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### Classification

and application in Remote Sensing

Overview

- Introduction to classification problem
- an application of classification in remote sensing: vegetation classification
- band selection
- multi-class classification

Introduction

- make program that automatically recognize handwritten numbers:

Introduction classification problem

- from raw data to decisions
- learn from examples and generalize
- Given: Training examples (x, f(x)) for some unknown function f.Find: A good approximation to f.

Examples

- Handwriting recognition
- x: data from pen motion
- f(x): letter of the alphabet
- Disease Diagnosis
- x: properties of patient (symptoms, lab tests)
- f(x): disease (or maybe, recommended therapy)
- Face Recognition
- x: bitmap picture of person’s face
- f(x): name of person
- Spam Detection
- x: email message
- f(x): spam or not spam

Steps for building a classifier

- data acquisition / labeling (ground truth)
- preprocessing
- feature selection / feature extraction
- classification (learning/testing)
- post-processing
- decision

Data acquisition

- acquiring the data and labeling
- data is independently randomly sample according to unknown distribution P(x,y)

Pre-processing

- e.g. image processing:
- histogram equalization,
- filtering
- segmentation
- data normalization

Feature selection/extraction

- This is generally the most important step
- conveying the information in the data to classifier
- the number of features:
- should be high: more info is better
- should be low: curse of dimensionality
- will include prior knowledge of problem
- in part manual, in part automatic

Feature selection/extraction

- User knowledge
- Automatic:
- PCA: reduce number of feature by decorrelation
- look which feature give best classification result

Classification

- learn from the features and generalize
- learning algorithm analyzes the examples and produces a classifier f
- given a new data point (x,y), the classifier is given x and predicts ŷ = f(x)
- the loss L(ŷ,y) is then measured
- goal of the learning algorithm: Find the f that minimizes the expected loss

Classification: Bayesian decision theory

- fundamental statistical approach to the problem of pattern classification
- assuming that the descision problem is posed in probabilistic terms
- using P(y|x) posterior probability, make classification (Maximum aposteriori classification)

Classification

- need to estimate p(y) and p(x|y), prior and class-conditional probability density using only the data: density estimation.
- often not feasible: too little data in to high-dimensional space:
- assume simple parametric probability model (normal)
- non-parametric
- directly find discriminant function

Post-processing

- include context
- e.g. in images, signals
- integrate multiple classifiers

Decision

- minimize risk, considering cost of misclassification : when unsure, select class of minimal cost of error.

no free lunch theorem

- don’t wait until the a “generic” best classifier is here!

Remote Sensing : acquisition

- image are acquired from air or space.

Feature extraction

- here: exploratory use: Automatically look for relevant features
- which spectral bands (wavelength) should be measured at what which spectral resolution (width) for my application.
- results can be used for classification, sensor design or interpretation

Feature extraction: Band Selection

With spectral response function:

Optimization

Minimize

Gradient descent is possible,

but local minima prevent it from giving good optimal values.

Therefore, we use global optimization : Simulated Annealing.

Multi-class Classification

- Linear Multi-class Classifier
- Combining Binary Classifiers
- One against all: K-1 classifiers
- One against one: K(K-1)/2 classifiers

Problem with max voting

- No Probabilities, just class labels
- Hard classification
- Probabilities are usefull for
- spectral unmixing
- post-processing

Combining Binary Classifiers :Coupling Probabilities

- Look for class probabilities pi:

with rij: probability class ωifor binary classifier i-j

- K-1 free parameters and K(K-1)/2 constraints !
- Hastie and Tibshirani: find approximations
- minimizing Kullback-Leibler distance

Remote Sensing: post-processing

- use contextual information to “adjust” classification.
- look a classes of neighboring pixels and probabilities, if necessary adjust pixel class

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