Description
. (Not to turn in, 0 pts) Use OpenCV or Matlab to compute the Sobel∇2 ∗edges of an image. In addition, compute the Marr- Hildreth edges (zero-crossings of ) for various values of σ. Try 1, 2, 4, 8, 16. Do you get closed contours?
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1. (9 pts) Show that |
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a. |
has Fourier Transform |
. Hint: Write out the |
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F.T. and change variables. |
( ⃗) = ( ⃗) × ( ⃗) |
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( ⃗) |
= ( ⃗) ∗ ℎ( ⃗) |
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b. |
In 1-D |
has Fourier Transform |
, assuming that |
as |
. |
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Hint: |
Integrate by parts. |
( ) |
( )→0 |
→ ±∞ |
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( )/ |
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c. In 2-D the Laplacian operator |
2 |
2 |
+ |
2 |
has Fourier Transform |
−| ⃗| |
2 |
. Hint: Use |
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2 |
2 |
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Part b. repeatedly and the fact that |
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∇ = |
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⃗ = � � |
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2. (9 pts) Ima Robot proposes an edge detector as follows: |
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Compute the Fourier Transform ( ⃗) of image |
( ⃗). |
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Multiply |
( ⃗) by 1 |
( ⃗) |
= − |
21 |
12| ⃗�|2 to form 1 |
( ⃗). |
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Multiply |
( ⃗) by 2 |
( ⃗) |
= − |
21 |
22| ⃗|2to form 2 |
( ⃗). |
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Compute |
3( ⃗) |
= |
2( ⃗)− 1( ⃗) |
. |
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2 |
− 1 |
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Compute ℎ3( ⃗) as the Inverse Fourier Transform of 3( ⃗). |
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Find zero-crossings of |
ℎ3 |
( ⃗). |
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a. |
Describe how |
can be computed by a single convolution with some kernel |
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What is the |
convolutional kernel |
? |
( ⃗) |
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ℎ3( ⃗) |
( ⃗) |
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( ⃗) = 1
3( �⃗) = 3( )
=
3( ⃗)
3( ⃗)
b.
show a slice through 3.
√ 2 + 2
is
If
, that is, the image has a “flat” spectrum, sketch
. Because
rotationally symmetric, that is,
( ⃗)
, where
, you only need to
c.
As
2
→ 1
2 → 1
ℎ3
occur at edges?
, is this a good edge detector, that is, do zero-crossings of
Why or why not? Hint: Consider
as
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3. (10 pts) Ima Robot proposes the following operators to detect diagonally oriented edges:
NE
NW
a.
How are these operators related to the Sobel H and V operators?
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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?
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Express the NW operator as the convolution of two different 2×2 operators.
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Show that |NW*I| + |NE*I| = Max(|H*I|,|V*I|)
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(5 pts) Read Canny’s PAMI article on Computational Edge Detection, available on Canvas.
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List the 3 criteria that his approach optimizes.
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Explain the drawback of using the Differences of Boxes edge operator.
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