INTRODUCCIÓN

Remote Sensing has many geoscience applica-tions which require satellite imagery displayingspatial and spectral high resolution simultaneously,in order to define thematic and spectral classes ableto properly characterising the surfaces being stu-died. However, the massive use satellite sensors(Landsat TM, AVHHR, ASTER, etc.) only provideone of these features, excluding the other.

Although nowadays there exist some satelliteslike Quickbird or IKONOS that produce high spa-tial and spectral resolution images, the costs toobtain these images are very high. One way ofobtaining images of high spatial and spectral reso-lution, at affordable costs, is through the imagefusion techniques.

Fusion of multiple sensors data presents someadvantages over obtaining data from an only sensor.Given that the information has been optimallyfused, one of the most important things is to have

Revista de Teledetección. 2005. 23: 5-12.

N.º 23 - Junio 2005 5

Tailored orthonormal bases for wavelet fusion

M. Lillo-Saavedra1, C. Gonzalo2, A. Arquero2 and E. Martínez2

matillo@udec.cl

1 Facultd de Ingeniería Agrícola Universidad de Concepción. Avda. Vicente Méndez 595 Casilla 537 Chillán, Chile

2 Facultad de Informática, DATSI, Universidad Politécnica de Madrid. Campus Montegancedo, Boadilla del Monte, 28660 Madrid, España

RESUMEN

Este trabajo propone un criterio objetivo para eldiseño de filtros FIR (Finite-duration Impulse Res-ponse), de manera que desde sus coeficientes seobtengan directamente las bases wavelet ortonorma-les (F) que son aplicadas para la fusión de imágenesde satélite. La determinación del orden del polinomioque compone el filtro FIR, esta basada en el cálculode la energía de la imagen a ser fusionada (imagenfuente) y la frecuencia de corte es obtenida de ladeterminación de un valor óptimo de entropía de estasimágenes. En este sentido, se ha determinado unabase wavelet ortonormal particular para cada imagenfuente. Por otro lado, este criterio ha permitido laintegración balanceada de la información contenidaen las imágenes fuente en la imagen fusionada.

El algoritmo propuesto, ha sido evaluado con imá-genes registradas por el sensor ETM+ (multiespec-tral, MULTI y pancromático, PAN) del Landsat 7.Los resultados obtenidos muestran que las baseswavelet ortonormales particularizadas para cada ima-gen fuente presentan una mejor relación señal-ruido(SNR), lo que redunda en una mejor integración,tanto espacial como espectral, de la información pro-veniente de las imágenes fuente en comparación conlas bases wavelet estándar.

PALABRAS CLAVE: filtro FIR, entropía de lainformación, fusión de imágenes, energía de la ima-gen, bases wavelet.

ABSTRACT

This work proposes an objective criterion for FIR(Finite-duration Impulse Response) digital filtersdesign, whose coefficients have been used to directlybuilt the orthonormal wavelet bases (F) to be appliedto satellite imagery fusion. The determination of thefilter order was based on the calculation of the imagespower to be fused (source images) and the cut-off fre-quency has been obtained from the optimum entropylevel of these images. In this way, a tailored ortho-normal basis has been determined for each sourceimages. Besides, this criterion has allowed to integra-te in a balanced way the amount of information pro-vided by each of the images to be fused.

The proposed algorithm has been tested with ima-ges registered by the Landsat 7 ETM+ (multispectral,MULTI, and panchromatic, PAN, images). Theresults obtained have shown that tailored orthonormalbases provides a better signal-to-noise ratio (SNR),what gives rise to a better integration of the spatialand spectral information from the sources imagesthan standard orthonormal bases.

KEY WORDS: FIR filters, information entropy,image fusion, power image, wavelet bases.

more than one observer of the phenomenon beinganalysed. In this way, more and better informationcan be integrated for further processing. There areseveral works that describe different techniques anddefinitions for image fusion (Wald, 1999; Pohl,1999; Shettigara, 1992; Bretschneider, 2000).

One of the most widely used methods is theimage fusion through a hierarchical decompositionby means of wavelet transform, such as the Mallat’salgorithm. An image is decomposed into a set ofmultiresolution images and their associated waveletcoefficients. The coefficients, corresponding toeach level, contain the spatial differences (detail)between two successive resolution levels (Ranchinand Wald, 2000; Din-Chang et al., 2001).

