Abstract
Pansharpening consists in merging a low-resolution multispectral image (MS) with a high spatial resolution panchromatic image (PAN) to produce a high resolution pansharpened MS image. It consists in enhancing spatially the low-resolution MS image by injecting the missing details provided by the high-resolution PAN image. In this paper, we propose a novel pansharpening approach based on decomposition/reconstruction processing using low-pass and high-pass filter banks. On the one hand, the low-pass approximation (taking into account the imaging system modulation transfer function MTF) of the pansharpened MS image is assumed to be equal to the original MS image in order to preserve the spectral quality. On the other hand, the high-pass filter allowing us to extract the high-frequency PAN details is designed as complementary filter to the low-pass one in order to provide perfect reconstruction in the ideal case. Quantitative assessment performed on reduced and full-resolution images are used to validate the proposed technique and compare it to state-of-art. Experimental results using Pléaides and GeoEye-1 data show that our proposed fusion schema outperforms the pre-existing methods visually as well as quantitatively.
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Aiazzi, B., Alparone, L., Baronti, S., & Garzelli, A. (2002). Context-driven fusion of high spatial and spectral resolution images based on oversampled multiresolution analysis. IEEE Transactions on Geoscience and Remote Sensing, 40(10), 2300–2312.
Aiazzi, B., Alparone, L., Baronti, S., Garzelli, A., & Selva, M. (2006). MTF-tailored multiscale fusion of high-resolution MS and pan imagery. Photogrammetric Engineering and Remote Sensing, 72(5), 591–596.
Aiazzi, B., Baronti, S., & Selva, M. (2007). Improving component substitution pansharpening through multivariate regression of MS+ Pan data. IEEE Transactions on Geoscience and Remote Sensing, 45(10), 3230–3239.
Aiazzi, B., Alparone, S. Land Baronti, Garzelli, A., & Selva, M. (2011). Twenty-five years of pansharpening: A critical review and new developments. In Signal and image processing for remote sensing (pp. 533–548). Boca Raton, FL: Taylor and Francis Books.
Aiazzi, B., Alparone, L., Baronti, S., Garzelli, A., & Selva, M. (2012). Advantages of Laplacian pyramids over ”à trous” wavelet transforms for pansharpening of multispectral images (Vol. 8537, pp. 853704–853704–10).
Alparone, L., Baronti, S., Garzelli, A., & Nencini, F. (2004). A global quality measurement of pan-sharpened multispectral imagery. IEEE Geoscience and Remote Sensing Letters, 1, 313–317.
Alparone, L., Wald, L., Chanussot, J., Thomas, C., Gamba, P., & Bruce, L. (2007). Comparison of pansharpening algorithms: Outcome of the 2006 GRS-S data-fusion contest. IEEE Transactions on Geoscience and Remote Sensing, 45(10), 3012–3021.
Alparone, L., Aiazzi, B., Baronti, S., Garzelli, A., Nencini, F., & Selva, M. (2008). Multispectral and panchromatic data fusion assessment without reference. Photogrammetric Engineering and Remote Sensing, 74(2), 193–200.
Amro, I., Mateos, J., Vega, M., Molina, R., & Katsaggelos, A. K. (2011). A survey of classical methods and new trends in pansharpening of multispectral images. EURASIP Journal on Advances in Signal Processing, 2011, 79.
Baronti, S., Aiazzi, B., Selva, M., Garzelli, A., & Alparone, L. (2011). A theoretical analysis of the effects of aliasing and misregistration on pansharpened imagery. IEEE Journal of Selected Topics in Signal Processing, 5, 446–453.
Bijaoui, A., Murtagh, F., & Starck, J. (1994). Restauration des Images Multi-Echelles par l’Algorithme a Trous. Traitement du Signal, 11(3), 229–243.
Blanc, P., Wald, L., & Ranchin, T. (1998). Importance and effect of co-registration quality in an example of “pixel to pixel” fusion process. In T. R. et L Wald (Ed.) 2nd international conference “ Fusion of earth data: Merging point measurements, raster maps and remotely sensed images ”, SEE/URISCA, Nice, France, Sophia Antipolis, France, pp. 67–74.
Blanchet, G., Moisan, L., & Rougé, B. (2005). A linear prefilter for image sampling with ringing artifact control. In IEEE International Conference Images Processing (pp. 577–580).
Blanchet, G., Lebeque, L., Fourest, S., Latry, C., Porez Nadal, F., Lacherade, S. & Thiebaut, C. (2012). Pleiades-HR innovative techniques for radiometric image quality commissioning (pp. XXXIX–B1:513–518).
