CS549-Homework 4 Solved

Not to turn in, 0 pts) Use OpenCV or Matlab to compute the Sobel edges of an image. In addition, compute the Marr-Hildreth edges (zero-crossings of ∇2𝑔𝑔𝜎𝜎 ∗ 𝑓𝑓) for various values of σ. Try 1, 2, 4, 8, 16. Do you get closed contours?
 

(9 pts) Show that
𝑓𝑓(𝑥𝑥⃗) = 𝑔𝑔(𝑥𝑥⃗) ∗ ℎ(𝑥𝑥⃗) has Fourier Transform 𝐹𝐹(𝜔𝜔⃗) = 𝐺𝐺(𝜔𝜔⃗) × 𝐻𝐻(𝜔𝜔⃗). Hint: Write out the
F.T. and change variables.

 

In 1-D 𝑑𝑑𝑓𝑓(𝑥𝑥)/𝑑𝑑𝑥𝑥 has Fourier Transform 𝑗𝑗𝜔𝜔𝐹𝐹(𝜔𝜔), assuming that 𝑓𝑓(𝑥𝑥) → 0 as 𝑥𝑥 → ±∞.
Hint: Integrate by parts.

 

In 2-D the Laplacian operator ∇2= 𝑑𝑑𝑥𝑥𝑑𝑑22 + 𝑑𝑑𝑦𝑦𝑑𝑑22 has Fourier Transform −|𝜔𝜔⃗|2. Hint: Use
𝑢𝑢

Part b. repeatedly and the fact that 𝜔𝜔⃗ =  .

𝑣𝑣

 

(9 pts) Ima Robot proposes an edge detector as follows:
 

Compute the Fourier Transform 𝐹𝐹(𝜔𝜔⃗) of image 𝑓𝑓(𝑥𝑥⃗).

 1 2|𝜔𝜔⃗|2

Multiply 𝐹𝐹(𝜔𝜔⃗) by 𝐺𝐺1(𝜔𝜔⃗) = 𝑒𝑒−2𝜎𝜎1 to form 𝐻𝐻1(𝜔𝜔⃗).

 −1 2|𝜔𝜔⃗|2

Multiply 𝐹𝐹(𝜔𝜔⃗) by 𝐺𝐺2(𝜔𝜔⃗) = 𝑒𝑒 2𝜎𝜎2 to form 𝐻𝐻2(𝜔𝜔⃗).

 

Compute 𝐻𝐻3(𝜔𝜔⃗) = 𝐻𝐻2(𝜔𝜔𝜎𝜎⃗2)−−𝐻𝐻𝜎𝜎11(𝜔𝜔⃗).

Compute ℎ3(𝑥𝑥⃗) as the Inverse Fourier Transform of 𝐻𝐻3(𝜔𝜔⃗).

Find zero-crossings of ℎ3(𝑥𝑥⃗).

 

Describe how ℎ3(𝑥𝑥⃗) can be computed by a single convolution with some kernel 𝑔𝑔(𝑥𝑥⃗). What is the convolutional kernel 𝑔𝑔(𝑥𝑥⃗)?
 

If 𝐹𝐹(𝜔𝜔⃗) = 1, that is, the image has a “flat” spectrum, sketch 𝐻𝐻3(𝜔𝜔⃗). Because 𝐻𝐻3(𝜔𝜔⃗) is rotationally symmetric, that is, 𝐻𝐻3(𝜔𝜔⃗) = 𝐻𝐻3(𝜌𝜌), where 𝜌𝜌 , you only need to show a slice through 𝐻𝐻3.
 

As 𝜎𝜎2 → 𝜎𝜎1, is this a good edge detector, that is, do zero-crossings of ℎ3 occur at edges? Why or why not? Hint: Consider 𝐺𝐺(𝜔𝜔⃗) as 𝜎𝜎2 → 𝜎𝜎1.
 

(10 pts) Ima Robot proposes the following operators to detect diagonally oriented edges:
                NE                                                                   NW

 

How are these operators related to the Sobel H and V operators?
 

Suggest two different ways in which to combine the NW and NE operators into a single measure of edge strength. What are the relative strengths and weaknesses of each?
Express the NW operator as the convolution of two different 2×2 operators.
 

Show that |NW*I| + |NE*I| = Max(|H*I|,|V*I|)
 

(5 pts) Read Canny’s PAMI article on Computational Edge Detection, available on Canvas.
 

List the 3 criteria that his approach optimizes.
 

Explain the drawback of using the Differences of Boxes edge operator.