The essential idea of the fusion based on thewavelet transform is to incorporate the high spatialfrequency information of the panchromatic image(PAN), contained in its wavelet coefficients, intothe degraded levels of the multispectral image(MULTI). This incorporation can be carried out bymeans of a direct replacement, addition or selectionof the corresponding wavelet coefficients (Núñez etal., 1999) as it is displayed Fig. 1. In particular, d HL

2,d LH

2 and d HH2 , represents the horizontal, diagonals

and verticals details of the image source, respectively,and they corresponds to the wavelet coefficients.

Wavelet fusion is a computationally intensiveprocess. This method allows that the colour distor-tion can be reduced to a certain extent, appearingthe fused image like a result of a high pass filteringfusion, e.g., the colour seems not being smoothlyintegrated into the spatial features (Zhang, 2002).

One of the reasons of the problems mentioned inthe previous paragraph, is the difficulty of establis-hing an objective methodology to determine tailo-red orthonormal bases (Coifman and Wickerhauser,1992) for each spectral band i of the MULTI image,FMULTI, and for the PAN image, FPAN. Therefore, themain goal of this work is to propose an objective

criterion that provides this kind of tailored ortho-normal bases, for the fusion of a MULTI image anda PAN image. In the next section, a discussionabout the determination of these bases for waveletfusion is presented.

AN OBJECTIVE CRITERION FORDETERMINING TAILORED WAVELETORTHONORMAL BASES

Fusion strategies based on hierarchical waveletdecomposition can be classified in two groups(Núñez et al., 1999): (1) Those that determinewavelet coefficients by selection rules, applied toeach spectral band of the MULTI image and thePAN image. (2) And those that partially replace thewavelet coefficients of the high resolution image bythe corresponding ones of the low resolution image.These two strategies allow obtaining the waveletcoefficients for the fused image, considering prede-fined orthonormal bases. Nevertheless, it is possi-ble to expect that tailored orthonormal bases, provi-de better quality results, since in this case, thewavelet coefficients will only depend on the featu-res of the images to be fused.

It should be also noted that the fusion of imagesthrough wavelet transform can be considered as thesuccessive application of low pass (LPF) and highpass (HPF) filters for the separation of the high andthe low frequencies of the image. Based on the pre-viously stated aspects, and as it has been alreadymentioned, a new strategy for the images fusion bymeans of the wavelet transform is proposed, basedon the definition of bases FMULTI and FPAN. Thesebases are built from the coefficients of the designedFIR filters (LPF and HPF). A critical aspect thatguarantees good spatial and spectral quality of thefused images through this technique is the appro-priate tune-up of the LPF and HPF filters. Thistune-up implies determining the order of filters (M)and the cut-off frequency (wc) (Proakis and Mono-lakis, 1995). For this aim, this work proposes anobjective criterion exclusively depending on theimages to be fused. The order of the filters, M, hasbeen determined through a power sensitivity analy-sis of the source images, with respect to the varia-tion of wc. And the optimum level of entropy of thesource images has been used to determine the cut-off frequency, wc, for the previously determinedvalue of M. This criterion has allowed a balancedintegration of the amount of information provided

M. Lillo-Saavedra, C. Gonzalo, A. Arquero y E. Martínez

6 N.º 23 - Junio 2005

Figura 1. Standard scheme for wavelet image fusion.

by each of the source images. This study must beperformed for all spectral bands of a particularMULTI image and the PAN image.

Fig. 2(a) and (b) show the curves given by the Mpower sensitivity analysis for the spectral band ETM1of the scene used in this work. It is obvious that forlarge values of M, the filtering results are almost inde-pendent of the cut-off frequency, preventing the wis-hed control of the amount of filtered information.Since a value of M=4 provides a good sensitivity asopposed to the rest of the other analysed values, thisvalue has been selected in this case. That means thatthe tailored wavelet bases are built by only five coef-ficients. And it implies a noteworthy reduction ofcomputational requirements. The same analysis mustbe performed to determine M for the rest of the bands.

Once the order (M) has been specified, the com-mon cut-off frequency of the lowpass and highpassFIR filters must be determined for each spectralband. Since these values will determine the amountof information that every source image will bring tothe fused image, the entropy of the source imageshas been calculated for different values of wc, accor-ding to equation (1) (Price, 1987):

Where pj represents the probability of the valuesof the digital numbers (DN) present in the image.

Fig. 3 shows the variation of entropy in the MULTI(ETM1, ETM2, ETM3, ETM4, ETM5 and ETM7)and PAN images for different wc values and for M=4.

