MSBD5010-Assignment 1 Solved

Overview
 
This assignment consists of two sections: programming section and written section. Both programming and written parts should be submitted via the Canvas system on or before the due date. If you would like to finish the written assignment with hand writing, you may scan and upload it as a PDF file. 

 

In this programming assignment, you will use MATLAB to compute gradient magnitude of the image, to program an additive (zero mean) uniform random noise generator, periodic noise generator, arithmetic mean filter, Alpha-trimmed filter, and Wiener filter. A set of M-files can be obtained from the course website. The routine found in the msbd5010_assign1.m file performs a series of image processing and display operations on a pre-defined grayscale image. You are asked to complete the missing implementations of functions in the given skeleton code. 

 

Programming assignment specifics (80%)
Part 1: Histogram equalization (15%) 
 

The function histogram_eq.m involves a series of operations on the histograms of images.

 

1.1 Given a gray level image G, compute and display its histogram. Note: You are not allowed to use the built-in histogram function.

 

1.2 Perform histogram equalization on G, then compute and display the histogram of the result image.

 

1.3 Given a color image C in RGB mode, perform histogram equalization on the R, G, B channels of C separately. Rebuild a RGB image from these histogram-equalized channels.

1.4 Compute the histograms on R, G, B channels separately and then calculate an average histogram from these three histograms. Use this average histogram as the basis to obtain a single-valued histogram equalization intensity transformation function. Apply this function to the R, G, B channels individually. Rebuild an RGB image from these processed channels. 

 
Part 2: Create a gradient magnitude image from a grayscale image. (5%) 
 

You need to complete the function in the file “grad_mag_image.m”.  The function “grad_mag_image” takes a grayscale image of type uint8 as input and returns a gradient magnitude image of the same data type.  𝑚𝑚𝑚𝑚𝑚𝑚(∇𝑓𝑓) at 𝑧𝑧5 can be calculated with the following equation:  

𝑚𝑚𝑚𝑚𝑚𝑚(∇𝑓𝑓) ≈ |(𝑧𝑧7 + 𝑧𝑧8 + 𝑧𝑧9) − (𝑧𝑧1 + 𝑧𝑧2 + 𝑧𝑧3)| +                                                  |(𝑧𝑧3 + 𝑧𝑧6 + 𝑧𝑧9) − (𝑧𝑧1 + 𝑧𝑧4 + 𝑧𝑧7)| 

  
Hint: you need to define the gradient mask in “grad_mag_image.m”.  

Part 3.1: An additive uniform random noise generator. (5%) 
 

You need to complete the implementation of an additive uniform random noise generator in the gen_uniform_noise.m file. The routine

gen_uniform_noise takes four variables as input: size of an image along the Xaxis and Y-axis, and the lower bound and upper bound.  You should generate an image of additive uniform random noise with the given lower bound and upper bound. The data type of the output image should be double.

 
Hint: You can find a MATLAB internal function to generate uniform random numbers.

Part 3.2: Arithmetic mean filter. (5%) 
 

You need to complete the implementation of the routine in the arithmetic_mean_filter.m file.  The routine takes a noisy image (in grayscale format of type uint8) as input and attempts to remove the noise with an arithmetic mean filter. A 5 × 5 window is used to filter the noisy image. The output image type should be uint8.

Part 3.3: Alpha-trimmed mean filter. (15%) 
 

You need to complete the atrimmed_mean_filter.m file. The routine takes a noisy image (in grayscale format of type uint8) as input and attempts to remove the noise with an alpha-trimmed mean filter. Different values of d will be applied to the filter. A 3x3 window is used to filter the noisy image. The output image type should be uint8.  

 
Part 4: High-boost filter (10%) 
 

High-boost filtering is a process that has been used for many years by the printing and publishing industry to sharpen images consists of subtracting an unsharp (smoothed) version of an image from the original image. The process is as follows,

a.       Blur the original image 𝑓𝑓(𝑥𝑥, 𝑦𝑦), denote it as 𝑓𝑓(𝑥𝑥,𝑦𝑦).

b.      Subtract the blur image 𝑓𝑓(𝑥𝑥,𝑦𝑦) from the original image (the result difference is usually called the mask), denote it as 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑥𝑥, 𝑦𝑦)

c.       Add the mask to the original.

