transform in this paper, we extend the pansharpening...

IET Image Processing

Research Article

Pansharpening scheme using filtering in two-dimensional discrete fractional Fouriertransform

ISSN 1751-9659Received on 8th September 2017Revised 22nd November 2017Accepted on 19th January 2018E-First on 15th February 2018doi: 10.1049/iet-ipr.2017.0961www.ietdl.org

Nidhi Saxena1 , Kamalesh K. Sharma1

1Department of Electronics and Communication Engineering, Malaviya National Institute of Technology, Jaipur 302017, Rajasthan, India E-mail: [email protected]

Abstract: The aim of the pansharpening scheme is to improve the spatial information of multispectral images using thepanchromatic (PAN) image. In this study, a novel pansharpening scheme based on two-dimensional discrete fractional Fouriertransform (2D-DFRFT) is proposed. In the proposed scheme, PAN and intensity images are transformed using 2D-DFRFT andfiltered by highpass filters, respectively. The filtered images are inverse transformed and further used to generate thepansharpened image using appropriate fusion rule. The additional degree of freedom in terms of its angle parametersassociated with the 2D-DFRFT is exploited for obtaining better results in the proposed pansharpening scheme. Simulationresults of the proposed technique carried out in MATLAB are presented for IKONOS and GeoEye-1 satellite images andcompared with existing fusion methods in terms of both visual observation and quality metrics. It is seen that the proposedpansharpening scheme has improved spectral and spatial resolution as compared to the existing schemes.

1 IntroductionMany of the remote sensing satellites, such as SPOT, IRS, Landsat7, IKONOS, QuickBird, Pleiades and Worldview-2, capture a lot ofimages for topographic mapping and map updating, land use,agriculture and forestry, flood monitoring, ice and snow monitoringand so on. These images can be categorised into multispectral (MS)and panchromatic (PAN) images. The MS images have highspectral resolution and low spatial resolution while PAN image hashigh spatial resolution and low spectral resolution of the samecaptured area [1]. It is difficult to obtain an image directly from thesatellite sensors having both high spatial and spectral resolution ofthe captured area due to some technical constraints [2]. Manypansharpening schemes have been proposed for achieving the goalof high spatial and spectral resolution in a single image [3]. Thepansharpening scheme refers to the fusion of information derivedfrom PAN and MS images captured simultaneously over the samearea [4, 5]. These pansharpening methods are most commonlyclassified into four categories: component substitution (CS),multiresolution analysis (MRA), hybrid and model-based methods[3]. The CS-based pansharpening methods based on intensity-hue-saturation (IHS) transformation, Gram–Schmidt (GS)orthogonalisation, principal component analysis and so on arediscussed in [6–8]. The MRA-based pansharpening schemes usingHilbert vibration decomposition (HVD), wavelet packet, non-subsampled shearlet transform have also been presented in [9–11].The hybrid pansharpening methods make use of both CS and MRAtechniques as discussed in [12, 13] and so on. The model-basedpansharpening methods based on online coupled dictionarylearning, spatial correlation modelling, MRF model, compressivesensing-based (CS) technique and so on are discussed in [14–17].The effect of aliasing and mis-registration on pansharpenedimagery is deeply discussed in [18].

Recently, HVD-based pansharpening scheme is proposed [9].This method decomposed the PAN image into the decreasing orderof the energy in terms of instantaneous amplitude and frequencycomponents and highest energy component of the PAN imageprovides the instantaneous frequency-based lowpass filtering.

A fast Fourier transform (FFT)-based pansharpening scheme isproposed in [19]. This method is based on IHS transform with FFTfiltering of both the PAN image and intensity image component ofthe original MS images, one of the drawbacks of this algorithm isthat original MS images have to contain only three bands. It