The intersection points between the curvescorresponding to each band of the MULTI imageand the curve of the PAN image (Fig. 3), give thevalues of wc, for the different spectral bands. Thatassures that the fused images contain the sameamount of information coming from each one of thesource images, and also it indicates that dependingon considered spectral band, the weight of the PANimage into the original MULTI image is different.Now, the required information (M and wc) to calcu-lated the FIR filters coefficients, Bi (equation. 2),are available (Proakis and Manolakis, 1995).

y(n) = B0x(n) + B1x(n–1)+...+BMx(n–M) (2)

Where x(i), with i varying from n to n-M, corres-ponds to the discrete components of the input sig-nal, y(n) corresponds to the output signal.

Since, it is well known that the Bi coefficientsdetermine orthonormal bases (Proakis and Monola-kis, 1995), these values will be employed to builtdirectly the tailored orthonormal bases to be used inthis wavelet image fusion method.

RESULTS

The data used for the evaluation of the proposedmethod were two 4.5km x 4.5km scenes located inMadrid, Spain. The images MULTI and PAN werecollected by the sensor ETM+ (Landsat 7) on 20thAugust, 1999. The ETM1, ETM2, ETM3, ETM4,ETM5 and ETM7 bands of the Thematic Mapper

Tailored orthonormal bases for wavelet fusion

N.º 23 - Junio 2005 7

(1)

Figura 3. FIR filters cut-off frequency (wc) per band.

(a)

(b)

Figura 2. FIR filter order determination for a particularspectral band: (a) Low pass, (b) HIgh pass.

ETM+ sensor, with a common 30m spatial resolu-tion, were fused with the PAN (ETM8) image, witha 15m spatial resolution, obtaining a new multis-pectral image distinguished by a high spatial (15m)and spectral (6 bands) quality.

The methodology described in section 2 must beapplied to each band of the multispectral image.The images to be fused must be radiometricallycorrected and spatially referenced. In order toobtain better results, it is advisable to reference thelow spatial resolution image in relation to the highspatial resolution image (Price, 1999).

Fig. 4 shows an overview of the fusion methodo-logy proposed in this work, applied to the ETM1and PAN images of the data described above. Thefirst step in the fusion process is to resize the mul-

tispectral image to the PAN image size, by means ofa interpolation methods. In the next step, the PANand MULTI images should be wavelet transformedthrough the tailored orthogonal bases previouslydetermined, in order to obtain one dyadic levelwavelet decomposition. Finally, MULTI high-fre-quency components are replaced by PAN high-fre-quency components, then the inverse wavelet trans-form is applied to obtain the ith band of the fusedimage.

Table 1 and Table 2 summarize the values of Mand wc (expressed as normalised frequency), as wellas, the coefficients of the tailored orthonormalbases FMULTIi

and FPANi, respectively, obtained for

the considered source images. Fig. 5(a), 5(b) and5(c) show three RGB compositions (R=ETM7,G=ETM4 and B=ETM1). A scene of the originalMULTI image (5a) and its corresponding scenesfused through the standard wavelet fusion method(5b) and the proposed method (5c). A visual analy-sis shows that both fused images presents a clearspatial resolution enhancement as compared to theoriginal image. Besides, it can be appreciated smo-other transitions in the high gradient areas in the

M. Lillo-Saavedra, C. Gonzalo, A. Arquero y E. Martínez

8 N.º 22 - Diciembre 2004

Figura 4. Overviwe of the proposed fusion scheme for aparticular band.

M

wc

ETM1

4

0.30

ETM2

4

0.25

4

0.18

4

0.22

4

0.14

4

0.16

ETM3 ETM4 ETM5 ETM7

Tabla 1. M and wc value, for FIR filters design

BAND 1

BAND 2

BAND 3

BAND 4

BAND 5

BAND 7

FMULTI1

FPAN1

FMULTI2

FPAN2

FMULTI3

FPAN3

FMULTI4

FPAN4

FMULTI5

FPAN5

FMULTI7

FPAN7

0.0121-0.0121

0.0127-0.0127

0.0115-0.0115

0.0125-0.0125

0.0097-0.0097

0.0107-0.0107

0.1386-0.1386

0.1215-0.1215

0.0919-0.0919

0.1090-0.1090

0.0725-0.0725

0.0828-0.0828

0.29880.7012

0.25000.7500

0.17970.8203

0.21870.7813

0.13870.8613

0.16020.8398

0.1386-0.1386

0.1215-0.1215

0.0919-0.0919

0.1090-0.1090

0.0725-0.0725

0.0828-0.0828

0.0121-0.0121

0.0127-0.0127

0.0115-0.0115

0.0125-0.0125

0.0097-0.0097

0.0107-0.0107

Coeff. 1 Coeff. 2 Coeff. 3 Coeff. 4 Coeff. 5

Tabla 2. Coefficients of FMULTIi

and FPANi

image fused with the tailored wavelet bases than inthe image fused with standard wavelet bases (Biort-hogonal). That means that the tailored bases provi-de a better integration of the spatial and spectralinformation form the source images.

