Principles of Satellite Image Processing Instructor: Prof. CSRE, IIT Bombay bkmohan@csre.iitb.ac.in Slot 5 Guest Lecture PCT and Band Arithmetic November 07, 2012 9.30 AM 10.55 AM
IIT Bombay Slide 1 November 07, 2012 PCT and Band Arithmetic Contents of the ure Principal Component Transform Alternative Interpretations Inverse PCT Band Arithmetic Motivation for band arithmetic Band ratio Vegetation indices
IIT Bombay Slide 2 Multiband Image Operations Operations performed by combining gray levels recorded in different bands for the same pixel Applications Data reduction through decorrelation Highlighting specific features with significant difference in response in different bands The transformed data may be viewed like enhanced versions compared to originals
IIT Bombay Slide 3 Principal Component Transform Highlights the redundancy in the data sets due to similar response in some of the wavelengths Original bands variables represented along different coordinate axes, redundancy implies variables are correlated, not independent Gray level in a band at a pixel can be predicted from the knowledge of the pixel gray level in other bands
IIT Bombay Slide 4 Example of Redundancy in Data Example: Highly correlated data Values along band b1 leads to knowledge along band b2 of the data element Linear variation (nearly) between b1 and b2 Often true in case of visible bands b2 b1
IIT Bombay Slide 5 Example of Redundancy in Data Points projected onto the line a small error in the position of the point. Points represented by only one coordinate b1 half data reduced For highly correlated data, this error will be minimal b2 b2 b1 b1
IIT Bombay Slide 6 Decorrelating Multispectral Remotely Sensed Data How do we identify the optimum axes along which the remotely sensed data should be projected so that the transformed data would be uncorrelated? What should be the way to rank the new axes so that we can discard the least important dimensions of the transformed data? Invertibility of the transformation?
IIT Bombay Slide 7 Useful band statistics M N i 1 j 1 Mean i M M N 1 j 1 k l ( g )( g ) ij k ij l M. N. N g k i j Variance M Covariance i 1 j 1 k ( g ) M. N N ij k 2
IIT Bombay Slide 8 Covariance Matrix C = {C kl k = 1,, K, l = 1,, K} K is the number of bands in which the multispectral dataset was generated C is a symmetric matrix C kl = C lk Diagonal elements of C are the intra-band variances Off-diagonal elements are the inter-band covariances
IIT Bombay Slide 9 Relation between correlation and Correlation R kl = covariance It can be shown that R kl = C kl + m k m l M i N 1 j 1 M For data with zero-mean, correlation and co-variance will be equal. g N k i j g l i j
IIT Bombay Slide 10 Principal Component Transformation Problem to solve: Find a transformation to be applied to the input multispectral image such that the covariance matrix of the result is reduced to a diagonal matrix Further, we should find an axis v k such that the variance of the projected coordinates (z k = v kt x) is maximum.
IIT Bombay Slide 11 Solution Given the transformed vector z k = v k t x The variance s z2 = This simplifies to s z2 = v t Cv (Dropping subscript k for a moment!) C, the covariance matrix is a positive, semi-definite, real symmetric matrix. M N i 1 j 1 t t v ( x )( x ) v ij k ij l M. N
IIT Bombay Slide 12 Finding vector v To maximize the projected variance s z2, find a v such that v t Cv is maximum, subject to the constraint v t v = 1. Combining the maximization function with the constraint, we can write v t Cv l(v t v 1) = maximum Differentiating w.r.t. v, t t C ( 1) 0 v v v v v
IIT Bombay Slide 13 Finding v The derivative results in Cv = lv (Verify!) Therefore, v is an eigenvector of C Given that s z2 = v t Cv v t Cv = v t (lv) = lv t v = l = s z 2 This implies that v is the eigenvector of C with the largest eigenvalue Therefore all the eigenvectors with decreasing eigenvalues lead to axes with decreasing variance along them.
