Edge Detection CS/BIOEN 4640: Image Processing Basics February 9, - - PowerPoint PPT Presentation

Edge Detection

CS/BIOEN 4640: Image Processing Basics February 9, 2012

Gaussian Blurring for Derivatives

◮ We have seen Prewitt and Sobel derivative

• perators

◮ They use averaging in the orthogonal direction to

the derivative

◮ Why not average in both directions? ◮ How about using Gaussian blurring for averaging? ◮ How do we know what width (σ value) to use?

Gaussian-Blurred Edges

We’ve seen in 1D that If we blur a step edge with a Gaussian, we get this function

• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 3
• 2
• 1

1 2 3

This is the error function: erf(x) =

x

−∞ gσ(t) dt

Derivatives of Edges

◮ Using the error function as

• ur model for an edge
• 0.2

0.2 0.4 0.6 0.8 1 1.2

• 3
• 2
• 1

1 2 3

Derivatives of Edges

◮ Using the error function as

• ur model for an edge

◮ The derivative of an edge is

a Gaussian

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3

Derivatives of Edges

◮ Using the error function as

• ur model for an edge

◮ The derivative of an edge is

a Gaussian

◮ The second derivative of an

edge is the first derivative

• f a Gaussian
• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25

• 3
• 2
• 1

1 2 3

Derivatives and Convolution

◮ Let D = [0.5

0 −0.5] be our central difference

kernel, and G a Gaussian kernel

◮ Remember, blurring and derivatives can be done in

either order

(I ∗ D) ∗ G = (I ∗ G) ∗ D

◮ Also, our blurring kernel and derivative kernel can

be combined first, then applied to the image

(I ∗ D) ∗ G = I ∗ (D ∗ G)

DOGs

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25

• 3
• 2
• 1

1 2 3

Image derivatives computed by convolution with a Derivative of Gaussian (DOG) kernel

D ∗ G

• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2

• 3
• 2
• 1

1 2 3

Second derivatives can also be computed as convolution with second derivative of Gaussian

D ∗ D ∗ G

Thresholding Edges

◮ Gradient magnitude image has floating point values ◮ High values where there are strong edges ◮ Low values where there are weak or no edges ◮ Thresholding can remove the weak edges and

leave just the ones we want

◮ Converts image into a binary edge image

Thresholding Edges

◮ At an edge, gradient

magnitude looks like a Gaussian

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3

Thresholding Edges

◮ At an edge, gradient

magnitude looks like a Gaussian

◮ Threshold at some value

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3

Thresholding Edges

◮ At an edge, gradient

magnitude looks like a Gaussian

◮ Threshold at some value ◮ Leaves behind a “fat” edge

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3

Zero-Crossings of the Second Derivative

◮ We would prefer to choose

just the peak edge response

◮ So, we want a local

maximum

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 3
• 2
• 1

1 2 3

Zero-Crossings of the Second Derivative

◮ We would prefer to choose

just the peak edge response

◮ So, we want a local

maximum

◮ This is where the image

second derivative is zero

◮ Second derivative of an

edge looks like 1st derivative of Gaussian

• 0.25
• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25

• 3
• 2
• 1

1 2 3

That’s Great, But What About 2D?

◮ In 2D we have an x and y derivative ◮ Gradient of the image ∇I is computed as before,

but now with DOG kernels for x and y

◮ Zero-crossings of the second derivative now looks

at the Laplacian of the image:

∆I(x, y) = ∂2 ∂x2I(x, y) + ∂2 ∂y2I(x, y)

2D Gaussian

• 3
• 2
• 1

1 2 3-3

• 2
• 1

1 2 3 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Gσ(x, y) = 1 2πσ2 exp

• −x2 + y2

2σ2

2D Gaussian X-Derivative

• 3
• 2
• 1

1 2 3-3

• 2
• 1

1 2 3

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.1

∂ ∂xGσ(x, y) = −x 2πσ4 exp

• −x2 + y2

2σ2

2D Gaussian Y-Derivative

• 3
• 2
• 1

1 2 3-3

• 2
• 1

1 2 3

• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.1

∂ ∂yGσ(x, y) = −y 2πσ4 exp

• −x2 + y2

2σ2

2D Gaussian Second X-Derivative

• 3
• 2
• 1

1 2 3-3

• 2
• 1

1 2 3

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1

∂2 ∂x2Gσ(x, y) = x2 − σ2 2πσ6 exp

• −x2 + y2

2σ2

2D Gaussian Second X-Derivative

• 3
• 2
• 1

1 2 3-3

• 2
• 1

1 2 3

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1

∂2 ∂y2Gσ(x, y) = y2 − σ2 2πσ6 exp

• −x2 + y2

2σ2

2D Edge Detection Algorithm with Laplacian

• 1. Compute gradient magnitude image using DOG

kernels

• 2. Threshold this image to include only edges with

high magnitude - binary image

• 3. Compute Laplacian image using second-DOG

kernels

• 4. Find zero crossings of Laplacian - binary image (1

at crossing, 0 elswhere)

• 5. AND operation between thresholded gradient

magnitude and Laplacian zero crossings

More Details on DOG Kernels

Let D = [0.5

0 −0.5] be our central difference

kernel, and Gσ a Gaussian kernel. Then we can compute our DOG kernel H = Gσ ∗ D as follows:

H(k) = 1 2(gσ(k + 1) − gσ(k − 1))

Here k goes from −R to R and gσ is the 1D Gaussian function

Second Derivative Finite Difference Operator

A second derivative finite difference looks like this:

δ2 f(x) = f(x − 1) − 2f(x) + f(x + 1)

This can be computed as a convolution with the kernel

D2 = [1 −2 1]

Be careful! δ2 f = δ(δf)

Second Derivative of Gaussian Kernels

Now let D2 = [1 −2 1] be our second derivative kernel, and Gσ be the Gaussian kernel. Then we can compute the second-DOG kernel H2 = Gσ ∗ D2 as follows:

H2(k) = gσ(k − 1) − 2gσ(k) + gσ(k + 1)

How To Compute Zero Crossings

Important step in the Laplacian edge detector. Do the following for each pixel I(u, v):

• 1. Look at your four neighbors, left, right, up and down
• 2. If they all have the same sign as you, then you are

not a zero crossing

• 3. Else, if you have the smallest absolute value

compared to your neighbors with opposite sign, then you are a zero crossing

FeatureJ

This is a nice package for computing derivatives, edge-detection, and more (ImageJ plugin):

www.imagescience.org/meijering/software/featurej/