GEOG 4110/5100 Advanced Remote Sensing Lecture 12. Classification (Supervised and Unsupervised) Richards: 6.1, ,

Transcription

1 GEOG 4110/5100 Advanced Remote Sensing Lecture 12 Classification (Supervised and Unsupervised) Richards: 6.1, , GEOG 4110/5100 1

2 Fourier Transforms Transformations in the Frequency Domain Sinusoid in horiz. direction with low spatial frequency Sinusoid in horiz. direction with high spatial frequency A Fourier transform encodes not just a single sinusoid, but a whole series of sinusoids through a range of spatial frequencies from zero (i.e. no modulation, i.e. the average brightness of the whole image) all the way up to the "nyquist frequency When you display a forward FFT image, ENVI displays the natural log of the magnitude of the complex pixel values. GEOG 4110/5100 2

3

4 Original Magnitude ( f(u,v) ) Phase (f) When you display a forward FFT image, ENVI displays the natural log of the magnitude of the complex pixel values. 4

5 Original Magnitude ( f(u,v) ) Phase (f) Fundamentals of Image Processing, Ian T. Young Jan J. Gerbrands Lucas J. van Vliet DelftUniversityofTechnology, 1998 Inversion of Magnitude Only Inversion of Phase Only 5

6 Original Magnitude ( f(u,v) ) Phase (f) Using the Complex Lookup Function For complex data types, use Complex Lookup Function to determine which image to display. 1. In the Header Info dialog, click Edit Attributes and select Complex Lookup Function. The Complex Data Lookup Function dialog appears. 2. Select the desired lookup function from the drop-down list: Real (real portion of number) Imaginary (imaginary portion) Power (ln(magnitude)) (default) Magnitude (square root of sum of the squares of the real and imaginary) Phase (arc tangent of imaginary divided by real). 6

7 Basic Terms Radiometric Enhancement: Modification of pixel brightness values to improve the visual impact of an image and facilitate extraction of certain information from within the image. Radiometric enhancement techniques are point operations New brightness value for a pixel is only generated by its pixel value Neighboring pixels have no influence Enhancement procedures in which they do are geometric enhancement Geometric Enhancement: Enhances geometric detail in an image, as opposed to radiometric detail. Changes in pixel brightness are driven by geometric considerations, and thus are directly influenced by the character of other surrounding pixels Scalar and Vector Images. Scalar image: each pixel has only a single brightness value associated with it; e.g black and white image Vector image: each pixel is represented by a vector (or combination) of brightness values GEOG 4110/5100 7

8 Multi-Spectral Space and Spectral Classes Multi-spectral data space: N-dimensional coordinate system in which spectral data can be plotted with each axis representing one spectral component (band) key tool for developing algorithms Can contains as many dimensions as there are spectral components (i.e. bands) Illustration of a 2-D multispectral space showing its relation to the spectral reflectance of ground cover types (Fig. 3.5 from Richards and Jia, 2006) GEOG 4110/5100 8

9 There are Spectral Classes Within Clusters Representation of information classes by sets of spectral classes (Fig 3.6 from Richards and Jia, 2006) Rarely this clean Additional dimensions help discriminate further, when there is overlap GEOG 4110/5100 9

10 x = x x! 1 2 x n Vectors and Classification Band Brightness x =

11 Mean Vector and Covariance Band 4 brightness Vegetation pixel Band 4 brightness 40 Pixel vector Band 3 brightness Band 3 brightness GEOG 4110/

12 Mean Vector and Covariance The mean vector (m) is the vector average of the individual components of a vector The covariance between two real-valued random variables describes how one variable varies in relation to another. Cov(X,Y ) = 1 n 1 n i=1 (X i x)(y i y) C = 1 x n 1 n i=1 (X i x)(x i x) T Which has high covariance and which has a low? GEOG 4110/

13 Mean Vector and Covariance The covariance matrix (S x ) is a matrix of covariance values that describes the scatter or spread between variables. = 1 x n 1 n i=1 (x i m)(x i m) t Computation of Covariance Matrix (Table 8.1 from Richards and Jia, 2006) m = GEOG 4110/

14 Two Dimensional Multi-Spectral Classes Probability of pixels belong to each class Probability tapers with distance from center Each class modeled by a normal distribution specified by a mean vector and covariance matrix, which determine location (mean), width, height, and robustness (covariance matrix) of detection boundaries Two-dimensional multi-spectral space with the spectral classes represented by Gaussian probability distributions. (Fig. 3.8 from Richards and Jia, 2006) GEOG 4110/

