Spectral Transforms 2

Transcription

1 Spectral Transforms Reading: Chapter 5 ECE/OPTI 531 Image Processing Lab for Remote Sensing Spectral Transforms Feature Spaces Spectral Band Ratios and VIs Principal and Tasseled-Cap Components Contrast Enhancement Spectral Transforms 2 1

2 Multispectral Data Spaces Three spaces associated with multispectral images: image space the b (x,y) space, i.e. an image in band b spectral (data) space the = ( 1, 2,..., K ) vector space feature space derived from the image or spectral space Spectral Transforms 3 Feature Spaces Good features reduce effects that hinder the extraction of information Nonlinear spectral transform multispectral ratios are one example Linear spectral transform corresponds to a coordinate rotation of the space to the space principal components transform is an example Spectral Transforms 4 2

3 Spectral Transforms Feature Spaces Spectral Band Ratios and VIs Principal and Tasseled-Cap Components Contrast Enhancement Spectral Transforms 5 Spectral Band Ratios Benefits reduce topographic shading emphasize spectral differences correlate with geophysical variables Multispectral ratios Simple Ratio: Modulation Ratio: Spectral Transforms 6 3

4 Calibrated Spectral Ratios Simple physical model for radiometric calibration Band-specific gain factor includes sensor gain solar spectral irradiance atmospheric transmittance (2-way) is the solar irradiance projection factor due to surface topography (Chapter 2) Band-specific bias includes atmospheric path radiance sensor offset Spectral Transforms 7 Calibrated Ratios (cont.) Partially-calibrated data estimate offset and subtract topographic shading due to goes away because it is the same in every band Fully-calibrated data proportionality constant also goes away Spectral Transforms 8 4

5 Partial Calibration TM1 uncalibrated 1 TM2 uncalibrated 2 1 b 1 2 b 2 Spectral Transforms 9 Shading Reduction Suppression of topographic shading by partial calibration band ratio TM2/TM1 retains topographic shading offset b k removed before ratio topographic shading reduced by b k removal Spectral Transforms 1 5

6 Geologic Discrimination Ratios for geologic discrimination in the SWIR TM5/TM4 (uncalibrated) partial calibration less important in NIR and SWIR TM7/TM5 (uncalibrated) Spectral Transforms 11 Vegetation Indices All VIs defined in terms of reflectance, not VIs therefore require scene calibrated data Ratio Vegetation Index: Normalized Difference Vegetation Index: Soil-Adjusted Vegetation Index (Huete) L is an empirical constant, typically about.5 reduces to NDVI for L = superior to NDVI for low vegetation cover Spectral Transforms 12 6

7 VI Interpretation NDVI correlates better with vegetation density than does RVI NDVI =.5.33! NIR RVI = CIR composite (Landsat TM) crops wet soil dry soil! red RVI NDVI Spectral Transforms 13 Spectral Transforms Feature Spaces Spectral Band Ratios and VIs Principal and Tasseled-Cap Components Contrast Enhancement Spectral Transforms 14 7

8 Principal Components Principal Components Transform (PCT) is a linear matrix transform Principal Component (PC) vector is K-dimensional, just like vector Each PC k is a weighted sum of all spectral bands; weights are the rows of K K W PC matrix Also known as the Karhonen-Loeve Transformation (KLT) and the Hotelling transformation Spectral Transforms 15 PCT Properties Covariance matrix of PC bands C PC is related to original covariance matrix C by, By matrix properties, W PC diagonalizes C, i.e. C PC is a diagonal matrix Since C PC is diagonal, the PC bands are uncorrelated Diagonal elements of C PC are the data eigenvalues Each eigenvalue λ k is equal to the variance of the corresponding PC k The total data variance is preserved, Spectral Transforms 16 8

9 PCT Calculation The rows of W PC are the eigenvectors of the data, Each eigenvector e k contains the weights applied to the original bands to obtain PC k and is found by solving the equation, Eigenvalues are found by solving the characteristic equation Spectral Transforms 17 PCT Example Original data in -space Pixel Spectral Transforms 18 9

