Edge in 1D-Introduction to Computer Vision-Lecture 05-Computer Science, Lecture notes of Computer Vision

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Intro to Computer Vision

Lecture 5

Greg Shakhnarovich

April 15, 2010

Review

Gaussians: x

p(x)

Gaussian blurring

Filter separability; integral image (^) A

B

C

D

Looking for edges

Let’s start with 1D problem

Edge in 1D - idealized

Different types of 1D edges:

step ramp

bar ridge

Edge in 1D - idealized

Different types of 1D edges:

step ramp

bar ridge

change in function value: magnitude of the derivative!

f (x)

∂f (x) ∂x

Edge in 1D - idealized

Different types of 1D edges:

step ramp

bar ridge

change in function value: magnitude of the derivative! (squared)

f (x)

∂f (x) ∂x

( (^) ∂f (x) ∂x

Edge in 1D - discrete

x − 1 x^ x + 1

I(x)

Edge in 1D - discrete

x − 1 x^ x + 1

I(x) I(x − 1)

I(x + 1)

Edge in 1D - discrete

x − 1 x^ x + 1

I(x) I(x − 1)

I(x + 1) Ix(x)^ ,^

∂I

∂x = (^) ∆limx→ 0 I(x + ∆x) − I(x) ∆x ≈ I(x + 1) − I(x) ≈ I(x + 1) − I(x − 1)

Edge in 1D - discrete

x − 1 x^ x + 1

I(x) I(x − 1)

I(x + 1) Ix(x)^ ,^

∂I

∂x = (^) ∆limx→ 0 I(x + ∆x) − I(x) ∆x ≈ I(x + 1) − I(x) ≈ I(x + 1) − I(x − 1)

Can we compute the derivative by convolution?

Edge in 1D - discrete

x − 1 x^ x + 1

I(x) I(x − 1)

I(x + 1) Ix(x)^ ,^

∂I

∂x = (^) ∆limx→ 0 I(x + ∆x) − I(x) ∆x ≈ I(x + 1) − I(x) ≈ I(x + 1) − I(x − 1)

Can we compute the derivative by convolution? Yes: filter mask Dx =

[

]

Barbara 1D profile:

I

Edge in 1D - discrete

x − 1 x^ x + 1

I(x) I(x − 1)

I(x + 1) Ix(x)^ ,^

∂I

∂x = (^) ∆limx→ 0 I(x + ∆x) − I(x) ∆x ≈ I(x + 1) − I(x) ≈ I(x + 1) − I(x − 1)

Can we compute the derivative by convolution? Yes: filter mask Dx =

[

]

Barbara 1D profile:

I

Ix = I ∗ Dx

Noisy edge in 1D

Smoothing the signal and then taking the derivative:

I

Noisy edge in 1D

Smoothing the signal and then taking the derivative:

I

I ∗ Gx,σ