## Description

## 1 Convolutions

(a) Compute the (continuous) convolution of g(x) and h(x) for the following three cases:

(i) The functions g(x) and h(x) are shown below (Figure 1)

(a): g(x)

(b): h(x)

Figure 1: Graphs of g(x) and h(x).

(ii) The functions g(x) = e

−(x−1)u(x − 1) and h(x) = u(x + 1) are shown below (Figure 2)

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(a): g(x) = e

−(x−1)u(x − 1)

(b): h(x) = u(x + 1)

Figure 2: Graphs of g(x) and h(x).

(iii) The functions g(x) and h(x) are shown below (Figure 3)

(a): g(x)

(b): h(x)

Figure 3: Graphs of g(x) and h(x).

(b) Let g(x) = sinc(ax) and h(x) = sinc(bx), where a, b are real-valued scalar quantities, with a ≤ b.

Using the convolution property of the Fourier transform, compute the convolution between the

functions g(x) and h(x). To do this:

(i) State the convolution property of the Fourier transform.

(ii) Using the result F{sinc(x)} = πrect(ω/2) (see problem 2(b) of Homework #2) and the scaling

property of the Fourier transform, find the Fourier transforms for g(x) and h(x).

(iii) Using your answers to part (ii) above, and the convolution property of the FT, compute

F(g(x) ? h(x)). (Remember that a ≤ b.)

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(iv) Take the inverse Fourier transform of your answer to (iii) above, to obtain g(x) ? h(x).

(c) Let g(x) = √

1

2πa

e

− x

2

2a2 and h(x) = √

1

2πb e

− x

2

2b2 .

(i) Compute the convolution between the functions g(x) and h(x), directly, that is by using the

definition of the 1D convolution integral. [Hint: At some point, you will need to use the fact

that the integral R ∞

−∞ e

−

(x−µ)

2

2σ2 =

√

2πσ, where µ and σ are scalars.]

(ii) Compute the convolution between the functions g(x) and h(x) by applying the convolution

property of the Fourier transform. [You may need the above hint.]

## 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

jλz e

jk

x

2+y

2

2z , (2)

and the corresponding transfer function,

H(fx, fy; z) = F{h(x, y; z)} = e

−jπλz(f

2

x+f

2

y

)

, (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

H(fx, fy; z) = Z +∞

−∞

Z +∞

−∞

h(x, y; z)e

−j2π(fxx+fyy) 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) Write down the mathematical expression for the operator that represents the imaging system, in

both standard form and spectral form.

(b) Find the range, null space, adjoint, and inverse of the system operator.

(c) Construct the forward models in both matrix form of the imaging system. 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 Mδ = N∆.

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Figure 4: Caption

(d) Find the range, null space, adjoint, and inverse of the discretized system.

(e) Write a Matlab/Python script for the forward model in (c).

(f) 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?

(Optional for Extra credit: 3D) Consider the 3D holographic imaging problem. The formation of the

hologram from an extended scene can be approximated using the following linear model

gout(x, y) = Z

zl

gin(x, y; zl) ? h(x, y; zl)dzl (5)

where ? denotes the 2D convolution in (x, y), gin(x, y; zl) is the object function at the distance zl

.

(g) 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.

(h) Is the system shift invariant? If so, plot the system transfer function. If not, plot the singular values

of the system.

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(i) Find the image space, object space, range, null space, and adjoint of the system. Does the inverse

exist?

(j) Write a Matlab/Python script for the forward model in part (g).

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