matlab questions, 1D and 2D convolution and digital holography


University of South Florida

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hey there!! i am looking for a computer science tutor or an electrical engineer. I need help with my entire assignment it has 1D and 2D convolution, matlab questions and digital holography questions .The name of the course is COMPUTATIONAL METHOD OF IMAGING. i do have slides which could help. please bid it's urgent.

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CIS 4930.006S20/CIS 6930.013S20: Computational Methods for Imaging and Vision Spring 2020 Homework #4 The University of South Florida Department of Computer Science and Engineering Tampa, FL Assigned: April 06, 2020 Due: April 19, 2020 (11:59 PM) 1 1D and 2D Convolutions (a) Write your own Matlab/Python function to implement 1D discrete convolution, between two vectors: • Use it to compute the convolution between the following two vectors:   −3    9  6    , and f = −5 4 h=   −10 3 9 • Verify the output of your result by computing the h ? f by hand. (b) (Optional) In this question, you will compute the convolution between an image of your choice, and the Gaussian filter. The PSF for the Gaussian filter is:   1 4 6 4 1 4 16 24 16 4  1  6 24 36 24 6 . H=   320  4 16 24 16 4 1 4 6 4 1 • For an image of your choice (please keep it PG!), load the image (read it in as a matrix), on Matlab you can use the function: ‘imread’. • Convolve that image with the matrix H. • Display the result of your discrete convolution. • Comment on the difference between the result and the original image. 1 2 Digital Holography Holography is a 3D imaging technique, in the sense that it allows recreate the 3D scene (optically or digitally) from a single intensity measurement. In this problem, we will explore the general idea of in-line (Gabor) holography and understand the unique feature about holography from the linear system perspective. A schematic of the in-line holography is shown in Fig. 2. To record a hologram, a coherent light source (e.g. laser) is needed to illuminate the 3D scene. The hologram (the intensity image captured by the camera) is the result from the interference between the unperturbed illumination (reference beam) and the light scattered from the object. The formation of the hologram from a 2D object can be approximated using the following linear shift invariant (LSI) model: gout (x, y) = gin (x, y; z) ? h(x, y; z), (1) where ? denotes the 2D convolution (i.e. over (x, y)), gout (x, y) is the output term of interest contained in the hologram, gin (x, y; z) is the object function, and h(x, y; z) is the point spread function (PSF), determined by the free-space propagation and diffraction theory, which has the following form, h(x, y; z) = 1 jk x2 +y2 e 2z , jλz (2) and the corresponding transfer function, 2 2 H(fx , fy ; z) = F{h(x, y; z)} = e−jπλz(fx +fy ) , (3) where k = 2π/λ is the wavenumber, λ is the wavelength of illumination, (x, y) denote the lateral coordinates and z denotes the axial direction (along which the light propagates) and (fx , fy ) denote the spatial frequency coordinates, according to the following Fourier transform relation Z +∞ Z +∞ H(fx , fy ; z) = −∞ h(x, y; z)e−j2π(fx x+fy y) dxdy, (4) −∞ note that we have used the substitutions (or change of variables) ωx = 2πfx and ωy = 2πfy in the usual definition of the Fourier transform we saw in class. (a) Construct the forward models in both operator and matrix forms. Assume square pixels with size ∆ and number of pixels in each spatial dimension is N . Discretization of the object space is performed on the standard basis with square pixel of size δ, and number of pixels in each spatial dimension M is chosen such that M δ = N ∆. (b) Find adjoint, and inverse of the system. (c) Write a Matlab/Python script for the forward model in (a). (d) Plot the system transfer functions with the following parameters: λ = 0.5 µm, N = 1000. (i) Keep z = 50 mm fixed, δ = ∆ = 2 µm, 5 µm, 10 µm, 20 µm. How does sampling affect the performance of digital holography? (ii) Keep δ = 5 µm fixed, while z ranges from 30 mm to 70 mm (use a small step-size). How does object distance affect the performance of digital holography? (iii) Keep ∆ = 5 µm and z = 50 mm fixed, while δ ranges from 1 µm to 10 µm in 2 µm step size. How does discretization of the object affect the performance of digital holography? 2 Figure 1: Digital holography schematic Digital holographic reconstruction: Inverse Write Matlab scripts to complete the following questions. Submit both your Matlab scripts as well as the output results. The object, ‘f2’ located at z1 = 50mm is given in the mat-file o2.mat. Use the following parameters throughout this problem for forming the hologram: λ = 0.5µm, N = 1000, and ∆ = δ = 5µm. Consider the object o2 is present. (e) Simulate the hologram using the discrete forward model (y = Ax) derived from. (f) Next, we will consider holographic reconstruction using the same parameters as in (e). • What is the condition number of the imaging matrix A? (g) Finally, we will investigate the effect of Gaussian noise. Gaussian noise is commonly encountered in imaging. The fundamental assumption is that the noise is independent of the measured intensity. The simplest case is the white Gaussian noise (WGN), which assumes the mean as the noise-free image Inoisefree , and the same variance nstd across all pixels. The noisy image under WGN, Inoisy , can be generated using the following Matlab script: Inoisy = normrnd(Inoisefree , nstd , Ny , Nx ); (where Ny , Nx are the dimensions of the image along vertical and horizontal directions.) Consider white Gaussian noise (WGN) with nstd = 0 + 0j (i.e. noise-free), 0.1 + 0.1j, 0.5 + 0.5j: • What is the direct deconvolution result? • What is the result from the least square solution? • What is the optimal Tikhonov deconvolved result? • Comment on what you have observed by comparing the three results. Why? 3 (Optional for Extra credit: 3D Digital Holography) Consider the 3D holographic imaging problem. The formation of the hologram from an extended scene can be approximated using the following linear model Z gout (x, y) = gin (x, y; zl ) ? h(x, y; zl )dzl (5) zl where ? denotes the 2D convolution in (x, y), gin (x, y; zl ) is the object function at the distance zl . (h) Construct the forward models in both operator and matrix forms. Assume that the discretization of the object space is performed on the standard basis with lateral spacing δx = δy = δ, and axial spacing δz . (i) Is the system shift invariant? If so, plot the system transfer function. If not, plot the singular values of the system. (j) Find the image space, object space, range, null space, and adjoint of the system. Does the inverse exist? (k) Write a Matlab/Python script for the forward model in part (e). 4 ...
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