Reference no: EM132157603

Assignment - Image processing & Electronics Questions

Q1. Let f(x, y) = 0.5 rect(4(x-0.5), 2(y-0.25)) and h(x, y) = rect(10x, 10y), where x and y are in ph.

(a) Sketch the region of support of f(x, y) and h(x, y) in the XY-plane (i.e., the area where these two signals are nonzero).

(b) Compute the two-dimensional convolution f(x, y) ∗ h(x, y) from the definition using integration in the spatial domain.

(c) Suppose that f(x, y) is the input to a two-dimensional system, and thea output of this system is computed as in (b). What can we say about this system?

(d) Determine the continuous-space Fourier transforms F(u, v), H(u, v) and G (u, v) of the above three signals. Make liberal use of Fourier transform properties. What are the units of u and v?

(e) Continuing with question (c), what is the interpretation of H(u, v)?

Q2. A two-dimensional continuous-space linear shift-invariant system has impulse response

(a) Sketch the region of support of the impulse response in the XY-plane, following the conventions used in the course for the labelling of axes. Express h(x, y) in terms of the circ function.

(b) Find the frequency response H(u, v) of this system, where u and v are in c/ph.

(c) The image f(x, y) = rect(5(x-.5), 2(y-.5)) is filtered with this system to produce the output g(x, y) = f(x, y)∗h(x, y). Determine the Fourier transform of the output, G(u, v).

Q3. Compute the two-dimensional continuous-space Fourier transform of the following signals:

(a) The separable signal f(x, y) = h_X⁽¹⁾(x)h_Y⁽¹⁾(y) where

(b) A Gaussian function f(x, y) = 1/(2πr₀²)e^{-(x^2+y^2)/2r_0^2}.

(c) A real zoneplate, f(x, y) = cos(π(x²+y²)/r₀²). (Hint: Find the Fourier transform of the complex zoneplate exp (jπ(x²+y²)/r₀²) and use linearity. You can use _-∞∫^∞e^{jy^2} dy = √πe^jπ/4.)

(d) Diamond-shaped pulse

(Hint: obtain this function from a rect function using an affine transformation.)

(e) Gabor function

f(x, y) = cos(2π(u₀x+v₀y))exp(-((x-x₀)² + (y-y₀)²)/2r₀²)

Q4. A two-dimensional continuous signal fc(x, y) has Fourier transform

for some real number W. The signal is sampled on a hexagonal lattice Λ with sampling matrix

to give the sampled signal f[x, y], (x, y) ∈ Λ, with Fourier transform F(u, v).

(a) What is the expression for F (u, v) in terms of Fc(u, v)?

(b) Find the largest possible value of X such that there is no aliasing?

Sketch the region of support of the Fourier transform of the sampled signal in this case (including all replicas), and also indicate a unit cell of the reciprocal lattice Λ∗.

Q5. The web color steelblue that we will denote [Q] is specified by the RGB values 70, 130, 180, on a scale from 0 to 255. Thus, they can be assumed to be Q'_R = 0.2745, Q'_G = 0.5098, Q'_B = 0.7059 on a scale from 0 to 1. We assume that these are gamma-corrected values, according to the Rec. 709 gamma law, and that the primaries are the Rec. 709 RGB primaries, normalized with respect to reference white D₆₅. The goal of this problem is to determine representations of this color in other color coordinate spaces. Determine the following showing all work:

a) the tristimulus values Q_R, Q_G, Q_R in the Rec. 709 RGB color space;

b) the luminance Q_L and the chromaticities q_R, q_G, q_R in the Rec. 709 space;

c) the XYZ tristimulus values Q_x, Q_y, Q_z and the corresponding chromaticities q_x, q_y, q_z;

d) the 1976 U'V'W' tristimulus values Q_U', Q_V', Q_W' and the corresponding chromaticities q_U', q_V', q_W'

e) the CIELAB coordinates Q_L*, Q_a*, Q_b*

f) the Luma and color differences Q_Y', Q_{P_B}, Q_{P_R}

You can visualize this color in any Windows program that lets you specify the RGB values of a color. For example, in Microsoft Word, draw a shape like a rectangle and set the fill color using "More Colors - Custom" and enter the red, green and blue values 70, 130, 180 in the boxes.

Q6. The NTSC primaries, now obsolete, have the following specification

	Red	Green	Blue	White C
x	0.67	0.21	0.14	0.310
y	0.33	0.71	0.08	0.316
Z	0.00	0.08	0.78	0.374

Assume that the reference white has unit luminance C_L = 1.0 and that [R]+[G]+[B] = [C].

(a) Find the XYZ tristimulus values of the reference white [C], i.e., C_X, C_Y and C_Z.

(b) Using [R] + [G] + [B] = [C], determine the luminances of the three primaries, R_L, G_L, and B_L.

(c) Now find the XYZ tristimulus values of the three primaries, i.e. if [R] = R_X[X] + R_Y[Y] + R_Z[Z], find R_X, R_Y, R_Z, and similarly for [G] and [B].

(d) If an arbitrary color [Q] is written

[Q] = Q_X[X] + Q_Y[Y] + Q_Z[Z] = Q_R[R] + Q_G[G] + Q_B[B]

determine the matrix relations to find Q_X, Q_Y, Q_Z from Q_R, Q_G, Q_B and vice-versa.

(e) Plot an xy chromaticity diagram showing the triangles of chromaticities reproducible with the NTSC RGB primaries and with the Rec. 709 RGB primaries. A matlab code is included on the moodle page for your convenience.

(f) The color matching functions of the XYZ primaries are available on the moodle. Compute and plot the color matching functions for the NTSC RGB primaries by transforming the XYZ color matching functions using the results of (d).