# Calculation of Minimum Distances. Minimum Distance to Means. Σi i = 1

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1 Minimum Distance to Means Similar to Parallelepiped classifier, but instead of bounding areas, the user supplies spectral class means in n-dimensional space and the algorithm calculates the distance between a candidate pixel and each class. The candidate pixel is assigned to the class with the smallest spectral Euclidian distance (minimum distance) to the candidate pixel. The distance is calculated using either an n-dimensional Pythagorean theorem, or a Round-the-Block measure. Pythagorean Theorem Round-the-Block D ab = nn ((a Σi i = i b i ) 2 ) /2 D ab = nn Σi i = Where: D ab = Distance between class a and pixel b. a i = Mean spectral value for class a, band i b i = Spectral value for pixel b, band I n = Number of spectral bands (a i b i ) A Calculation of Minimum Distances 63 (A A 2 ) 2 + (B B 2 ) 2 + (C C 2 ) = D 2 33 DA 3 22 DB 2 DA 2 DB DB 3 D D 2 D B DA = Min D(,2,3) Where Min D = Minimum Distance to class, 2, or 3 Minimum Distance to Means (cont.) As with all classification algorithms, every pixel in the image is evaluated to determine their class assignments. Depending on file size, this can be time consuming. There are modifications to the standard MDM classification that increase the computational efficiency. Under a normal MDM classification, all pixels are assigned to the nearest spectral class. No pixel is unassigned. Some algorithms allow the user to specify a threshold distance from the class means beyond which a pixel will not be assigned and therefore will remain unclassified. A common threshold to use is the combined n-dimensional standard deviation of the spectral class. When used properly, this method is just as accurate as more robust, computationally intensive algorithms like the Maximum Likelihood.

2 Raw ETM Image of Utah County Area Collected on Sep. 2, 2 Mean Values of 2 Unsupervised, Standard Deviations of 2 Unsupervised, Signature Plots of 2 Unsupervised, 2

3 Feature space plot of the Red and reflectance bands from Utah County area ETM image GREEN BRIGHT 2 Spectral Cluster MDM Map of ETM Image of Utah County Area DARK WATER Thematic Feature space plot of Red and Reflectance bands from Utah County area ETM image GREEN BRIGHT Scatter plot of the Red and Reflectance Bands from Utah County Area ETM Image DARK WATER Discrete colors correspond with colors given to individual clusters Points refer to the mean values for red and nir brightness values for each of the 2 clusters 3

4 Overlay of spectral cluster means with feature space plot for red and nir brightness values 2 Overlay of spectral cluster means with thematic feature space plot for red and brightness values Euclidian distance image of individual ETM pixels and their associated MDM spectral cluster Frequency distribution of MDM distance image values Mean = Standard Deviation =

5 Binary map of distance values showing pixels that are greater than 2 standard deviations above the mean Simple logical model that extracts original BV s from ETM data based on distance value threshold Distance File Original Image Either if Distance LE 978 or Original File Otherwise Output File Masked raw ETM image showing pixels that were assigned to specific clusters, but were farther than the established threshold distance from that cluster mean Maximum Likelihood Uses spectral class probabilities to determine class ownership of a particular pixel. Uses mean and variance and co-variance estimates for each spectral class. The probability is calculated for each class. The pixel is assigned to the class with the largest probability Therefore, if MDM uses D ab as the measure for association, Maximum Likelihood uses P ab which is the probability of pixel b belonging to class a Assumes that the statistics for each spectral class have a Gaussian (normal) distribution. Spectral classes with bi- or tri-modal distributions in any of the n bands imply that more than one ground class is represented in the training data. 5

6 Maximum Likelihood (cont.) Most computationally intensive algorithm discussed so far. Theoretically the best classification, but has shown to be very similar to the MDM depending on spectral cluster generation strategy. Like the MDM classifier, all pixels in the scene are assigned to one class. As with MDL a threshold can be used to exclude pixels with a low probability of association with any of the spectral classes. A-priori probabilities can be assigned to each class as another means of controlling output through a Bayesian decision rule which does not assume equal probabilities for each class. As a further measure of pixel associations to spectral classes, the a posteriori probabilities can be output for each pixel. Raw ETM Image of Utah County Area Collected on Sep. 2, 2 Mean Values of 2 Unsupervised, 6

7 Standard Deviations of 2 Unsupervised, Signature Plots of 2 Unsupervised, Feature space plot of the Red and reflectance bands from Utah County area ETM image GREEN BRIGHT Scatter plot of the Red and Reflectance Bands from Utah County Area ETM Image DARK WATER Points refer to the mean values for red and nir brightness values for each of the 2 clusters 7

8 2 Spectral Cluster Maximum Likelihood Map of ETM Image of Utah County Probability distance image of individual ETM pixels and their associated spectral cluster Maximum Likelihood Classification using 2 unsupervised clusters Difference image between minimum distance and maximum likelihood classifications Minimum Distance Classification using 2 unsupervised clusters 8

9 Fuzzy Classification Traditional classification methodology follows classical set theory which would assign one pixel to one (and only one) cover class. Fuzzy set theory allows ownership of a cell with a specific spectral signature by any of n cover classes. The concept of fuzzy classification can take on many meanings depending on how it s applied to mapping land cover.. Identification of ecotones between uniform and discrete sets 2. Accounting for spectral confusion between disparate cover types (different cover types with similar spectral properties) 3. Evaluation of spectral cluster assignment of pixels during the classification process. Confusion Between Disparate Cover Types Class Class 2 Class 2 to Class Confusion Decision Boundary Class to Class 2 Confusion Fuzzy set theory would assign the probability of a particular pixel belonging to a particular surface cover class. Ecotonal Fuzzy Sets Fuzzy Classification Water Forested Wetland Forested Upland Best Association Second Best Association.5 Water Forested Wetland Forested Upland.5 Using classical set theory, pixels representing each of these three land cover types are given either a or. probability of belonging to that type. Under fuzzy set theory pixels are provided with a probability of belonging to any type. Depending on the type, probabilities can be equal to 9

10 Best Association Distance Image Second Best Association Raw ETM Image of Utah County Area Collected on Sep. 2, 2 BEST SECOND 5 4 THIRD FOURTH 2

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