters are specified by an index matrix. The values in an index matrix indicate how likely a dot will be turned on. For example, an index matrix is given by �:(�,�) = ; 1 2 3 0 >
where 0 indicates the pixel that is the most likely to be turned on, and 3 is the least likely one. This index
matrix is a special case of a family of dithering matrices firstly introduced by Bayer [5]. The Bayer index
matrices are defined recursively using the formula:
�:?(�,�) = @
4 × �?(�,�) + 1 4 × �?(�,�) + 2
4 × �?(�,�) + 3 4 × �?(�,�) D
The index matrix can then be transformed into a threshold matrix T for an input grayscale image with
normalized pixel values (i.e. with its dynamic range between 0 and 255) by the following formula:
�(�, �) = �G(�, �) + 0.5
�: × 255
where �: denotes the total number of pixels in the threshold matrix, and (�, �) is the location in the
matrix. Since the image is usually much larger than the threshold matrix, the matrix is repeated
periodically across the full image. This is done by using the following formula:
�(�,�) = J 0 �� �(�,�) ≤ �(� ��� �,� ��� �)
255 ��ℎ������
Please create �:, �R, �S: threshold matrices and apply them to halftone Light House (Fig. 4). Compare your
results.
(b) Error Diffusion (15%)
Convert the 8-bit Light House image to a half-toned one using the error diffusion method. Show the
outputs of the following three variations and discuss these obtained results. Compare these results with
Dithering Matrix method in part (a). Which method do you prefer? Why?
(1) Floyd-Steinberg’s error diffusion with the serpentine scanning, where the error diffusion matrix is:
1
16 T
0 0 0
0 0 7
3 5 1
V
(2) Error diffusion proposed by Jarvis, Judice, and Ninke (JJN), where the error diffusion matrix is:
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 6 of 8
1
48

0 0 0 0 0
0 0 0 0 0
0 0 0 7 5
3 5 7 5 3
1 3 5 3 1⎦

(3) Error diffusion proposed by Stucki, where the error diffusion matrix is:
1
42

0 0 0 0 0
0 0 0 0 0
0 0 0 8 4
2 4 8 4 2
1 2 4 2 1⎦

Describe your own idea to get better results. There is no need to implement it if the time is limited.
(c) Color Halftoning with Error Diffusion (20%)
Figure 5: The Rose image [6]
(1) Separable Error Diffusion
One simple idea to achieve color halftoning is to separate an image into CMY three channels and apply
the Floyd-Steinberg error diffusion algorithm to quantize each channel separately. Then, you will have
one of the following 8 colors, which correspond to the 8 vertices of the CMY cube at each pixel:
W = (0,0,0), Y = (0,0,1), C = (0,1,0), M = (1,0,0),
G = (0,1,1), R = (1,0,1), B = (1,1,0), K = (1,1,1)
Note that (W, K), (Y, B), (C, R), (M, G) are complementary color pairs. Please show and discuss the result
of the half-toned color Rose image. What is the main shortcoming of this approach?
(2) MBVQ-based Error diffusion
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 7 of 8
Shaked et al. [7] proposed a new error diffusion method, which can overcome the shortcoming of the
1) Describe the key ideas on which the MBVQ-based Error diffusion method is established and give
reasons why this method can overcome the shortcoming of the separable error diffusion method.
2) Implement the algorithm using a standard error diffusion process (e.g. the Floyd-Steinberg error
diffusion) and apply it to Fig. 5. Compare the output with that obtained by the separable error
diffusion method.
EE 569 Digital Image Processing: Homework #2
Professor C.-C. Jay Kuo Page 8 of 8
Appendix:
RGB to grayscale conversion:
� = 0.2989 ∗ � + 0.5870 ∗ � + 0.1140 ∗ �
Problem 1: Edge detection
Gallery.raw 321x481x3 (Height×Width×Band) 24-bit color (RGB)
Dogs.raw 321x481x3 24-bit color (RGB)
Gallery.jpg 321x481x3 24-bit color (RGB)
Dogs.jpg 321x481x3 24-bit color (RGB)
Gallery_GT.mat (containing 5 ground truth annotations)
Dogs_GT.mat (containing 5 ground truth annotations)

(The following three are examples)
Helicopter.jpg 321x481x3 24-bit color (RGB)
Helicopter_prob.png 321×481 8-bit gray
Helicopter_binary.png 321×481 8-bit gray
Problem 2: Digital Half-toning
LightHouse.raw 500×750 8-bit gray
Rose.raw 480x640x3 24-bit color (RGB)
Reference Images
Images in this homework are from the BSDS 500 [2] and the Online images [4][6].
References
[1] J. Canny, “A computational approach to edge detection,” IEEE Transactions on pattern analysis and
machine intelligence, no. 6, pp. 679–698, 1986.
[2] P. Arbelaez, M. Maire, C. Fowlkes, and J. Malik, “Contour detection and hierarchical image
segmentation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, no. 5, pp. 898–916, May 2011.
[Online]. Available: http://dx.doi.org/10.1109/TPAMI.2010.161
[3] P. Dollár and C. L. Zitnick, “Structured forests for fast edge detection,” in Proceedings of the IEEE
International Conference on Computer Vision, 2013, pp. 1841–1848.
[4] [Online] https://unsplash.com/photos/-kEr-QltARg
[5] B. E. Bayer, “An optimum method for two-level rendition of continuous-tone pictures,” SPIE MILESTONE SERIES MS, vol. 154, pp. 139–143, 1999
[6] [Online] https://pixabay.com/photos/rose-nature-flowers-spring-174817/
[7] D. Shaked, N. Arad, A. Fitzhugh, I. Sobel, “Color Diffusion: Error-Diffusion for Color Halftones”,
HP Labs Technical Report, HPL-96-128R1, 1996.