IIT Bombay Slide 14 Alternative Explanation to PCT Let the transformed pixel vector y = D t x Covariance matrix of y = S y = D t S x D (Note that S y = E{(y m y )(y-m y ) t } = E{(D t x D t m x )(D t x D t m x ) t } This simplifies to S y = D t E(x m x )(x m x ) t D D is a set of vectors independent of x)
IIT Bombay Slide 15 Alternative Explanation Covariance matrix of y = S y = D t S x D It is desired that S y be diagonal, i.e., the data in the transformed domain is uncorrelated Let S y = 1 0... 0 0 2... 0... 0 0... n
IIT Bombay Slide 16 Let S y = Alternative Explanation 1 0... 0 0 2... 0... 0 0... n Then S y = D t S x D is a similarity transformation with D containing eigenvectors of S x We can order l i in such a way that they are in descending order. Given that y = D t x, y 1 corresponds to direction given by e 1, that is the first row of D t, Each transformed pixel vector y is obtained from scalar products of eigenvectors of S x and x
IIT Bombay Slide 17 Sample Eigenvectors and Eigenvalues Covariance Matrix 34.89 55.62 52.87 22.71 55.62 105.95 99.58 43.33 52.87 99.58 104.02 45.80 22.71 43.33 45.80 21.35 Eigenvalues 253.44 7.91 3.96 0.89 Eigenvectors 0.34 0.61 0.71 0.06 0.64 0.40 0.65 0.06 0.63 0.57 0.22 0.48 0.28 0.38 0.11 0.88
IIT Bombay Slide 18 Sample Eigenvectors and Eigenvalues
IIT Bombay Slide 19 Transformation New component value = dot product of eigenvector and pixel vector (i,j) pixel position n eigenvectors for n principal components 1 st principal component dot product of pixel vector with eigenvector corresponding to largest eigenvalue
IIT Bombay Slide 20 Principal Components For n input bands, n principal components are computed The utility of the principal components gradually decreases from 1 st towards the last e.g., For Landsat TM, last three PCs are generally of very little value
IIT Bombay Slide 21 Visualization of PCT From J.R. Jensen s ure notes at Univ. South Carolina; used with permission
IIT Bombay Slide 22 Comments on PCT For IRS / IKONOS images, out of four bands, 2-3 principal components capture most of the useful information. The last 1-2 bands are redundant. Advantages Smaller data volume to handle Principal components appear to be enhanced versions of the originals, having contributions from all the four input bands Application scientists use composites of PC 1-2-3 for interpretation of various features such as geology
IIT Bombay Slide 23 Band 1 (Blue)
IIT Bombay Slide 24 Band 2 (Green)
IIT Bombay Slide 25 Band 3 (Red)
IIT Bombay Slide 26 Band 4 (NIR)
IIT Bombay Slide 27 Band 5 (SWIR)
IIT Bombay Slide 28 Band 7 (SWIR)
IIT Bombay Slide 29 PC1
IIT Bombay Slide 30 PC2
IIT Bombay Slide 31 PC3
IIT Bombay Slide 32 PC6
IIT Bombay Slide 33 Input Image FCC
IIT Bombay Slide 34 Decorrelation Stretch
IIT Bombay Slide 35 Inverse PCT Inverse PCT is used to generate the bands in the original domain If ALL PCTs are retained, inverse will give back the original bands If any PCTs are dropped, inverse will give new bands in the original domain that may be close to the original bands depending on how many PCTs are discarded
IIT Bombay Slide 36 Inverse PCT From the principle of PCT, we have y = D t x D t contains eigenvectors of S x, covariance matrix from the original image Since D t is an orthonormal matrix, (D t ) t = (D t ) -1 From each pixel vector in PC domain, x = (D t ) t y
IIT Bombay Slide 37 Inverse PCT For k band image, matrix D is square, of size k x k If m principal components are dropped, we are left with a matrix (D 1 ) of size k x (k-m) The vector y is reduced to y 1 of size k-m x 1 Therefore the modified vector x 1 is given by x 1 = D 1 y 1 The difference between x and x 1 is a measure of the loss of information due to removal of some of the PCs
IIT Bombay Slide 38 Comments on PCT One of the other important applications of PCT is data fusion Replace first PC by the image from another sensor Apply inverse PC
Band Arithmetic
IIT Bombay Slide 39 Motivation
IIT Bombay Slide 40 Multiband Arithmetic In a given pair of bands the response of two objects is generally different. Pixel by pixel comparison between images can highlight pixels that have very high difference in ref ance in those bands Operations like band difference and band ratio or combinations of them are popularly used for this purpose
IIT Bombay Slide 41 Band Ratio Very common operation Ratio i,j (m,n) = Band i (m,n) / Band j (m,n) If Band j (m,n) = 0, suitable adjustment has to be made (e.g., add +1 to the denom.) Minimum ratio will be 0; Maximum ratio will be 255
IIT Bombay Slide 42 Input Image
IIT Bombay Slide 43 Input Image FCC
IIT Bombay Slide 44 IR/R
IIT Bombay Slide 45 Band Ratio For fast computing, approximations can be made such as: 0 Ratio i,j (m,n) 1, Ratio i,j (m,n) scaled = Round [Ratio i,j (m,n)x127] 1 < Ratio i,j (m,n) 255, Ratio i,j (m,n) scaled = Round [127 + Ratio i,j (m,n)/2] Advantage in one pass image is generated in range 0-255
IIT Bombay Slide 46 Band Difference Similar to band ratio, band difference can also be used to account for difference in ref ance by objects in two wavelengths Band ratio - more popular in practical applications such as geological mapping Topographic effects on the images are reduced by ratioing.
IIT Bombay Slide 47 Band Multiplication Pixel by pixel multiplication of two images Not used to multiply gray levels in one band with corresponding gray levels in another band Used in practice to mask some part of the image and retain the rest of it by preparing a mask image and performing image to image multiplication of pixels
IIT Bombay Slide 48 Band Addition Similar to Band Multiplication, band addition has no direct practical application in adding gray levels of two bands of an image This method too can be used to mask a portion of the image and retain the remaining part.