10 PCT Example (cont.) Step 1. Find eigenvalues Solve the characteristic equation (two solutions) Therefore, Note that PC 1 contains 89% of the total data variance, and PC 2 contains 11%, i.e. Spectral Transforms 19 PCT Example (cont.) Step 2. Find eigenvectors Substitute eigenvalues into First eigenvalue yields dependent equations Solving either equation PCT requires orthonormality of the eigenvectors Solve simultaneous equations to yield eigenvectors, and the PCT weight matrix Spectral Transforms 2 1

11 PCT Example (cont.) Step 3. Transform data to PC-space For Pixel 1: Final PC data and scatterplot Pixel PC1 PC PC PC1 Spectral Transforms 21 PCT Benefits Why use the PCT? Decorrelates spectral data Multispectral bands are often highly-correlated because of material spectral correlation topography sensor band overlap Decorrelation separates independent components into separate bands Compresses the variance PC 2 2 PC 2 Correlated PC Uncorrelated PC 1 3 Compression can potentially reduce data computation burden variance TM bands PC bands band or PC index Spectral Transforms 22 11

12 PCT Decorrelation Non-vegetated scene PCT removes spectral redundancy TM 1 TM 2 TM 3 PC 1 PC 2 PC 3 TM 4 TM 5 TM 7 PC 4 PC 5 PC 6 Spectral Transforms 23 PCT Contrast Extraction Vegetated scene PCT extracts contrast between bands 3 and 4 (red and NIR) due to the vegetation red edge TM 1 TM 2 TM 3 PC 1 PC 2 PC 3 TM 4 TM 5 TM 7 PC 4 PC 5 PC 6 Spectral Transforms 24 12

13 PCT Noise Detection PCT can isolate spectrally-uncorrelated noise TM 2 TM 3 TM 4 PC 1 PC 2 PC 3 Spectral Transforms 25 PCT Drawbacks Why not use the PCT? It is data-dependent W coefficients change from scene-to-scene Makes consistent interpretation of PC images difficult Spectral details, particularly in small areas, may be lost if higher-order PCs are ignored Computationally expensive for large images or for many spectral bands Calculation of covariance matrix is the culprit Spectral Transforms 26 13

14 Tasseled-Cap Components Linear spectral transform like the PCT Tasseled-cap components for MSS and TM In this case, the W TC matrix is fixed for a given sensor Spectral Transforms 27 TCT Benefits Why use the TCT? It is a fixed reference Same reference for every image from a given sensor permits consistent interpretation Components are related to geophysical properties of the scene First component is soil brightness Second component is greeness Greeness TCT axes align better with the soil and vegetation directions PC 2 4 Brightness PC 1 3 Spectral Transforms 28 14

15 TCT Drawbacks Why not use the TCT? Nonoptimal compression of data Derivation of W TC requires multitemporal data for each sensor Comparison of PC and TC images PC 1 PC 2 PC 3 TC 1 TC 2 TC 3 PC 4 PC 5 PC 6 Spectral Transforms 29 TC 4 TC 5 TC 6 Spectral Transforms Feature Spaces Spectral Band Ratios and VIs Principal and Tasseled-Cap Components Contrast Enhancement Spectral Transforms 3 15

16 Contrast Enhancement Two problems Most images do not fill the dynamic range of the sensor a b optional high gain b signal range required at A/D input for full range output standard gain b anticipated range of detected signals offset b low radiance scenes e b all scenes Most images also do not fill the dynamic range of the display system Contrast enhancement means stretching the data range to fill the display system range GL = T() Parameters of transformation T based on global or local image statistics Spectral Transforms 31 Global Single-Band Transforms Linear stretch min-max scale range of image s to range of display GLs sensitive to outliers saturation scale a smaller range of s to range of display GLs saturation of 1 3% pixels at each end usually acceptable Nonlinear stretch piecewise-linear different contrast gain over different ranges histogram equalization use scaled CDF of original image as the transformation Spectral Transforms 32 16

17 -to-gl Transformations Spectral Transforms 33 Contrast Stretch Examples Application to GOES image of North America number of pixels number of pixels original min-max original min-max number of pixels number of pixels % saturation 8% saturation % saturation 8% saturation number of pixels number of pixels piecewise linear histogram equalization piecewise linear histogram equalization Spectral Transforms 34 17