reduces the general applicability of this algorithm to very few caseslike LANDSAT and SPOT satellite images in which MS sensor hasthree bands of the visible spectrum and a fourth band on theinfrared. The infrared sensor has a spatial resolution much lowerthan the three visible sensors, and it is usually discarded for taskrelated to the production of visual products or applications [12]. Itmay be mentioned here that due to the finite spatial size of MS andPAN images, these images will not be band limited in FFT domainand therefore FFT-based pansharpening may not be suitable in suchcases. In addition, the PAN and MS images collected throughdifferent sensors satellite environment suffer from different typesof noise which may not be stationary in nature. Therefore,conventional Fourier transform is not suitable for handling suchnon-stationary signals and noise [20, 21]. The fractional Fouriertransform (FRFT) is known to handle such non-stationary noise ina more effective way than the conventional Fourier transform [22].It would, therefore, be interesting to investigate the intermediatedomains (known as FRFT domains) between spatial and FFTdomains for pansharpening purpose. The FRFT is a generalisedversion of the conventional Fourier transform. The angle parameterof two-dimensional discrete FRFT (2D-DFRFT) can be varied toprovide infinite representations of the given signal in different 2D-DFRFT domains each corresponding to the different value of theangle parameter. The 2D-DFRFT provides a free degree offreedom in terms of its angle parameters. The 2D-DFRFT has beenapplied in many image processing applications [23–31] but the useof discrete FRFT (DFRFT) in pansharpening has not beeninvestigated so far.

In this paper, we extend the pansharpening scheme using theconventional Fourier transform presented in [19] to the DFRFTdomains. Simulations are carried out on IKONOS and GeoEye-1satellite datasets. The results are compared with the existingpansharpening schemes based on HVD (HVD_F) [9], optimal filter(OMF) [32], adaptive IHS (AIHS) [6], generalised Laplacianpyramid with modulation transfer function matched filter [33],decimated wavelet transform using an additive injection model(Indusion) [34] and a trous wavelet transform using the model-2(ATWTM2) [35]. It is seen that the proposed scheme showsimprovement in the quality of the fused image as compared to theexisting schemes which may be attributed to fractional domainfiltering provided by 2D-DFRFT-based approach. We have alsoinvestigated the effects of aliasing and mis-registration on our

IET Image Process., 2018, Vol. 12 Iss. 6, pp. 1013-1019© The Institution of Engineering and Technology 2018

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proposed method and compared it with other existingpansharpening methods using the methodology discussed in [18].

The rest of the paper organised is as follows. In Section 2, thedetails of the FRFT are explained; Section 3 provides the details ofthe proposed pansharpening scheme; Section 4 describes thesimulation results and comparative analysis of the proposedscheme with existing schemes. Conclusions are drawn in Section 5.

2 Review of FRFTThe FRFT is a generalisation of the conventional Fourier transform[36] and has been applied in different applications [25–27]. TheFRFT is a linear operator which provides a representation of thesignal along the axis making an angle α with the time axis [37] andit reduces to conventional Fourier transform for an angle parameterequal to π /2.

The algorithms for 2D-DFRFT have also been proposed [36]and have found similar results as of 2D continuous FRFT. The 2D-DFRFT transform of input image h(i, j) of size U × V is computedas [36]

Hα, β(m, n) = ℱα, β[h(i, j)] = ∑i = 0

U − 1∑j = 0

V − 1h(i, j)ℛα, β(i, j, m, n) (1)

where ℛα, β(i, j, m, n) is the 2D transform kernel defined as [38, 39]

ℛα, β = ℛα ⊗ ℛβ, (2)

where ℛα, ℛβ are the 1D-DFRFT transformation matrix andsymbol ⊗ denotes Kronecker product of matrices given in [38, 39].The angle parameters α and β can be varied for obtaining bettersignal representation.

For α = β = π /2, the 2D-DFRFT reduces to the conventional2D-discrete Fourier transform while for α = β = 0, the 2D-DFRFTprovides the original signal itself.

The inverse 2D-DFRFT (2D-IDFRFT) is computed as [36]

h(i, j) = ∑m = 0

U − 1∑n = 0

V − 1Hα, β(m, n)ℛ−α, − β(i, j, m, n), (3)

3 Proposed pansharpening schemeThe images captured from different sensors in remote sensingapplications may suffer from spatial and spectral distortionproblems [40]. Some pansharpening schemes improve the spatialquality and reduce the spectral distortion in the pansharpenedimage obtained from the MS and PAN images by performingfiltering in the time domain or joint time–frequency domains [35,41]. These filtering-based pansharpening schemes, however,assume that the image is band-limited in conventional Fourierdomains. However, the finite size of the images acquired precludesthis assumption to hold true. In addition, the PAN and MS imagescollected through different sensors satellite environment sufferfrom different types of noise which may not be stationary in nature.Therefore, conventional Fourier transform is not suitable forhandling such non-stationary signals and noise [20, 21]. The FRFTis known to handle such non-stationary noise in a more effectiveway than the conventional Fourier transform [22]. The angleparameter of 2D-DFRFT can be varied to provide infiniterepresentations of the given signal in different DFRFT domainseach corresponding to the different value of the angle parameter.The DFRFT provides a free degree of freedom in terms of its angleparameters. Therefore, it would be interesting to extend thefiltering-based pansharpening approach to intermediate domainscalled FRFT domains.

