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EE640 Project 1: Generating Stochastic Images and Noise, Lecture notes of Mechanical Engineering

University of Kentucky (UK)Mechanical Engineering

A project for ee640: stochastic systems, where students are required to generate various types of stochastic images and noise. The project involves generating uniform and gaussian noise from uniform data, creating control noise from deterministic data, and generating digital stochastic signal images. The document also includes instructions for visualizing the results and demonstrating symmetry conditions in the frequency domain.

Typology: Lecture notes

2019/2020

Uploaded on 06/05/2020

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bg1
2/12/2009
EE640 PROJECT 1
1
EE640 STOCHASTIC SYSTEMS
SPRING 2008
COMPUTER PROJECT 1A
PART A: SYNTHESIS(4-8-08)
A.1 GENERATING GAUSSIAN NOISE FROM UNIFORM NOISE
Let Nx=My=128:
1. Uniform pseudo-random numbers. Generate 6 random images, each with a different seed.
The images are all My x Nx where each element is uniformly distributed between 0 and 1.
Each element is independent from the others. Let N = My x Nx. Mathematically refer to the
images as N x 1 vectors in lexicographical form as:
2. Prove the parametric transformation equations for converting from a uniform distribution to a
gaussian distribution are correct.
where n=0,1,2, …., (N/2 -1).
3. Generate six My x Nx gaussian random images from the associated images in part A.1.1. Use
the transformation developed in A.1.2. Generate them with a 0 mean and unity variance and
store as you did in A.1.1. Refer to them in lexicographical form as
4. Linear combinations of r.v.s. Generate five My x Nx images such that
Display your results in a way that is suitable for visualizing them.
In the next project 1B, you will analyze your synthetic data.
(A-1)
123456
u,u,u,u,u,u
[]
[
]
[
]
222cos12ln212 ++−=+ nunung ii
i
π
(A-2)
[]
[
]
[
]
222sin12ln222 ++−=+ nunung ii
i
π
(A-3)
1234 56
g,g,g,g,g,g (A-5)
112 4 12345
2123 512 6
31234
s=u+u s=u+u+u+u+u
s=u+u+u s=u+u+...u
s=u+u+u+u
(A-6)
pf3
pf4

Partial preview of the text

Download EE640 Project 1: Generating Stochastic Images and Noise and more Lecture notes Mechanical Engineering in PDF only on Docsity!

EE640 PROJECT 1

EE640 STOCHASTIC SYSTEMS

SPRING 2008

COMPUTER PROJECT 1A

PART A: SYNTHESIS(4-8-08)

A.1 GENERATING GAUSSIAN NOISE FROM UNIFORM NOISE

Let Nx=My=128:

  1. Uniform pseudo-random numbers. Generate 6 random images, each with a different seed. The images are all My x Nx where each element is uniformly distributed between 0 and 1. Each element is independent from the others. Let N = My x Nx. Mathematically refer to the images as N x 1 vectors in lexicographical form as:
  2. Prove the parametric transformation equations for converting from a uniform distribution to a gaussian distribution are correct.

where n =0,1,2, …., ( N /2 -1).

  1. Generate six My x Nx gaussian random images from the associated images in part A.1.1. Use the transformation developed in A.1.2. Generate them with a 0 mean and unity variance and store as you did in A.1.1. Refer to them in lexicographical form as
  2. Linear combinations of r.v.s. Generate five My x Nx images such that

Display your results in a way that is suitable for visualizing them. In the next project 1B, you will analyze your synthetic data.

u ,u ,u ,u ,u ,u 1 2 3 4 5 6^ (A-1)

g i [ 2 n + 1 ] = − 2 ln ui [ 2 n + 1 ] cos 2 π ui [ 2 n + 2 ] (A-2)

g i [ 2 n + 2 ] = − 2 ln ui [ 2 n + 1 ] sin 2 π ui [ 2 n + 2 ] (A-3)

g , g , g , g , g , g 1 2 3 4 5 6 (A-5)