Burt, P., & Adelson, E. (1983). The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, 31(4), 532–540.
Carla, R., Santurri, L., Aiazzi, B., & Baronti, S. (2015). Full-scale assessment of pansharpening through polynomial fitting of multiscale measurements. IEEE Transactions on Geoscience and Remote Sensing, 53(12), 6344–6355.
Carper, W., Lillesand, T., & Kiefer, P. (1990). The use of intensity-hue-saturation transformations for merging spot panchromatic and multispectral image data. Photogrammetric Engineering and Remote Sensing, 56(4), 459–467.
Chavez, P., Sides, S., & Anderson, J. (1991). Comparison of three different methods to merge multiresolution and multispectral data: Landsat TM and SPOT panchromatic. Photogrammetric Engineering and Remote Sensing, 57(3), 295–303.
Cheng, M., Wang, C., & Li, J. (2014). Sparse representation based pansharpening using trained dictionary. IEEE Geoscience and Remote Sensing Letters, 11(1), 293–297.
Chibani, Y., & Houacine, A. (2002). The joint use of IHS transform and redundant wavelet decomposition for fusing multispectral and panchromatic images. International Journal of Remote Sensing, 23, 3821–3833.
CNES. (2012). Pléaides user guide. Toulouse: ASTRUIM Services.
Croisier, A., Esteban, D., & Galand, C. (1976). Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques. International Conference on Information Sciences and Systems, Patras, Greece, 2, 443–446.
Dai, X., & Khorram, S. (1998). The effects of image misregistration on the accuracy of remotely sensed change detection. IEEE Transactions on Geoscience and Remote Sensing, 36(5), 1566–1577.
Daubechies, I. (1988). Adjustable model-based fusion method for multispectral and panchromatic image. Communications on Pure and Applied Mathematics, 41, 909–996.
Delleji, T., Kallel, A., & Hamida, A. B. (2014). Multispectral image adaptive pansharpening based on wavelet transformation and NMDB approaches. International Journal of Remote Sensing, 35(19), 7069–7098.
Ehlers, M., Klonus, S., Johan Åstrand, P., & Rosso, P. (2010). Multi-sensor image fusion for pansharpening in remote sensing. International Journal of Image and Data Fusion, 1(1), 25–45.
Esteban, D., & Galand, C. (1977). Application of quadrature mirror filters to split band voice coding schemes. In: IEEE international conference on acoustics, speech, and signal processing, ICASSP’77 (Vol. 2, pp. 191–195).
Ghahremani, M., & Ghassemian, H. (2015). Remote sensing image fusion using Ripplet transform and compressed sensing. IEEE Geoscience and Remote Sensing Letters, 12(3), 502–506.
González-Audícana, M., Saleta, J. L., Catalán, R. G., & García, R. (2004). Fusion of multispectral and panchromatic images using improved IHS and PCA mergers based on wavelet decomposition. IEEE Transactions on Geoscience and Remote Sensing, 42, 1291–1299.
Harikumar, V., Gajjar, P., Joshi, M., & Raval, M. (2014). Multiresolution image fusion: use of compressive sensing and graph cuts. Selected Topics in Applied Earth Observations and Remote Sensing, IEEE Journal of, 7(5), 1771–1780.
He, X., Condat, L., Bioucas-Dias, J., Chanussot, J., & Xia, J. (2014). A new pansharpening method based on spatial and spectral sparsity priors. IEEE Transactions on Image Processing, 23(9), 4160–4174.
Jiang, C., Zhang, H., Shen, H., & Zhang, L. (2012). A practical compressed sensing-based pan-sharpening method. IEEE Geoscience and Remote Sensing Letters, 9(4), 629–633.
Jing, L., & Cheng, Q. (2011). An image fusion method for misaligned panchromatic and multispectral data. International Journal of Remote Sensing, 32(4), 1125–1137.
Kallel, A. (2015). MTF-adjusted pansharpening approach based on coupled multiresolution decompositions. IEEE Transactions on Geoscience and Remote Sensing, 53(6), 3124–3145.
Khan, M., Alparone, L., & Chanussot, J. (2009). Pansharpening quality assessment using the modulation transfer functions of instruments. IEEE Transactions on Geoscience and Remote Sensing, 47(11), 3880–3891.
Laben, C., & Brower, B. (2000). Process for enhancing the spatial resolution of multispectral imagery using pan-sharpening. US Patent 6011875.
