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An Evaluation of Matching Spectral by Using Wavelet Transforms in Hyper spectral Imagery SOURI A.H.(1), PARANG S.(2) & SHAHRISVAND M.(3) (1 , 3)

Remote Sensing MSc student, Dept. of Surveying and Geomatic Eng., College of Eng. University of Tehran, Tehran, Iran. (Souri_rs, m.shahrisvand)@ut.ac.ir (2) Geodesy MSc student, Dept. of Surveying and Geomatic Eng., College of Eng. University of Tehran, Tehran, Iran. [email protected]

Abstract. One of the methods of classification in hyperspectral remotely sensed imagery is using of Spectral Matching. Till this day many methods including SAM and Correlation coefficient are used for matching spectral. Users can obtain features by using mentioned methods and laboratory data without any need of field observation. Although above methods are reliable from a mathematical point of view, in presence of aerosols, water vapor and gases in atmosphere, radiance does not interact with reflectance which is a property of ground feature. In this paper, we try to emphasize the generalities and neglect details by dividing radiance and reflectance spectra derived from Wavelet transforms into various levels to detect features precisely. Three basis functions, Haar, Doubshi and Meyer have been selected. After classifying the vegetation in study area by using this method, subsequently the resultant class has been compared to another result that is retrieved from SVM method. The results show Haar is more accurate than the others. Keywords: Hyperspectral, Spectral Matching, Wavelet, SVM 1. Introduction. The hyperspectral remotely sensed imagery is used in vast applications, especially in agriculture, mineralogy, geology, ecology and surveillance. Although it can give us abundant information with high spectral resolution, the presence of atmosphere with gases and aerosols alters the signal, leading to reduce and scatter the energy so that radiance does not interact with the ground surface. To clearly specify this matter we have to take into account how atmosphere hinders having accurate data. 1.1 Atmospheric effect. Although hyperspectral sensors are generally airborne, it’s necessary to heed atmospheric effect in order to have reliable and accurate result. According to figure 1 the main absorbing gases in the atmosphere are water vapor, ozone, carbon dioxide, and oxygen. Atmospheric water vapor focused at 0.94, 1.14, 1.38 and 1.88 µm, the oxygen gas at 0.76 µm, and the carbon dioxide near 2.08 µm .We can conclude that approximately more than half of the spectral region is affected by atmospheric gas absorptions. The shorter wavelength region below 1 µm is also affected by molecular and aerosol scattering. (Reference [6])

Fig. 1. Atmospheric transmittance in the 0.4μm to 2.5μm.

For illustrating more comprehensively, the radiance and reflectance curve of three specific features have been showed in the following:

Fig. 2. Radiance curve (left) and Reflectance curve (right).

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The main reason which we merely don’t rely on typical methods of spectral matching is that radiometric errors such as atmospheric effect can lead us to discriminate between features mistakenly Although the typical methods of spectral matching can distinguish water from vegetation, they can’t exactly indicate that which kind of vegetation could be in our image. 1.2 Wavelet Transform. The name “wavelet” comes from the requirement that a function should integrate to zero, waving above and below the axis. The diminutive connotation of wavelet suggests the function has to be well localized. (Reference [2]) At this point, some students might ask, why not use traditional Fourier methods? Fourier basis functions are localized in frequency, but not in time so that Fourier analysis is the ideal tool for the efficient representation of very smooth, stationary signals. Wavelet basis functions are localized in time and frequency. So, wavelet analysis is an ideal tool for representing signals that contain discontinuities or for signals that are not stationary. (Reference [1]) The discrete wavelet transform of g(t) with respect to wavelet ψ(t) is defined by :

(1) Here f [n],

and

are discrete functions defined in [0,M-1],totally M points. We can construct a basis

from the scaling function and wavelet function with two parameters: scaling and translating. (2) (3) Where j is the parameter about dilation, or the visibility in frequency and k is the parameter about the position. In practice, we may want to see the whole data with "desired" resolution, i.e. for some resolution j. We define the subspace: (4) (5) For example, Haar scaling function is: (6)

Fig. 3.The relationship between scaling and wavelet function spaces.

Expand the notation of

, we have (7)

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In general, this equation is called refinement equation, multi-resolution analysis equation, or dilation equation : (8) Again, for Haar wavelets: (9) These two filters are related by

(10) Therefore, we can decompose any function in functions.

which is almost in all cases, using the scaling function and wavelet

(11) An analogy of this concept is the cross-section of a cabbage. A cabbage is made of a kernel and a lot of leaves. A cabbage on the market can be considered as a scaling function at the finest scale J. As we take a layer of leaves off, the remaining cabbage is like the original cabbage in shape, but different in size. This is the concept of the scaling function at different scales with different supports but with similar shape. The leaves of a layer are the differences between different scales as the wavelet function. It can be used in cookery, or in mathematical analysis. We can peel the leaves until the leaves are all taken of just like the multi-resolution analysis. (Reference [11]) 2. Developing the problem. At first, the reflectance spectra have been decomposed to 8 levels by wavelet transform with three basis function. The three basis functions are: • Haar • Doubshi • Meyer Every pixel in our hyperspectral image has already been decomposed to 8 levels by mentioned basis functions. In order to compare the decomposed radiance and reflectance we utilize the simple Correlation coefficient method. The spectral correlation measure (Reference[5]) is calculated as the correlation coefficient of the pixel (portrayed as vector T T T in a n-dimensional feature space) ri =(ri1, ..., riL) and rj =(rj1, ..., rjL) and their respective reflectance si =(si1, ..., siL) and sj T =(sj1, ..., sjL) as: (12)

