CS 549 – Computer Vision-HW #4 Solved

Description

 

 

 

1. (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 ∇²𝑔𝑔_𝜎𝜎 ∗ 𝑓𝑓) for various values of σ. Try 1, 2, 4, 8, 16. Do you get closed contours?

 

 

1. (9 pts) Show that
2. 𝑓𝑓(𝑥𝑥⃗) = 𝑔𝑔(𝑥𝑥⃗) ∗ ℎ(𝑥𝑥⃗) has Fourier Transform 𝐹𝐹(𝜔𝜔⃗) = 𝐺𝐺(𝜔𝜔⃗) × 𝐻𝐻(𝜔𝜔⃗). Hint: Write out the

F.T. and change variables.

 

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

Hint: Integrate by parts.

 

2. In 2-D the Laplacian operator ∇²= 𝑑𝑑𝑥𝑥^𝑑𝑑2₂ + _{𝑑𝑑𝑦𝑦}^𝑑𝑑2₂ has Fourier Transform −|𝜔𝜔⃗|². Hint: Use

𝑢𝑢

Part b. repeatedly and the fact that 𝜔𝜔⃗ =  .

𝑣𝑣

 

 

2. (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⁻₂^𝜎𝜎2 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).

 

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

 

3. 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.

 

1. 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.

 

 

3. (10 pts) Ima Robot proposes the following operators to detect diagonally oriented edges:

 

NE                                                                   NW

 

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

 

1. 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?

 

1. Express the NW operator as the convolution of two different 2×2 operators.

 

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

 

 

 

4. (5 pts) Read Canny’s PAMI article on Computational Edge Detection, available on Canvas.

 

1. List the 3 criteria that his approach optimizes.

 

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