Lecture Notes on Computer Vision - Binary Operations | CAP 5415, Exams of Computer Science

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Lecture 8: More on Classification,

Lines from Edges, Interest Points,

Binary Operations

CAP 5415: Computer Vision

Fall 2009

Non-Linear Classification

x y

• We've focused on linear decision boundaries
• What if that's not good enough?

Non-Linear Classification

x y

• What would a quadratic decision boundary look like?
• This is the decision boundary from x 2 + 8xy + y 2 > 0

Non-Linear Classification

x y

• We can generalize the form of the boundary
• ax 2 + bxy + cy 2 +d > 0 Notice that this is linear in a,b,c, and d!

(Assume this is the training data)

What's wrong with this decision

boundary?

What's wrong with this decision

boundary?

● What if you tested on this data? ● The classifier has over-fit the data

● Boosted Classifiers and SVM's are probably the two most popular classifiers today ● I won't get into the math behind SVM's, if you are interested, you should take the pattern recognition course (highly recommended)

The Support Vector Machinecd ..

The Support Vector Machine

The Support Vector Machine

● Margin – minimum distance from a data point to the decision boundary

The Support Vector Machine

● The SVM finds the boundary that maximizes the margin

Non-Linear Classification in SVMs

• We will do the same trick as before x y This is the decision boundary from x 2

• 8xy + y 2

0 This is the same as making a new set of features, then doing linear classification

Non-Linear Classification in

SVMs

The decision function can be expressed in terms of dot-products Each α will be zero unless the vector is a support vector

The Kernel Trick

Let Φ(x) be a function that transforms x into a different space A kernel function K is a function such that

Example (Burges 98)

If Then This is called the polynomial kernel