15 Classification Unsupervised classification: The assigning of pixels of an image to spectral classes without the knowledge of their existence and names. Performed using clusters. The methods determine the location and the number of classes in the data and the class of each pixel. Identifying the classes using a reference data (maps, field). It is useful in identifying the spectral classes of an image before further analysis (e.g. supervised). Supervised classification A number of statistical and non-statistical methods are available. Statistical methods assume that each spectral class has a particular probability distribution (Gaussian) function in multispectral space. Consists of three broad phases : (1) Selection of training pixels (field data, maps, ), (2) Compute the mean and covariance matrix, and (3) Assigning each pixel to a class using the highest probability. GEOG 4110/

16 Supervised Classification Types of Supervised Classification Maximum Likelihood Minimum Distance Parallelepiped (par al lel e pi ped) Context Classification Others Two underlying principles Probability distribution models for classes of interest Partitioning of multi-spectral space into class-specific regions using optimally located surfaces GEOG 4110/

17 Six Steps in Supervised Classification 1. Decide on set of ground cover types into which the image is classified. 2. Choose representative pixels or training data from each class. Based on knowledge of the region acquired either through ancillary information, or interpretation of the imagery 3. Use the training data to estimate the parameters of the particular classifier algorithm to be used. Properties that define a probability model Equations that define partitions in multi-spectral space Signature of that class 4. Use the trained sets to classify every pixel in the image into one of the information classes. 5. Produce tables or thematic maps that summarize the results of the classification. 6. Assess the accuracy of the classification using a testing dataset. GEOG 4110/

18 Maximum Likelihood Classification Most common supervised classification with remote sensing imagery. We define a vector (x) that is the set of brightness values of a pixel in multi-spectral space. Band Brightness This vector has a certain probability of being in one of M spectral classes (w i ) in an image p(w i x), i = 1, 2, M x = x is classified as follows x, if p(w i i x) > p(w j x) for all j i w i.e. the probability of a given pixel is greatest that it falls into class i rather than any other class [pixel in a class] GEOG 4110/

19 Maximum Likelihood Classification Conversely, we can define the likelihood that class w i will be found at pixel location x in spectral space to be: p(x w i ). [class at a pixel] p(w i x) and p(x w i ) are related by Bayes theorem Where: p(w i x) p(x w i ) p(w i ) p(x) p(w i x) = p(x w i ) p(w i )/p(x) = the probability of a pixel at position x belonging to class w i = the probability of finding a pixel at position x from class w i (known from training data) = the probability of class w i in the image (e.g if 15% of pixels in an image belong to a class; based on a priori knowledge or assumptions about the image) = the probability of finding a pixel from any class at location x (remember, x is in multi-spectral space, not geometric space) x i, if p(x w i )p(w i ) > p(x w j )p(w j ) for all j i w Note: p(x) dropped out because it is common to both i and j The probabilities are now expressed in terms that can be learned from training data. GEOG 4110/

20 Maximum Likelihood Classification For mathematical convenience, we use the log of each of these to yield: g i (x) = ln[p(x w i )p(w i )] or ln[p(x w i ) + ln[p(w i )] x w i if g i (x) > g j (x) for all j i Pixels at every point in multispectral space will be classified into one of the available classes w i, even for extremely small probabilities, resulting in some very poor classifications. We apply a threshold (T i ) to avoid poor classifications. x w i if g i (x) > g j (x) and g i (x) > T i for all j i GEOG 4110/

21 Maximum Likelihood Classification Number of pixels needed for good classification Depends on purity of those pixels V Minimum of N+1 pixels for N spectral bands Rule of thumb for: minimum of 10N, preferably 100N Difficult with hyperspectral imagery, so we sometimes employ other approaches. F W U 21 GEOG 4110/5100

22 Mean Vector and Covariance in Classification W V False Color Composite of Landsat MSS image containing four types of surfaces: water, fire burn, vegetation, and urban Training areas for each are violet, red, green, and dark blue polygons respectively CV x = 1 x n 1 For each training area: n i=1 F m (x i m)(x i m) t = U With four bands in MSS, a 4 element mean vector and a 4x4 covariance matrix can be calculated DN DN DN DN DN 1 DN 2 DN 3 DN 4 DN 1 CV 1,1 CV 2,1 CV 3,1 CV 4,1 DN 2 CV 1,2 CV 2,2 CV 3,2 CV 4,2 DN 3 CV 1,3 CV 2,3 CV 3,3 CV 4,3 DN 4 CV 1,4 CV 2,4 CV 3,4 CV 4,4 GEOG 4110/

23 Signature Class Mean Vector Water Comprised of mean values in each band from a sample population Fire Burn Vegetation Urban GEOG 4110/

24 Mean Vector and Covariance Signatures Class Mean Vector Covariance Matrix Water Fire Burn Vegetation Urban Class signatures generated from the training pixels (adapted from Richards, 2013). GEOG 4110/