In this section, we present a generalisation of FFT-basedpansharpening scheme proposed in [19]. This method is based on2D-DFRFT filtering of both the PAN image and intensity imagecomponent of the original MS images.

The block diagram of the proposed method is given in Fig. 1. Inthis method, low-resolution MS input images are up-sampled to thesize of PAN image by using interpolation scheme describe in [42].These up-sampled MS images are converted into the intensityimage (I) using averaging operation. The intensity image (I) istransformed using 2D-DFRFT defined in (1) as given below

Iα1, β1(m, n) = ∑i = 0

U − 1∑j = 0

V − 1I(i, j)ℛα1, β1(i, j, m, n), (4)

where Iα1, β1 is the 2D-DFRFT domain representation of image Icorresponding to angles α1 = a1π /2 and β1 = b1π /2.

The image Iα1, β1 given in (4) is filtered by the highpass filter andthen filtered image I^α1, β1 is transformed using the 2D-IDFRFTgiven in (3) to obtain

Iα1, β1′ (i, j) = ∑m = 0

U − 1∑n = 0

V − 1I^α1, β1(m, n)ℛ−α1, − β1(m, n, i, j) . (5)

where image Iα1, β1′ is the 2D-IDFRFT domain representation offiltered image I^α1, β1.

The high-resolution PAN image P is transformed using the 2D-DFRFT given in (1) to obtain

Pα1, β1(m, n) = ∑i = 0

U − 1∑j = 0

V − 1P(i, j)ℛα1, β1(i, j, m, n), (6)

where Pα1, β1 is the 2D-DFRFT domain representation of image Pcorresponding to angles α1 and β1.

The image Pα1, β1 given in (6) is filtered by highpass filter andthen filtered image P^

α1, β1 is transformed using the 2D-IDFRFTdefined in (3) as given below

Pα1, β1′ (i, j) = ∑m = 0

U − 1∑n = 0

V − 1P^

α1, β1(m, n)ℛ−α1, − β1(m, n, i, j), (7)

Fig. 1 Block diagram of the proposed panshapening method

1014 IET Image Process., 2018, Vol. 12 Iss. 6, pp. 1013-1019© The Institution of Engineering and Technology 2018

where image Pα1, β1′ is the 2D-IDFRFT domain representation offiltered image P^

α1, β1.The high-frequency filtered Pα1, β1′ image given in (7) and Iα1, β1′

image given in (5) are added for obtaining the new image Iα1, β1′′ asgiven by

I′α1, β1′ (i, j) = Pα1, β1′ (i, j) + Iα1, β1′ (i, j), (8)

In (8), image Iα1, β1′′ has spatial information of both the PAN and MSimages.

Using image Iα1, β1′′ , we propose the following pansharpeningrule as

MS(α1, β1)r = MSr + βGrIα1, β1′′ , r = 1, 2, …, N, (9)

where MS(α1, β1)r is the high spatial and spectral pansharpenedimage, MSr is the rth interpolated MS images in the given totalnumber of N band images, β is the tuning factor obtained by thesimulation trials and Gr are the injection coefficients obtained fromthe regression between each MS band image and image Iα1, β1′′ [40].The injection coefficients are calculated using [43]

Gr =cov(MSr, I′′α1, β1)

var(I′′α1, β1), r = 1, …, N, (10)

where var(Iα1, β1′′ ) is the variance of Iα1, β1′′ image and cov(MSr, Iα1, β1′′ )indicates the covariance between two images MSr and Iα1, β1′′ . Algorithm 1: 2D-DFRFT-based pansharpening algorithm

1: Obtain up-sampled MS images of the size of PAN image;2: Obtain the IHS images using up-sampled MS images by RGB toIHS conversion;3: Compute the image Iα1, β1 using 2D-DFRFT of I imagecorresponding to angles α1 and β1;4: Compute the image I^α1, β1 using lowpass filtering;

5: Obtain the image Iα1, β1′ using 2D-IDFRFT of I^α1, β1 image;

6: Obtain histogram matched image P using input PAN and Iimage;7: Compute the image Pα1, β1 using 2D-DFRFT of P imagecorresponding to angles α1 and β1;8: Compute the image P^

α1, β1 using highpass filtering;

9: Obtain the image Pα1, β1′ using 2D-IDFRFT of P^α1, β1 image;

10: Obtain the Iα1, β1′′ by adding Iα1, β1′ and Pα1, β1′ images;11: Calculate the injection coefficients Gr;12: Obtain the high spatial and spectral pansharpened imageMS(α1, β1)r using pansharpening rule

MS(α1, β1)r = MSr + βGrI′α1, β1′ , r = 1, 2, …, N,

where β is the tuning factor.