The high-boost filtering can be represented by the following equation,

𝑚𝑚(𝑥𝑥, 𝑦𝑦) = 𝑓𝑓(𝑥𝑥, 𝑦𝑦) + 𝑘𝑘∗𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑥𝑥, 𝑦𝑦),

Where 𝑘𝑘 > 1.

 

Perform high-boost filtering on the given image. The averaging part of the process should be done using the filter in Part 3.2. Choose a 𝑘𝑘 as you see fit.  

 

Part 5.1: Periodic noise (5%) 
 

You need to complete the implementation of an additive periodic noise generator in the gen_periodic_noise.m file. One of the periodic noise model with the size of 𝑀𝑀 × 𝑁𝑁 in 2D is the sine wave, which is given by 𝑟𝑟(𝑥𝑥, 𝑦𝑦) =

𝐴𝐴𝑥𝑥 sin(2𝜋𝜋𝑢𝑢0(𝑥𝑥 + 𝐵𝐵𝑥𝑥)/𝑀𝑀) + 𝐴𝐴𝑦𝑦 sin(2𝜋𝜋𝑣𝑣0(𝑦𝑦 + 𝐵𝐵𝑦𝑦)/𝑁𝑁)) ,where 𝐴𝐴𝑥𝑥and 𝐴𝐴𝑦𝑦 are amplitudes, 𝑢𝑢0 and 𝑣𝑣0 are the frequencies along x-axis and y-axis. The noise generator will be invoked in the following sub-tasks.

 

Part 5.2: Denoising in frequency domain (10%) 
 

                                                                      2𝑀𝑀                   2𝑁𝑁

Let 𝐴𝐴𝑥𝑥 = 𝐴𝐴𝑦𝑦 = 10, 𝑢𝑢0 =  𝜋𝜋 , 𝑣𝑣0 =  𝜋𝜋 , 𝐵𝐵𝑥𝑥 = 0, 𝐵𝐵𝑦𝑦 = 0, add the above periodic noise to the given image. You are required to implement the function band_reject_filter.m which performs bandreject filtering on a gray level image, returning a gray scale image. You can choose to implement either the Butterworth bandreject filter or the Gaussian bandreject filter, which are given as follows,

1

                                 𝐻𝐻𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵ℎ(𝑢𝑢, 𝑣𝑣) =  2𝑛𝑛

𝐷𝐷(𝑢𝑢, 𝑣𝑣)𝑊𝑊

1 + 𝐷𝐷2(𝑢𝑢, 𝑣𝑣) −𝐷𝐷02

 

1 𝐷𝐷2(𝐵𝐵,𝑣𝑣)−𝐷𝐷022

− 

                                         𝐻𝐻𝐺𝐺𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚𝐺𝐺𝑚𝑚𝑛𝑛(𝑢𝑢, 𝑣𝑣) = 1 −𝑒𝑒 2 𝐷𝐷(𝐵𝐵,𝑣𝑣)𝑊𝑊  

 

where 𝐷𝐷(𝑢𝑢, 𝑣𝑣) = (𝑢𝑢−𝑢𝑢0)2 + (𝑣𝑣−𝑣𝑣0)2 is the distance from the origin of the centered frequency rectangle, 𝑊𝑊 is the band width and 𝐷𝐷0 is its radial center. Please determine the suitable band width and its radial center for the periodic noise given by the above parameters.

 

Part 6: Wiener filtering with the power spectra of noise and un-degraded image. (10%) 
 

You need to complete the Wiener filter, suppose we know the power spectra of the noise 𝑆𝑆𝜂𝜂 and the un-degraded image 𝑆𝑆𝑓𝑓. You need to complete the implementation in the wiener_filter.m file. The data type of the output image should be uint8. You are not allowed to use the internal MATLAB function “wiener2” for this task.

 
Sample run of the programming assignment 
The main routine of the assignment including displaying the final results is well written in the msbd5010_assign1.m file. After completing all the functions, you are supposed to get three figures on the screen when you run the command below in the MATLAB environment.  

 
>> msbd5010_assign1; 

             
Written assignment specifics (20%)
The Fourier spectrum (Fourier Representation) in Fig.(b) corresponds to the original image Fig.(a) and the Fourier spectrum in Fig.(d) was obtained after the image Fig.(a) was padded with zeros (Shown in the Fig.(c)).  

a)      Explain the significant increase in signal strength along the vertical and horizontal axes of Fig.(d) compared with Fig.(b).

b)      Explain the significant increase in signal strength in the low frequency region of Fig.(d) compared with Fig.(b).