Where n is number of the overlapped band. The correlation coefficient has advantage that it takes into account the relative shape of radiance curve as well as reflectance curve. (Reference [5]) The main idea is that we note the generalities (high levels) by giving them higher weight and pay less attention to the details (low levels) by giving them lower weight. We assume that low levels has noisy signal which is created by radiometric errors such as atmosphere. In simple word, the first level which has low generalities is scored 0.05 and the last level that has high generalities is scored 0.5. Clearly, the score of middle levels has been given with linear stretching. 2.1 Study area and used data. The study area is ProSpecTIR-VS image located on Reno, NV, USA that’s retrieved from SpecTIRCompany.TheProSpecTIR-VS instrument has dual sensors individually covering visible/near-infrared (VNIR) wavelengths of 400-1000nm and short-wave infrared (SWIR) in the 1000-2500nm wavelength range within 360 channels and 2m spatial resolution. The image located in Reno is covered by urban and mixed environment (Fig4).

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Fig. 4.Reno, NV, USA.

Fig. 5. The reflectance curve of vegetation.

For classifying the vegetation, we retrieved reflectance of this feature which is shown below: 2.2 Results and discussion The reflectance curve has been decomposed by wavelet transform. The following figure indicates the curve levels with Haar basis function:

Fig. 6.Decomposed reflectance curve of vegetation by Haar basis function. The generalities belong to high levels and vice versa. After assigning high weight for higher levels and low weight for lower levels, the vegetation area in our case study will be obtained the following:

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Fig. 7. From the left to right, vegetation area calculated by Haar, Db6, Meyer. In order to evaluate the capability of the method, resultant vegetation area will be compared to SVM which is one the most accurate methods in supervised classification. Figure 8 illustrates the classification image obtained by SVM.

Fig. 8. Left: classified image by SVM, right: vegetation area. For comparing vegetation area retrieved by SVM to wavelet method, we utilize change detection ability in ENVI software. In figure 9, the red areas are shown as the highest difference, grey areas show no difference and the middle difference regions are shown as blue:

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Fig9. From left to right: the difference between vegetation area calculated by wavelet and SVM, basis functions are: Haar, Db6. Meyer.

Overall accuracy Kappa coefficient

Haar 0.956 0.774

Db6 0.935 0.429

Meyer 0.914 0.090

Table 1. comparison of basis function’s accuracy. 3. Conclusions. The marked observation which emerged from the data comparison was that the Haar basis function can classify our image more precisely than the other functions. It is possible that wavelet transform may have erroneous result and our work clearly has some limitations;firstly we should analyze the spectrum of radiance in order to figure out where can be destructed and changed by atmosphere and other source of radiometric errors; secondly, we have to know in which levels it occurs. Therefore, to have better result we need to use a threshold to discard the noise and errors or to give noisy levels a lower weight. Nevertheless, we believe our work could be a starting point for analyzing radiance spectra more precisely and accurately. Acknowledgements We would like to thank the following people for their support, without whose help this work would never have been possible: Mr.Haghshenas and Eng.Soofbaf, we also thank Dr.Safari for his support and giving us valuable advice. References [1] I. Daubechies, Ten lectures on wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), (1992). [2] S. Mallat, December, A Wavelet Tour of Signal Processing, 3rd ed., Third Edition: The Sparse Way. Academic Press, 3 Ed, (2008). [3] N.-C. Shen, “Sectioned convolution for discrete wavelet transform”, Master’s thesis, (2008). [4] P. M. A. K. Louis and A. Rieder, Wavelets Theory and Applications. John Wiley & Sons,(1997). [5] John, A.Richards, Xiuping, Jia, Remote Sensing Digital Image Analysis, Australia (2006). [6] Bo-Cai,Gao,Curtiss,O. Davis,Alexander F. H. Goetz,A Review of Atmospheric Correction Techniques for Hyperspectral Remote Sensing of Land Surfaces and Ocean Color,IEEE. [7] P, Auscher, G. Weiss, and M, V. Wickerhauser, Local sine and cosine basis of Coifmarn and Meyer and the construction of smooth wavelets, Wavelets: A Tutorial in Theory and Applications (C. K. Chui, ed.),Academic Press, Boston, 1992,pages(pp. 237-256). [8] Y. T. Chan, An introduction o Wavelets, Kluwer Academic Publishers, Boston (1992). [9] G. W. Wornell, Signal Processing with Fractals: A Wavelet-Based Approach, Prerntice Hall, Upper SaddIe River, NJ (1996). [10] R. C. Gonzalez and R. E. Woods, Digital Image Processing (2nd Edition). Prentice Hall (January 2002). [11] Chun-Lin, Liu, A Tutorial of the Wavelet Transform (February 23, 2010). International Conference of GIS-Users, Taza GIS-Days, May 23-24, 2012 Proceeding Book