Different quality indices, proposed in the literature,have been calculated with the aim of quantifying thequality of the images fused by the proposed method:the signal-to-noise ratio (SNR), correlation andERGAS (Wald et al., 1997). Besides, those indiceshave allowed to compared the fusion performances ofthe proposed method versus the standard waveletfusion method, described in previous section.

The SNR has been calculated (Table 3) for allbands of the MULTI, by means of the equations (3).

Where f(x,y) and f̂ (x,y) are the ith band of the ori-ginal MULTI and the fused image, respectively.

Tailored orthonormal bases for wavelet fusion

N.º 23 - Junio 2005 9

(3)

*Figura 5. RGB compositions (R=ETM7, G=ETM4, B=ETM1) (a) Scene of the original MULTI image, (b) fused scenethrough the standard orthonormal basis and (c) fused scene through the tailored orthonormal bases.

It can be noted that, except for band ETM2, theproposed method provides a better signal-to-noiserate than the standard method, that confirms theaspects discussed in the visual analysis.

Table 4 shows the statistical correlation values bet-ween the spectral bands of the original image and thecorresponding spectral bands of the fused image,obtained through tailored and standard bases. The sta-tistical correlation is defined by equation (4):

Where Am and Bm are the averages of the digitalnumbers of the corresponding images and Npix is thenumber of pixels in the image.

It can be observed at Table 4 that correlationvalues of the fused images by both method are veryclose to 1 for all spectral bands, meaning that bothmethods provides similar good performances.

Finally, the ERGAS index has been used to quan-tify the spectral quality of fused images. It has beenestablished that images with an ERGAS valuelower than 3 exhibit a good spectral quality (Waldet al. 1997). Table 5 summarizes the index valuescalculated and averaged for the six bands of the

(a) (b) (c)

SNR(dB)

STANDARD

BASIS

TAILORED

BASES

ETM1

28.38

29.92

18.41

17.47

21.64

22.14

24.36

24.90

5.22

8.99

20.11

20.58

19.69

20.67

ETM2 ETM3 ETM4 ETM5 ETM7 Average

Tabla 3. Signal-to-Noise Ratio (SNR)dB)).

(4)

CORR ETM1 ETM2 ETM3 ETM4 ETM5 ETM7

STANDARD

BASIS0.971

0.979

0.979

0.983

0.985

0.985

0.991

0.991

0.989

0.989

0.987

0.987TAILORED

BASES

Tabla 4. Correlation between original and fused images.

Todas las figuras precedidas de asterisco se incluyen en el cuadernillo anexo de color

fused image. The low ERGAS value given by theproposed method, evidence a light improvement inthe spectral quality respect to the standard waveletfusion method.

CONCLUSION

A new methodology, to determine tailored ortho-normal wavelet bases for multispectral imagesfusion, has been proposed in the present work.These bases depend only upon features of the sour-ce images, freeing the process from other subjecti-ve factors. The methodology is based on an objecti-ve criterion for determining the parameters (M, wc)implied in the tune up of the FIR filters.

In this study, it has been proved that low values ofM provides a high power sensitivity of the image tobe fused with respect to the variation of wc. Thishigh sensitivity has allowed to control the amount ofinformation that each source image contributes tothe fused image. On the other hand, the low valuesof M imply that a small number of coefficients isrequired to define the tailored orthonormal bases.This diminishes the computational requirements.Moreover the calculation of a common entropyvalue for the source images has allowed to determi-ne an unique cut-off frequency for the low and highpass filters, but different to each band, guaranteeingthat the same amount of the information is integra-ted from the source images in each band.