Li, S., & Yang, B. (2011). A new pan-sharpening method using a compressed sensing technique. IEEE Transactions on Geoscience and Remote Sensing, 49(2), 738–746.
Li, S., Yin, H., & Fang, L. (2013). Remote sensing image fusion via sparse representations over learned dictionaries. IEEE Transactions on Geoscience and Remote Sensing, 51(9), 4779–4789.
Li, Z., & Leung, H. (2009). Fusion of multispectral and panchromatic images using a restoration-based method. IEEE Transactions on Geoscience and Remote Sensing, 47(5), 1482–1491.
Liu, J. G. (2000). Smoothing filter-based intensity modulation: A spectral preserve image fusion technique for improving spatial details. International Journal of Remote Sensing, 21(18), 3461–3472.
Luo, B., Khan, M., Bienvenu, T., Chanussot, J., & Zhang, L. (2013). Decision-based fusion for pansharpening of remote sensing images. IEEE Geoscience and Remote Sensing Letters, 10(1), 19–23.
Mallat, S. (1989). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 674–693.
Nencini, F., Garzelli, A., Baronti, S., & Alparone, L. (2007). Remote sensing image fusion using the curvelet transform. Information Fusion, 8(2), 143–156.
Nunez, J., Otazu, X., Fors, O., Prades, A., Pala, V., & Arbiol, R. (1999). Multiresolution-based image fusion with additive wavelet decomposition. IEEE Transactions on Geoscience and Remote Sensing, 37(3), 1204–1211.
Oppenheim, A. V., & Schafer, R. W. (2009). Discrete-time signal processing (3rd ed.). Upper Saddle River, NJ: Prentice Hall Press.
Otazu, X., González-Audícana, M., Fors, O., & Nunez, J. (2005). Introduction of sensor spectral response into image fusion methods. Application to wavelet-based methods. IEEE Transactions on Geoscience and Remote Sensing, 43(10), 2376–2385.
Palsson, F., Sveinsson, J., & Ulfarsson, M. (2014). A new pansharpening algorithm based on total variation. IEEE Geoscience and Remote Sensing Letters, 11(1), 318–322.
Palsson, F., Sveinsson, J., Ulfarsson, M., & Benediktsson, J. (2015). Model-based fusion of multi- and hyperspectral images using PCA and wavelets. IEEE Transactions on Geoscience and Remote Sensing, 53(5), 2652–2663.
Patra, S. K., Mishra, N., Chandrakanth, R., & Ramachandran, R. (2002). Image quality improvement through MTF compensation—A treatment to high resolution data. Indian Cartographer, 86–93.
Pohl, C., & Van Genderen, J. (1998). Review article multisensor image fusion in remote sensing: Concepts, methods and applications. International Journal of Remote Sensing, 19(5), 823–854.
Ranchin, T., & Wald, L. (2000). Fusion of high spatial and spectral resolution images: The ARSIS concept and its implementation. Photogrammetric Engineering and Remote Sensing, 66(1), 49–61.
Shah, V. P., Younan, N. H., & King, R. L. (2008). An efficient pan-sharpening method via a combined adaptive PCA approach and contourlets. IEEE Transactions on Geoscience and Remote Sensing, 46, 1323–1335.
Starck, J. L., Fadili, J., & Murtagh, F. (2007). The undecimated wavelet decomposition and its reconstruction. IEEE Transactions on Image Processing, 16(2), 297–309.
Thomas, C., Ranchin, T., Wald, L., & Chanussot, J. (2008). Synthesis of multispectral images to high spatial resolution: A critical review of fusion methods based on remote sensing physics. IEEE Transactions on Geoscience and Remote Sensing, 46(5), 1301–1312.
Vaidyanathan, P. (1990). Multirate digital filters, filter banks, polyphase networks, and applications: A tutorial. Proceedings of the IEEE, 78(1), 56–93.
Vetterli, M. (1986). Filter banks allowing perfect reconstruction. Signal Processing (Elsevier), 10(3), 219–244.
Vetterli, M., & Kovačević, J. (1995). Wavelets and subband coding. Prentice-Hall signal processing series. Prentice Hall PTR.
Vicinanza, M., Restaino, R., Vivone, G., Dalla Mura, M., & Chanussot, J. (2015). A pansharpening method based on the sparse representation of injected details. IEEE Geoscience and Remote Sensing Letters, 12(1), 180–184.
