Homework 3 Solution

Description

General Instructions:

1. Read Homework Guidelines and MATLAB Function Guidelines for the information about homework programming, write-up and submission. If you make any assumptions about a problem, please clearly state them in your report.

2. You need to understand the USC policy on academic integrity and penalties for cheating and plagiarism. These rules will be strictly enforced.

Problem 1: Geometric Modification (50%)

(a) Geometric Transformation (20%)

Three sub-images “lighthouse1.raw” ~ “lighthouse3.raw” as shown in Figure 1(a) were cut out of an image of a scene containing a lighthouse. The three holes in the image “lighthouse.raw” as shown in Figure 1(b) are all of size 256×256. Each sub-image is from one of the hole in the image, but with different scales and orientations. You are asked to fill the holes in “lighthouse.raw”. Write a program to implement a geometric transformation algorithm to create the filled image of size 512×512.

1. 31. 569 Digital Image Processing: Homework #3

3. For scaling-up, implement the interpolation function Θ(·) such that Θ( F_i[X_i (x, y)]) is of size 160 x 160. Drop redundant pixels when scaling-down.

3. Find the coordinates of the holes (specifically the top-left corners). This must be done

by your program.

5. Fill {Θ(F_i[X_i (x, y)]), i = 1,2,3} into the holes to get the final image. You can assume

you know “lighthouse1.raw” goes to the left hole, “lighthouse2.raw” goes to top, and

“lighthouse3.raw” goes to bottom-right.

Hint: Bi-linear interpolation may be needed to generate the pixel value at fractional positions.

(b) Spatial Warping (20%)

You can use the spatial warping technique to create fantastic images. Figure 2 shows an example:

We can warp a square-shaped image (on the left) into another shape (on the right).

(a) square-shaped image (b) warped image

Figure 2. Image warping

Develop a spatial warping algorithm as the one demonstrated in Figure 2 and apply it to the 512 x 512 gray scale image “hat.raw” shown in Figure 3 to obtain a warped image. The height of each arc in Figure 2(b) is 128, and the tip of the arc should be located at the center line of the image. Please indicate the reference points to be used and derive the reverse address mapping function that converts Figure 2(a) to (b).

Figure 3. hat.raw

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1. 31. 569 Digital Image Processing: Homework #3

3. Lens Distortion Correction (10%)

Radial distortion often happens when an image is captured on a non-flat camera’s focal plane. The relationship between the actual image and its distortion is as follows:

“ = x(1 + )	+	+ +	,	+ –	.	)	[1]
	+		,		.
“ = y(1 + )		+ +		+ –		)

where x, y are undistorted pixel locations, K₁, K₂, K₃ are called radial distortion coefficients of the lens, and ⁺ = ⁺ + ⁺.

To recover the undistorted image, the (x, y), given the (x_d, y_d), the key lies on finding the inverse function so that:

= ( “, “)	[2]
= ( “, “)

However, there is no exact inverse function for Eq. [1] since the forward mapping from (x, y) to

(x_d, y_d) is not linear. A common solution is first to project (x, y) to (x_d, y_d) which ends up in triplets of (x, y, x_d) and (x, y, y_d). We then use linear regression (refer to https://www.mathworks.com/help/matlab/data_analysis/linear-regression.html for further readings about linear regression) to find the best approximate inverse functions.

You are given the following distorted image (Figure 4.), and given that K₁ = -0.3536, K₂ = 0.1730, K₃ = 0, please implement the aforementioned method to correct the distortion. You can use third party tools (MATLAB, Python Scipy, etc.) for the linear regression part.

Figure 4. classroom.raw

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EE 569 Digital Image Processing: Homework #3

Problem 2: Morphological Processing (50%)

In this problem, you will implement three morphological processing operations: shrinking, thinning, and skeletonizing. A pattern table (patterntables.pdf) is attached for your reference. Please show outputs for all following parts in your report and discuss them thoroughly. Please state any assumptions you make in your solution.

(a) Basic Morphological Process Implementation (20%)

Please apply the “shrinking”, “thinning”, and “skeletonizing” filters to the 4 pattern images (pattern1-4.raw), respectively. Show your results and give reasonable explanation.

Figure 5. pattern1~4.raw

2. Defect Detection and Correction (15%)

One of the applications of morphological processing technique is defection detection. Suppose a deer image in Figure 6 is designed for product decoration, and it will be enlarged to fit multiple product sizes later.

Figture 6. deer.raw

For consistent product appeal, image defect detection to insure no black dots in the main deer body is necessary. Apply morphological process to help determine

1. Whether deer image is defectless.

2. If it is not defectless, where are defect regions? Correct them to white dots then.

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EE 569 Digital Image Processing: Homework #3

3. Object Analysis (15%)

Rice grain inspection is a procedure to define rice quality in the marketplace. Figure 7 is an example image for rice grain type examination.

Note: You need to convert the incoming RGB image to grayscale image first and then apply morphological processes. Some pre-processing or post-process may be needed to deal with unwanted dots and holes.

Figure 7. rice.raw

1. Count the total number of rice grains.

2. Compare the size of rice grains. Rank the grain’s size from small to large in terms of type.

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	EE 569 Digital Image Processing: Homework #3
Appendix:
Problem 1: Geometric image modification
lighthouse.raw	512×512	8-bit	gray
lighthouse1.raw	256×256	8-bit	gray
lighthouse2.raw	256×256	8-bit	gray
lighthouse3.raw	256×256	8-bit	gray
hat.raw	512×512	8-bit	gray
classroom.raw	1072×712	8-bit	gray
Problem 2: Digital Half-toning
pattern1-4.raw	375×375	8-bit	gray
deer.raw	691×550	8-bit	gray
rice.raw	500x690x3	24-bit	color(RGB)

Reference Images

All images in this homework are from the USC-SIPI [1] image database and MPEG-7 Shape Dataset [2].

Reference

1. http://sipi.usc.edu/database/

2. http://www.dabi.temple.edu/~shape/MPEG7/dataset.html

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