4 Simulation resultsTo test the proposed 2D-DFRFT-based pansharpening scheme,datasets collected by the IKONOS and GeoEye-1 satellites areused. The site location selected for the IKONOS satellite is China-Sichuan (58208_0000000.20001108) collected on May 2008 andfor GeoEye-1 satellite is Hobart, Tasmania, Australia collected onFebruary 2009. The spatial resolutions of MS and PAN images forIKONOS are 4 and 1 m while for GeoEye-1 are 2 and 0.5 m,respectively. The 20 image pairs of each satellite datasets are usedfor the simulation purpose in which size of each image pairs (MSand PAN images) are 320 × 320 and 1280 × 1280, respectively.

For obtaining the pansharpening results, we follow the Wald'sprotocol [44] in which the PAN and MS images are degraded to thelower resolution to compare the pansharpened image with thereference original MS images [40]. The quality metrics forevaluating the quality of the pansharpened images considered inthis paper are Q-index (Q4) [45], spectral angle mapper (SAM)[46], relative dimensionless global error (ERGAS) [44] and qualitywith no-reference (QNR). The QNR is composed of a spectral (Dλ)and spatial (DS) distortion indices, without requiring a high-resolution reference MS images [47].

In the block diagram of the proposed scheme shown in Fig. 1,the highpass filters are chosen as first-order Butterworth filterswith cutoff radius 40 and 350 at degraded and full scale,respectively, and the values of the tuning factor β are 2 and 4 forIKONOS and GeoEye-1 satellite images, respectively. Theselection of cutoff radius and tuning factor β has been done usingthe simulation trials. The input PAN and MS images for IKONOSand GeoEye-1 satellite dataset are shown in Figs. 2a and b andFigs. 2c and d, respectively. The value of N is four in the MSimages; it combines red, blue, green and infrared componentimages.

Simulation results of the spatial information image I′α1, β1′ givenin (8) in the proposed scheme using different values of the angleparameters (α1, β1) of the 2D-DFRFT for IKONOS satellite datasetare shown in Figs. 3a–f and the pansharpened images usingdifferent values of the angle parameters (α1, β1) of the 2D-DFRFTare shown in Figs. 4a–f. It is seen from Figs. 3c and 4c for thevalues of the angle parameters α1 = β1 = 0.98π /2 of the 2D-DFRFT are provided the images with maximum spatialinformation as compared to the other images and Figs. 3d and 4dfor the angle parameters α1 = β1 = π /2 of the 2D-DFRFT conditionof the conventional Fourier transform are provided the images withsome spatial distortion.

The quality assessment of the pansharpened images of theproposed scheme on the 20 image pairs of the IKONOS satellitedatasets using different values of the angle parameters (α1, β1) ofthe 2D-DFRFT is tabulated in Table 1 in terms of Q4, SAM andERGAS with mean and standard deviation (SD). It is observedfrom Table 1 that the values of the angle parametersα1 = β1 = 0.98π /2 of the 2D-DFRFT quality metrics Q4, SAM andERGAS provided the best results as compared to the other valuesof the angle parameters.

Fig. 2 Input dataset images(a) PAN image for IKONOS satellite, (b) MS images for IKONOS satellite, (c) PANimage for GeoEye-1 satellite, (d) MS images for GeoEye-1 satellite 2


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Simulation results are also performed to compare thepansharpened images in terms of mean and SD of variousperformance parameters of IKONOS and GeoEye-1 satellitedataset images using the proposed pansharpening scheme for thevalues of angle parameters α1 = β1 = 0.98π /2 of the 2D-DFRFTwith existing pansharpening schemes based on HVD (HVD_F) [9],OMF [32], AIHS [6], Indusion [34] and ATWTM2 [35].