Vivone, G., Alparone, L., Chanussot, J., Dalla Mura, M., Garzelli, A., Licciardi, G., et al. (2015a). A critical comparison among pansharpening algorithms. IEEE Transactions on Geoscience and Remote Sensing, 53(5), 2565–2586.
Vivone, G., Simões, M., Dalla Mura, M., Restaino, R., Bioucas-Dias, J., Licciardi, G., et al. (2015b). Pansharpening based on semiblind deconvolution. IEEE Transactions on Geoscience and Remote Sensing, 53(4), 1997–2010.
Wald, L. (1999). Some terms of reference in data fusion. IEEE Transactions on Geoscience and Remote Sensing, 37(3), 1190–1193.
Wald, L., Ranchin, T., & Mangolini, M. (1997). Fusion of satellite images of different spatial resolutions: Assessing the quality of resulting images. Photogrammetric Engineering and Remote Sensing, 63(6), 691–699.
Wang, Z., & Bovik, A. (2002). A universal image quality index. IEEE Signal Processing Letters, 9, 81–84.
Wei, Q., Bioucas-Dias, J., Dobigeon, N., & Tourneret, J. Y. (2015). Hyperspectral and multispectral image fusion based on a sparse representation. IEEE Transactions on Geoscience and Remote Sensing, 53(7), 3658–3668.
Xu, Q., Zhang, Y., & Li, B. (2014). Recent advances in pansharpening and key problems in applications. International Journal of Image and Data Fusion, 5(3), 175–195.
Yin, H. (2015). Sparse representation based pansharpening with details injection model. Signal Processing, 113, 218–227.
Yocky, D. (1996). Artifacts in wavelet image merging. Optical Engineering, 35, 2094–2101.
Zhang, G., Fang, F., Zhou, A., & Li, F. (2015). Pan-sharpening of multi-spectral images using a new variational model. International Journal of Remote Sensing, 36(5), 1484–1508.
Zhang, L., Shen, H., Gong, W., & Zhang, H. (2012). Adjustable model-based fusion method for multispectral and panchromatic images. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42(6), 1693–1704.
Zhang, Y., De Backer, S., & Scheunders, P. (2009). Noise-resistant wavelet-based bayesian fusion of multispectral and hyperspectral images. IEEE Transactions on Geoscience and Remote Sensing, 47(11), 3834–3843.
Zhou, J., Civco, D. L., & Silander, J. A. (1998). A wavelet transform method to merge Landsat TM and SPOT panchromatic data. International Journal of Remote Sensing, 19(4), 743–757.
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Appendices
Appendix 1: Downsampling/upsampling processes
This appendix contains the equations defining the downsampling and upsampling processes.
For downsampling, a considered discrete input sequence x[n] having z-transform noted X(z) is subsampled by an integer factor M yielding to the output y[n] . The latter is obtained by discarding those samples of x[n] whose indices are not multiple of the integer M :
The z -transform of y[n] is Y(z) given by:
where, the term \( X(z^{\frac{1}{2}}) \) is the expanded version of X(z) and the second term \( X(-z^{\frac{1}{2}}) \) is a shifted version of the first term. The term \( X(-z^{\frac{1}{2}}) \) causes an overlap in the frequency domain, which induces the aliasing effect. When aliasing occurs, the original signal, x[n], cannot be recovered perfectly from y[n] . To overcome this problem, the input signal should be bandlimited using a low-pass filtering. Therefore, ideal decimator is recommended before performing downsampling (Oppenheim and Schafer 2009).
Interpolating by a factor L consists in inserting \(L-1\) ( L is an integer) zeros between successive samples. It is expressed by:
The z -transform of the upsampled signal is:
The term Y(z) contains \( L-1 \) replica of the original spectrum, that causes spectral periodization effects. This phenomenon may be seen as the dual of the aliasing effect. In practice, in order to eliminate the replica produced by the upsampler, one may introduce a filtering step after the upsampling process.
Aliasing and spectral periodization may simultaneously occur, when downsamplers and upsamplers are cascaded (Vaidyanathan 1990). In this case, it is not possible to restore the input signal from its downsampled and upsampled version. Such distortions should be canceled (or at least reduced): the signal x[n] is filtered using low-pass filtering followed by a decimator, then it is interpolated and filtered by an ideal filter aiming at rejecting the spectral periodizations and retaining only the desired signal.
Appendix 2: Two-channel filter banks
Remark Let the z -transform of a sequence x[n] be X(z) .
