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 operators ◮ 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 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -3 -2 -1 0 1 2 3 � x This is the error function: erf ( x ) = −∞ g σ ( t ) dt

Derivatives of Edges ◮ Using the error function as our model for an edge 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -3 -2 -1 0 1 2 3

Derivatives of Edges ◮ Using the error function as our model for an edge 0.4 ◮ The derivative of an edge is 0.35 0.3 a Gaussian 0.25 0.2 0.15 0.1 0.05 0 -3 -2 -1 0 1 2 3

Derivatives of Edges ◮ Using the error function as our model for an edge 0.25 ◮ The derivative of an edge is 0.2 0.15 0.1 a Gaussian 0.05 0 ◮ The second derivative of an -0.05 -0.1 -0.15 edge is the first derivative -0.2 -0.25 of a Gaussian -3 -2 -1 0 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 0.2 0.1 0.15 0.1 0 0.05 0 -0.1 -0.05 -0.2 -0.1 -0.15 -0.3 -0.2 -0.25 -0.4 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Image derivatives computed by Second derivatives can also be convolution with a Derivative of computed as convolution with Gaussian (DOG) kernel second derivative of Gaussian D ∗ G 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 0.4 ◮ At an edge, gradient 0.35 0.3 magnitude looks like a 0.25 0.2 Gaussian 0.15 0.1 0.05 0 -3 -2 -1 0 1 2 3

Thresholding Edges ◮ At an edge, gradient 0.4 0.35 magnitude looks like a 0.3 0.25 Gaussian 0.2 0.15 ◮ Threshold at some value 0.1 0.05 0 -3 -2 -1 0 1 2 3

Thresholding Edges ◮ At an edge, gradient 0.4 0.35 magnitude looks like a 0.3 0.25 Gaussian 0.2 0.15 ◮ Threshold at some value 0.1 0.05 ◮ Leaves behind a “fat” edge 0 -3 -2 -1 0 1 2 3

Zero-Crossings of the Second Derivative ◮ We would prefer to choose just the peak edge response 0.4 ◮ So, we want a local 0.35 0.3 maximum 0.25 0.2 0.15 0.1 0.05 0 -3 -2 -1 0 1 2 3

Zero-Crossings of the Second Derivative ◮ We would prefer to choose just the peak edge response 0.25 ◮ So, we want a local 0.2 0.15 0.1 maximum 0.05 0 ◮ This is where the image -0.05 -0.1 -0.15 second derivative is zero -0.2 -0.25 -3 -2 -1 0 1 2 3 ◮ Second derivative of an edge looks like 1st derivative of Gaussian

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 ∂ x 2 I ( x , y ) + ∂ 2 ∂ y 2 I ( x , y )

2D Gaussian 0.16 0.14 0.12 0.1 0.08 0.06 0.04 3 0.02 2 1 0 -3 0 -2 -1 -1 0 1 -2 2 3-3 − x 2 + y 2 � � 1 G σ ( x , y ) = 2 πσ 2 exp 2 σ 2

2D Gaussian X-Derivative 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 3 2 -0.08 1 -0.1 -3 0 -2 -1 -1 0 1 -2 2 3-3 − x 2 + y 2 � � ∂ xG σ ( x , y ) = − x ∂ 2 πσ 4 exp 2 σ 2

2D Gaussian Y-Derivative 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 3 2 -0.08 1 -0.1 -3 0 -2 -1 -1 0 1 -2 2 3-3 − x 2 + y 2 � � ∂ yG σ ( x , y ) = − y ∂ 2 πσ 4 exp 2 σ 2

2D Gaussian Second X-Derivative 0.1 0.05 0 -0.05 -0.1 3 -0.15 2 1 -0.2 -3 0 -2 -1 -1 0 1 -2 2 3-3 − x 2 + y 2 ∂ x 2 G σ ( x , y ) = x 2 − σ 2 ∂ 2 � � 2 πσ 6 exp 2 σ 2

2D Gaussian Second X-Derivative 0.1 0.05 0 -0.05 -0.1 3 -0.15 2 1 -0.2 -3 0 -2 -1 -1 0 1 -2 2 3-3 − x 2 + y 2 ∂ y 2 G σ ( x , y ) = y 2 − σ 2 ∂ 2 � � 2 πσ 6 exp 2 σ 2

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 G σ a Gaussian kernel. Then we can compute our DOG kernel H = G σ ∗ D as follows: H ( k ) = 1 2 ( g σ ( k + 1 ) − g σ ( k − 1 )) Here k goes from − R to R and g σ is the 1D Gaussian function

Second Derivative Finite Difference Operator A second derivative finite difference looks like this: δ 2 f ( x ) = f ( x − 1 ) − 2 f ( x ) + f ( x + 1 ) This can be computed as a convolution with the kernel D 2 = [ 1 − 2 1 ] Be careful! δ 2 f � = δ ( δ f )

Second Derivative of Gaussian Kernels Now let D 2 = [ 1 − 2 1 ] be our second derivative kernel, and G σ be the Gaussian kernel. Then we can compute the second-DOG kernel H 2 = G σ ∗ D 2 as follows: H 2 ( k ) = g σ ( k − 1 ) − 2 g σ ( k ) + g σ ( 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/