1 1 2 4 1 2 3 4 5 2 1 2 3 5 1 2 6 3 1 2 3 4

s = u +u s = u +u +u +u +u s = u +u +u s = u +u +...u s = u +u +u +u

(A-6)

EE640 PROJECT 1

A.2 GENERATING CONTROL NOISE FROM DETERMINISTIC DATA

There are 23 target and clutter images which are saved in target.zip and clutter.zip, respectively. The size of each image is 128 x 128 pixels. Choose one target and one clutter image from the two classes. Fig. 1 shows an example of 5 target training and Fig. 2 shows 5 examples of clutter images.

Figure 1: Five samples of the 23 target images.

Figure 2: Five samples of the 23 clutter images.

  1. Find the PSD envelope of your target and clutter images by taking the 2-D Discrete Fourier Transform, then select the magnitude of the resulting spectra and refer to it has Ht for the target and Hc for the clutter image. We want to generate noise with an equivalent PSD.
  2. Generate two independent My x Nx uniform noise images, At and Ac, with element ranges between 0 and 2π. Combine these with Ht and Hc such that the magnitude of each element is the same as Ht and Hc but the phase angle of the complex elements is determined by the two random images At and Ac, respectively. Inverse the result back to space domain and display the noise images.
  3. Generate two independent My x Nx Gaussian noise images, gt and gc with element values having zero mean and unit variance. 2-D DFT these images to get Gt and Gc spectra. Elementwise multiply these spectra by Ht and Hc, respectively. Then inverse DFT the result back to the space domain and display results.
  4. Are the noise images pairs, in task 2 and 3, statistically equivalent? Show why or why not, mathematically.

then real(B(m,n))=real(B(m,n1)) where n1=2+N-n. The same is true for the column index such

that if n=1 and m>M/2 then m1=2+M-m, so that real(B(m,n))=real(B(m1,n)). then imag(B(m,n)) = -imag(B(m,n1)) where n1=2+N-n. The same is true for the column index

  • 2/12/ - EE640 PROJECT
    • 33.5960 2.5077 1.4570 -2.7182 -0.8693 -2.7182 1.4570 2.
      • -1.1868 -2.5328 -1.0442 -0.3453 0.8548 -0.5695 -0.9498 0.
        • 2.0612 -0.2416 1.2408 2.2483 0.8064 -1.4776 1.4948 0.
        • 1.5700 0.1783 -1.1027 3.3849 0.9106 -0.3712 -1.7902 3.
        • 0.0120 1.3609 1.9355 2.6287 -0.6061 2.6287 1.9355 1.
        • 1.5700 3.3390 -1.7902 -0.3712 0.9106 3.3849 -1.1027 0.
        • 2.0612 0.0927 1.4948 -1.4776 0.8064 2.2483 1.2408 -0.
      • -1.1868 0.5963 -0.9498 -0.5695 0.8548 -0.3453 -1.0442 -2.
  • Note that the for a column index n=1,2,…N and a row index m=1,2,…M, that if m=1 and n>N/ - 0 0.5449 -1.3813 3.1988 0 -3.1988 1.3813 -0. imag(B)= - -1.0333 -1.0398 0.6684 1.8551 -0.7279 -0.2027 -0.8158 0. - 0.1559 -2.0416 -2.9714 -0.5650 -1.3007 2.7827 0.6965 -0. - 0.3264 -0.9061 0.4175 1.0483 -1.8740 -0.7414 2.5313 0. - 0 0.3909 -3.4593 2.7975 0 -2.7975 3.4593 -0. - -0.3264 -0.7601 -2.5313 0.7414 1.8740 -1.0483 -0.4175 0. - -0.1559 0.6664 -0.6965 -2.7827 1.3007 0.5650 2.9714 2. - 1.0333 -0.4521 0.8158 0.2027 0.7279 -1.8551 -0.6684 1.
  • Note that the for a column index n=1,2,…N and a row index m=1,2,…M, that if m=1 and n>N/