The analysis filter bank (Fig. 1) decomposes the input signal x[n] into low and high frequency components \(x_1[n]\) and \( x_2[n]\), respectively, which are expressed as functions of X(z) by:
Then, the signals \( x_1[n]\) and \( x_2[n]\) in both channels are subsampled by a factor two according to [cf. (28), (31)]:
The second configuration aims at reconstructing the original signal from the two smoothed and subsampled signals \(v_1[n]\) and \(v_2[n]\). These latter are upsampled by a factor two. Based on (30) and (32), the two components \(U_1(z)\) and \(U_2(z)\) are given by:
Then, they are interpolated using low-pass and high-pass filtering: \(h_r[n]\) and \(g_r[n]\), respectively, obtaining the following components \(Y_1(z)\) and \(Y_2(z)\):
Finally, the two components \(Y_1(z)\) and \(Y_2(z)\) are summed allowing the reconstruction of the output signal \(x_r[n]\):
where E(z) and F(z) are called distortion transfer function and aliasing transfer function (Vetterli and Kovačević 1995), respectively.
In (35), the component involving X(z) represents the desired signal, whereas the term involving \(X(-z)\) is the unwanted aliased component caused by downsampling that must be minimized.
Appendix 3: 4-reduction versus 2 -reduction Gaussian filters
For many remote sensing imaging systems, the resolution-scale between PAN and MS images is four. In order to downsample directly the PAN image to reach the resolution of MS image and to take into account the MTF of the MS sensors, the 4 -reduction filter (\( r_4 \)) is used in this case. To respect the MTF Gaussian shape, Kallel (2015) defined it as a centered Gaussian distribution of standard deviation, \( \sigma _4 \), given by:
where \( \alpha _{cf} \) is the cutoff value at the Nyquist frequency.
Now, in our study, the two-channel filter bank is considered, the scale-ratio is equal two. The downsampling operation must be then applied twice. It consists of applying low-pass filter (\( r_2 \)) followed by sub-sampling (\( \downarrow 2 \)).
The question is how to compute the adequate filter \( r_2 \) such that when 2 -reduction downsampling is applied twice it fits the 4 -reduction one. At the one hand, according to Kallel (2015), the first level ( 2 -reduction) and the second level ( 4 -reduction) filters (respectively, \( r_2 \) and \( r_4 \) ) are linked together as follows:
At the other hand, to conserve the radiometric properties, a reduction filter, r , must verify the averaging filter equality:
Using the latter two equalities, we will show in the following that \( r_2 \) can be Gaussian-shaped.
Lemma 1
Let us consider the couple of filters (h, g) , satisfying the averaging property. Then, the filter v defined by:
satisfies it, too.
Proof
Summing all coefficients of the filter v :
Then, substitute p for \( n - 2 i \) in (40):
\(\square \)
Theorem 1
Given a Gaussian filter, \( v_2 \), of standard deviation \( \sigma \), defined by:
where \( a_{\sigma } \) allows to verify the averaging property, i.e.:
Then, the filter \( v_4 \) defined by:
is a Gaussian filter of standard deviation \( \sqrt{5} \sigma \).
Proof
(44) can be written as follows:
Then, replacing variable \( \frac{2n}{5} -k \) by p, (45) can be written as:
It is clear that \( a_{\sigma }^2 \sum _p \exp \left( \frac{-5 p^2}{2 \sigma ^2}\right) \) is a constant. Moreover, according to Lemma 1, \( v_4 \) satisfies the averaging property, therefore:
Replacing (47) in (46), one find:
Finally, it is possible to conclude that \( v_{4} \) is a Gaussian distribution with a standard deviation equal \( \sqrt{5} \sigma \). \(\square \)
According to Theorem 1, applying twice a Gaussian 2 -reduction downsampling of sdv \( \sigma \) is equivalent to apply a Gaussian 4 -reduction filter of sdv \( \sqrt{5} \sigma \). Now, in our case, we need to find \( r_2 \) such as \( r_4 \) is Gaussian of sdv \( \sigma _4 \). Therefore, \( r_2 \) can be chosen Gaussian of sdv \( \sigma = \frac{\sigma _4}{\sqrt{5}} \).
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Hallabia, H., Kallel, A., Hamida, A.B. et al. High spectral quality pansharpening approach based on MTF-matched filter banks. Multidim Syst Sign Process 27, 831–861 (2016). https://doi.org/10.1007/s11045-016-0421-4
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DOI: https://doi.org/10.1007/s11045-016-0421-4