The pansharpened images obtained using the proposed methodfor value of angle parameters α1 = β1 = 0.98π /2 of the 2D-DFRFT

and existing methods are shown in Figs. 5a–f and 6a–f. It can beobserved from the middle portion of the image shown in Fig. 5athat this fused image reveals additional spatial information than thefused images obtained by the existing methods. Similarly, spectralquality of the fused image shown in Fig. 5a is better than thesimulation results obtained by the other existing methods. Similarremarks hold good for the other set of images shown in Fig. 6,particularly the left corner portion of it. Using the abovepansharpened images shown in Figs. 5a and 6a, the quality metrics

Fig. 3 Spatial information image I′α1, β1′ for IKONOS satellite dataset at different values of angle parameters of the 2D-DFRFT(a) α1 = β1 = 0.94π /2, (b) α1 = β1 = 0.96π /2, (c) α1 = β1 = 0.98π /2, (d) α1 = β1 = 1π /2, (e) α1 = β1 = 1.02π /2, (f) α1 = β1 = 1.04π /2

Fig. 4 Pansharpend images of the proposed scheme for IKONOS satellite dataset at different values of angle parameters of the 2D-DFRFT(a) α1 = β1 = 0.94π /2, (b) α1 = β1 = 0.96π /2, (c) α1 = β1 = 0.98π /2, (d) α1 = β1 = 1π /2, (e) α1 = β1 = 1.02π /2, (f) α1 = β1 = 1.04π /2


Q, SAM, ERGAS for degraded scale assessment and Dλ, DS, QNRfor full-scale assessment are computed and the results are tabulatedin Tables 2 and 3. The best values of the performance measures arehighlighted as boldface numerals. It can be seen from thequantitative results given in Table 2 for IKONOS dataset that theproposed method outperforms the other methods in terms of SAM,DS and QNR quality metrics while in Q4, ERGAS and Dλ arecomparable in values obtained for other existing methods.Similarly from the results given in Table 3 for the GeoEye-1 imagedataset, it can be observed that the proposed method outperformsthe other methods in terms of SAM, ERGAS, DS and QNR qualitymetrics while the Q4 and Dλ are comparable in values obtained forother existing methods.

The pansharpening results are obtained using the proposedmethod for the values of angle parameters (α1, β1) of the 2D-DFRFT for IKONOS and GeoEye-1 satellite datasets and it isobserved that pansharpened images of the proposed method for thevalues (0.98π /2, 0.98π /2) provided best quality and have therobustness against the effect of aliasing and misregistration errorsas compared to the existing schemes.

5 Conclusion

In this paper, a 2D-DFRFT-based pansharpening scheme isproposed. The additional degree of freedom in terms of its angleparameters associated with the 2D-DFRFT is exploited forobtaining better results in the proposed pansharpening scheme. Thequalitative and quantitative analyses of the presented simulationresults show that the proposed technique provides improvedspectral and spatial quality fused image as compared to some of theexisting pansharpening techniques for the IKONOS and GeoEye-1satellite datasets.

Table 1 Quality assessment of the pansharpened images of the proposed scheme using different values of the angleparameters (α1, β1) of the 2D-DFRFT for IKONOS satellite dataset

Degraded scale(α1, β1) Q4 SAM ERGAS

Mean SD Mean SD Mean SDRef.Val. 1 0 0(0.94π /2, 0.94π /2) 0.429 0.0951 5.091 1.562 6.697 3.182(0.96π /2, 0.96π /2) 0.536 0.095 3.654 0.865 3.693 1.388(0.98π /2, 0.98π /2) 0.648 0.095 2.881 0.435 2.363 0.507(π /2, π /2) 0.574 0.129 3.267 0.623 2.979 0.756(1.02π /2, 1.02π /2) 0.633 0.099 2.971 0.514 2.491 0.630(1.04π /2, 1.04π /2) 0.509 0.112 3.982 1.076 4.285 1.870

Fig. 5 Pansharpened images using different pansharpening methods for IKONOS satellite dataset(a) Proposed MS(α1, β1)r method, (b) HVD_F method, (c) OMF method, (d) AIHS method, (e) Indusion method, (f) ATWTM2 method


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Table 2 Spectral quality assessment of the pansharpened images for IKONOS dataset at degraded and full-scale optimised2D-DFRFT angle parameters α1 = 0.98π /2 and β1 = 0.98π /2

Degraded scale Full scaleQ4 SAM ERGAS Dλ DS QNR

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Table 3 Spectral quality assessment of the pansharpened images for GeoEye-1 dataset at degraded and full-scale optimised2D-DFRFT angle parameters α1 = 0.98π /2 and β1 = 0.98π /2

Degraded scale Full scaleQ4 SAM ERGAS Dλ DS QNR

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transform in this paper, we extend the pansharpening...

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