18 Normalization Stretch Linear scale of mean and sigma to specified values, followed by saturation Consistent behavior (robust) over wide range of images original µ = 128, σ = 32 µ = 128, σ = 48 µ = 128, σ = 64 Spectral Transforms 35 Reference Stretch Match the CDF of the image being processed to a reference CDF, for example from another image useful for multitemporal or multisensor radiance matching matching image to reference contrast Reference stretch is a double transformation CDF CDF ref 1 1 ref Spectral Transforms 36 18

19 Multitemporal Normalization Reference number of pixels Dec 31, 1982 Aug 12, number of pixels dark-light linear stretch TM band 3, Dec 31, 1982 TM band 3, Aug 12, 1983 GL number of pixels dark-light target linear stretch CDF reference stretch Spectral Transforms 37 GL number of pixels CDF reference stretch Thresholding Binary clipping of s to low and high values Useful for segmentation of certain images, e.g. clouds/water, land/water T = 5 T = 1 T = 15 Spectral Transforms 38 19

20 Formulas Mathematical formulas for contrast enhancement techniques Spectral Transforms 39 Color Images Techniques used for single-band imagery can be extended to color, but... Sensitivity of the human vision system to shifts in color and saturation require special attention Min-max stretch Stretch the s in each band over their respective minmax range Good news: Easy to calculate and implement No data lost by saturation Bad news: Sensitive to outlier s Color balance can change unpredictably Spectral Transforms 4 2

21 Color Normalization Normalization stretch Standardized stretch Good news: Average color is grey Contrast controlled by single parameter, the desired output standard deviation Bad news: Some data are lost in saturation Normalization Stretch Linearly stretch each band to same mean ( typically 128) and same standard deviation (typically 32 48) Clip to [,255] RGB linear stretch to µ,! linear stretch to µ,! linear stretch to µ,! clip at [,255] clip at [,255] clip at [,255] RGB Spectral Transforms 41 Color Decorrelation Decorrelation stretch Enhance small spectral deviations in highlycorrelated spectral bands Commonly used in geology Good news: Decorrelates bands Emphasizes differences among bands Can be applied to any number of bands Bad news: Produces highly saturated colors PCT transform Stretch each PC component to equalize variances Inverse PCT transform Clip to [,255] clip at [,255] image data space PCT PCT -1 Decorrelation Stretch equalize variances PC space Spectral Transforms 42 21

22 Color Spaces HSI color coordinate system Hexcone model similar to a cylindrical coordinate system, but based on RGB color cube value = max(r,g,b) used instead of intensity efficient CST from RGB to Hue- Saturation-Value (HSV) cylindrical coordinates hue saturation intensity project subcube faces onto orthogonal plane intersection at the vertex of subcube H yellow magenta P black S white I blue green red magenta (,,255) blue red black GL 1 (255,,) white yellow (,255,) green GL 2 cyan Spectral Transforms 43 GL 3 cyan Color-Space Transforms Color-space transforms Human vision system perceives hue (H), saturation (S) and intensity (I), not RGB Therefore, control over color appearance is best done in HSI space Color-Space Transform Transform RGB to HSI Modify HSI components as desired Inverse transform modified HSI to RGB Clip to [,255] CST modify clip at [,255] CST -1 RGB space HSI space Spectral Transforms 44 22

23 Example Ramp Spectrum CSTs H S I RGB linear hue 1% saturation 1% intensity linear hue 5% saturation 1% intensity linear hue 1% saturation 5% intensity cycle hue 1% saturation 1% intensity Spectral Transforms 45 CST for Contrast Enhancement Intensity stretch Good news: Improves contrast without changing hue or saturation Based on human vision system model Bad news: Can be applied only to color (3-band) images Based on human vision system model Do CST Stretch intensity component as desired Inverse CST Clip to [,255] Intensity Enhanced CST Spectral Transforms 46 23

24 Color Contrast Enhancement TM bands 3,2,1 TM bands 7,5,4 min-max stretch normalization stretch decorrelation stretch Spectral Transforms 47 Examples (cont.) original min-max stretch histogram equalization stretch normalization stretch decorrelation stretch HSI intensity stretch Spectral Transforms 48 24

25 3-D Scatterplots original min-max histogram equalized normalized decorrelated Spectral Transforms HSI 49 25