IIT Bombay Slide 49 Specialized Indices Combination of band differences, ratios and additions can result in useful outputs that can highlight features like green vegetation One such feature is Normalized Difference Vegetation Index (NDVI) NDVI(m,n) = BandIR ( m, n) BandR ( m, n) Band ( m, n) Band ( m, n) IR R
IIT Bombay Slide 50 NDVI NDVI results in high values where IR dominates red wavelength. This happens where vegetation is present Range of NDVI is [-1 +1] NDVI has been widely used in a wide ranging of agricultural, forestry and biomass estimation applications It is also used to measure the length of crop growth and dry-down periods by comparing NDVI computed from multidate images
IIT Bombay Slide 51 Input Image
IIT Bombay Slide 52 NIR
IIT Bombay Slide 53 RED
IIT Bombay Slide 54 NDVI
IIT Bombay Slide 55 Other Vegetation Indices Simple Ratio = Red/NIR NDVI6 = (Band 6 Band 5)/(Band 6 + Band 5) NDVI7 = (Band 7 Band 5)/(Band 7 + Band 5) Standard NDVI TM = (TM4 TM3)/(TM4 + TM3) These are applicable when seven band data like Landsat Thematic Mapper data are available For IRS LISS3 imagery, NDVI IRS = Band ( m, n) Band ( m, n) 4 3 Band ( m, n) Band ( m, n) 4 3
IIT Bombay Slide 56 IRS L4- NDVI
IIT Bombay Slide 57 Fast Computation of NDVI Range of NDVI [-1, +1] Scale suitably to generate an NDVI image For example, NDVI scaled =127(1+NDVI) This ensures that the resultant NDVI has a range of [0 254]
IIT Bombay Slide 58 Se ed Ref ance Curves From J.R. Jensen s ure notes at Univ. South Carolina Used with permission
IIT Bombay Slide 59 Time Series of 1984 and 1988 NDVI Measurements Derived from AVHRR Global Area Coverage (GAC) Data Region around El Obeid, Sudan, in Sub-Saharan Africa From J.R. Jensen s ure notes at Univ. South Carolina Used with permission
IIT Bombay Slide 60 Simple Ratio v/s NDVI From J.R. Jensen s ure notes at Univ. South Carolina Used with permission
IIT Bombay Slide 61 Infrared Index Traditional NDVI does not work very well when the soil is moist, as in case of wetlands. The Infrared Index (II) can tackle this situation better II NIR NIR MIR TM 4 TM 5 MIR TM 4 TM 5 Several bands needed in the infrared region, as in case of Landsat TM
IIT Bombay Slide 62 Soil Line From J.R. Jensen s ure notes at Univ. South Carolina Used with permission
IIT Bombay Slide 63 Perpendicular Vegetation Index PVI is defined as 2 2 PVI S, R V, R S, NIR V, NIR A vegetation index that assumes that the ref ance in the NIR and red varies with increasing vegetation density (such as leaf area index) and that these variations are parallel to the soil baseline. Therefore, the perpendicular distance from the baseline in a NIR-red plot determines the vegetation density. See http://www.ccrs.nrcan.gc.ca/glossary/index_e.php?id=2179 for more definitions of various indices in remote sensing including PVI
IIT Bombay Slide 64 Soil Adjusted Vegetation Index The soil adjusted vegetation index (SAVI) introduces a soil calibration factor, L, to the NDVI equation to minimize soil background influences resulting from first order soilplant spectral interactions: SAVI (1 L) NIR red NIR red L Ref:A. R. Huete, A soil-adjusted vegetation index (SAVI), Rem. Sens. Env., vol. 25, pp. 295-309, 1988.
IIT Bombay Slide 65 Atmospherically Adjusted Vegetation Index (ARVI) The atmospheric effects are accounted for in ARVI ARVI p* nir p* rb p* nir p* rb p* rb p* red p* blue p* red p* indicates the atmospherically corrected versions of NIR, Red and Blue bands for molecular scattering and ozone absorption (Ref. J.R. Jensen s notes)
IIT Bombay Slide 66 Enhanced Vegetation Index EVI is a mixture of SAVI and ARVI, in that both atmospheric effects and soil effects are accounted for. EVI p* nir p * red p * nir C p* red C p* blue L 1 2 C1 and C2 describe the use of the blue band in correction of the red band for atmospheric aerosol scattering. The coefficients, C1, C2,, and L are empirically determined as 6.0, 7.5, and 1.0, respectively for MODIS. This algorithm has improved sensitivity to high biomass regions and improved vegetation monitoring through a de-coupling of the canopy background signal and a reduction in atmospheric influences Source: Yoram J. Kaufman and Didier Tanre, Atmospherically Resistant Vegetation Index (ARVI) for EOS-MODIS, IEEE Trans. GERS, vol. 30, no. 2, pp. 261-270, 1992
IIT Bombay Slide 67 From J.R. Jensen s ure notes at Univ. South Carolina; used with permission
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