Convolution Operators
Overview of a few convolution operators
Sobel Operator for Edge Detection
The Sobel Operator calculates the pixels first order derivatives in the X and Y direction.
If the is little derivative, there is no unstructured area
If the derivative high in one direction, we have an edge in that direction
If the derivative high in two directions , we have a corner / point of interest
Gaussian Smoothing
Replace each pixel by it’s local average
.It is used for scale Space
The Laplace Operator for
The differnce compard to the Sobel operator is, that it uses the second order derrivative.
This makes the Laplace operator very sensitive to noisy.
Edges are, where the second derivative is crossing over to 0 ( Zweite ableitung = 0 Hochpunkt!)
Laplacian of Gaussian Filter :

First smooth with gaussian filter.

Then find zero crossings with the Laplacian filter.
 Can also combine one into one LOG convolution
Double of Gaussian
LoG does not have to be calculated, it can be also approximated by calculating the difference between two Gaussian Filters at different scales . DoG
SIFT Detector
 Multiple DoG filters are applied to the image at different scales
 The resulting images are stacked on top each other to create a 3D volume
 Points that are local extrema inside of the 3D voluma, are considerd points of interest.
 Remove bad points, like candidates in smooth regions or directly on top of edges
SIFT Discriptor
 For a detected point of interest, choose 16×16 region around the point
 Compute each gradient for each pixel
 Subdivide into 16 * 4×4 groups
 Compute orientation histogram
 Glue the histograms together to get a 128 element feature vector
Characteristics
 SIFT is very resillient to changes (invariant) to constant intensity changes, as itt is based on gradients
 Very invariant to contreast changes , as the histogram binning provides normalization
 invarient to small deformations
Scale invariance is caused by the SIFT detector