In other words, the influence of the PAN image ineach of the fused bands is different. From the eva-luation of the quality of the fused images, it is rea-sonable to conclude that tailored orthonormal basesprovides a better integration of spatial and spectralinformation than image fused by the standard wave-let method. Since it has been obtained an ERGASvalue lower than 3, it can assure that the fusedimage by the proposed method preserves the spec-tral features of the MULTI image. Finally, the eva-luation of the signal-to-noise ratio (SNR) hasshown that tailored orthonormal bases provides abetter integration of the spatial and spectral infor-mation from the sources images than standard ort-honormal bases.

BIBLIOGRAPHY

BRETSCHNEIDER, T., KAO, O. 2000. Image fusionin remote sensing, Proceedings of the 1st OnlineSymposium of Electronic Engineers.(www.ntu.edu.sg/home/astimo/Publications/ Docu-ments/OSEE2000.pdf).

CHAVEZ, P.S., SIDES, S.C., ANDERSON, J.A.1991. Comparison of the three methods to mergemultiresolution and multispectral: LANSAT TMand SPOT panchromatic. Photogrammetric Engi-neering and Remote Sensing. 57: 295-303.

COIFMAN, R.R. and WICKERHAUSER, M.V.1992. Entropy-based algorithms form best basesselection. Transaction on Information Theory,IEEE. 38(2): 713-718.

DIN-CHANG, T., YI-LING, C., LIU, M.S.C. 2001.Wavelet-based multispectral image fusion. Geos-cience and Remote Sensing Symposium,IGARSS. 4: 1956-1958.

NÚÑEZ, J., OTAZU, X., FORS, O., PRADES A.,PALÁ, V. and ARBIOL, R. 1999. Multiresolu-tion-based image fusion with additive waveletdecomposition. Transactions on Geoscience andRemote Sensing. IEEE. 37(3): 1204-1211.

POHL, A. 1999. Tools and methods for fusion ofimages of different spatial resolution. Internatio-nal Archives of Photogrammetry and RemoteSensing. 32, Part 7-4-3 W6.

PRICE, J.C. 1987. Combining Panchromatic andMultispectral Imagery from Dual ResolutionSatellite Instruments. Remote Sensing of Environ-ment. 21: 119-128.

PRICE, J.C. 1999. Combining multispectral data ofdiffering spatial resolution. Transactions onGeoscience and Remote Sensing. IEEE, 37 (3):1199-1203.

PROAKIS, J.G. and MANOLAKIS, D.G. 1995.Digital Signal Processing: Principles. Algorithmsand Applications. Prentice Hall.

RACHIN, T. and WALD, L. 2000. Fusion of highspatial and spectral resolution image: TheARSIS concept and its implementation. Photo-grammetric Engineering and Remote Sensing.66(1): 49-61.

SHETTIGARA, V.K. 1992. A generalized compo-nent substitution techniques for spatial enhance-ment of multispectral images using a higher reso-lution data set. Photogrammetric Engineeringand Remote Sensing. 58 (5):561-567.

WALD, L., RANCHIN, T. and MANGOLINI, M.1997. Fusion of satellite images of different spa-

M. Lillo-Saavedra, C. Gonzalo, A. Arquero y E. Martínez

10 N.º 23 - Junio 2005

BASES

ERGAS

STANDARD

2.93

TAILORED

2.92

Tabla 5. ERGAS index.

tial resolutions: assessing the quality of resultingimages. Photogrammetric Engineering andRemote Sensing. 63 (6): 691-699.

WALD, L. 1999. Some terms of reference in datafusion. Transactions on Geoscience and RemoteSensing. IEEE. 37: 1190-1193.

ZHANG, Y. 2002. Problems in the fusion of com-mercial high-resolution satellite images as well asLandsat 7 images and initial solution. Internatio-nal Archives of Photogrammetry and RemoteSensing (IAPRS). 34, Part 4.

Tailored orthonormal bases for wavelet fusion

N.º 23 - Junio 2005 11

El artículo 15º de los estatutos de la AsociaciónEspañola de Teledetección (A.E.T.) contempla laposibilidad de que los socios numerarios (residentesen territorio español) y correspondientes (no residen-tes en territorio español), estudiantes de Facultades,Escuelas Técnicas y Universitarias, gocen de un 50%de bonificación en sus cuotas.

Animamos a todos los estudiantes, con interés enconocer las investigaciones y técnicas de teledetec-ción, para que se integren como socios de la Aso-ciación Española de Teledetección (A.E.T.) por unacuota anual de 18 euros.

Socios numerariosy correspondientes estudiantes

NNNNOOOOTTTTIIIICCCCIIIIAAAASSSS