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High-resolution satellite image fusionusing regression krigingQingmin Meng a , Bruce Borders a & Marguerite Madden ba Warnell School of Forestry and Natural Resources , University ofGeorgia , Athens, GA, 30602, USAb Department of Geography , University of Georgia , Athens, GA,30602, USAPublished online: 28 Apr 2010.

To cite this article: Qingmin Meng , Bruce Borders & Marguerite Madden (2010) High-resolutionsatellite image fusion using regression kriging, International Journal of Remote Sensing, 31:7,1857-1876, DOI: 10.1080/01431160902927937

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High-resolution satellite image fusion using regression kriging

QINGMIN MENG*†, BRUCE BORDERS† and MARGUERITE MADDEN‡

†Warnell School of Forestry and Natural Resources, University of Georgia,

Athens, GA 30602, USA

‡Department of Geography, University of Georgia, Athens, GA 30602, USA

(Received 26 September 2007; in final form 21 January 2009)

Image fusion is an important component of digital image processing and quanti-

tative image analysis. Image fusion is the technique of integrating and merging

information from different remote sensors to achieve refined or improved data. A

number of fusion algorithms have been developed in the past two decades, and

most of these methods are efficient for applications especially for same-sensor and

single-date images. However, colour distortion is a common problem for multi-

sensor or multi-date image fusion. In this study, a new image fusion method of

regression kriging is presented. Regression kriging takes consideration of correla-

tion between response variable (i.e., the image to be fused) and predictor variables

(i.e., the image with finer spatial resolutions), spatial autocorrelation among pixels

in the predictor images, and the unbiased estimation with minimized variance.

Regression kriging is applied to fuse multi-temporal (e.g., Ikonos, QuickBird, and

OrbView-3) images. The significant properties of image fusion using regression

kriging are spectral preservation and relatively simple procedures. The qualitative

assessments indicate that there is no apparent colour distortion in the fused images

that coincides with the quantitative checks, which show that the fused images are

highly correlated with the initial data and the per-pixel differences are too small to

be considered as significant errors. Besides a basic comparison of image fusion

between a wavelet based approach and regression kriging, general comparisons

with other published fusion algorithms indicate that regression kriging is compar-

able with other sophisticated techniques for multi-sensor and multi-date image

fusion.

1. Introduction

Remotely sensed data recorded by most Earth observation satellite systems – for

example, Ikonos, QuickBird, SPOT, and Landsat TM – are panchromatic imagery

with higher spatial resolution but lower spectral resolution and multi-spectral imagerywith lower spatial resolution but higher spectral resolution. Image analysis often

require high spatial and high spectral information simultaneously in a single image,

so that image fusion integrating and merging information from different remote

sensors to achieve refined or improved data is an important step in digital image

processing for the data collected from these multiple satellite sensors.

*Corresponding author, currently at the Center for Applied GIScience, Department of

Geography and Earth Sciences, University of North Carolina – Charlotte, 9201 University

City Blvd. Charlotte, NC 28223, USA. Email: qmeng1@uncc.edu

International Journal of Remote SensingISSN 0143-1161 print/ISSN 1366-5901 online # 2010 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/01431160902927937

International Journal of Remote Sensing

Vol. 31, No. 7, 10 April 2010, 1857–1876

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Image fusion is important for digital image processing and makes it possible to

obtain both high spatial and spectral resolution. For example, the achievement of

both high spatial and spectral resolutions is significant to characterize the Earth

objects in image analysis, mapping, and decision making for environmental and

resource management. One important aim of image fusion is to merge multiple imagesobtained through different remote sensors to achieve more information than can be

extracted from only a single sensor (i.e. ‘1 þ 1 ¼ 3’) (Pohl and Van Genderen 1998).

Many methods including the IHS (Intensity, Hue, Saturation), the PCS (Principal

Component Substitution) methods (e.g., Cliche et al. 1985, Price 1987, Welch and

Ehlers 1987, Chavez et al. 1991, Ehlers 1991, Shettigara 1992, Yesou et al. 1993, Zhou

et al. 1998, Zhang 1999, Li et al. 2002, Chen et al. 2003, Zhang and Hong 2005), Gram

Schmidt fusion, modified IHS fusion, and CN (i.e. colour normalized) spectral

sharpening (Klonus and Ehlers 2007) have been explored for image fusion. Amongthese methods, the IHS transform and PCS methods are the most commonly used

algorithms in digital image processing (Zhang 1999, Tu et al. 2001). The IHS and PCS

are effective for fusing Radar or SPOT panchromatic images with Landsat imagery

and other multiple images (Ling et al. 2007); however, the spectral characteristics of

the original multi-spectral images are distorted to different extents using the most

commonly available methods (Chavez et al. 1991, Shettigara 1992, Wald et al. 1997,

Klonus and Ehlers 2007, Ling et al. 2007). Wavelet transformation can be an effective

image fusion method, but it is complex and computation consuming, which is neces-sary for maintaining higher numerical precision through the whole fusion process to

finally obtain a correct result (Carr 2004).

Given different spatial and spectral resolution images of the same area produced by

remote sensors, the main objective is to fuse them to produce a single image with as

much of the information from the original images preserved as possible. It is evident

that the spectral and spatial effects of the fused images are two of the most important

criteria for assessing fusion methods. However, one basic problem in the available

fusion techniques reported by many studies is that the fused image typically hasnotable deviations in visualization and in spectral values from the original image

(Chavez et al. 1991, Pellemans et al. 1993, Van Der Meer 1997, Wald et al. 1997,

Zhang 2002). These deviations are often colour distortions that affect further image

interpretation.

Therefore, this paper aims to develop a methodological framework for image

fusion based on the theory and techniques of regression kriging (RK). This frame-

work can then be applied for image fusion with available remote sensing data. The

main objective of image fusion using regression kriging in this study is to fuse high-resolution images with lower-resolution multi-spectral data to achieve high-

resolution multi-spectral images while maintaining the spectral characteristics of the

multi-spectral data in the fused imagery. Regression kriging with the results of

unbiased estimation with minimum variance takes into consideration correlation

between response and predictor variables and spatial autocorrelation among predic-

tor variables. Regression kriging fusion approaches in this study include: (1) multi-

date (same sensor or multi-senor) band-to-panchromatic fusion – for example, Ikonos

image fusion using either Ikonos or Quickbird panchromatic band as predictor; (2)multi-date and multi-sensor band-to-band fusion – for instance, Ikonos band fusion

using one Quickbird band as predictor; and (3) multi-date and multi-sensor band-to-

multibands fusion, such as Ikonos band fusion using Quickbird multibands as pre-

dictor. Finally, fused images are evaluated visually, spectrally, and spatially, and a

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general comparison of the results between RK fusion and recently published research

of image fusion is discussed in terms of spectral conservation, computation efficiency,

and reproducibility.

2. Methodology

2.1 Regression kriging for image fusion

Odeh et al. (1994, 1995) originally suggested using the term regression kriging (RK) to

employ correlation with auxiliary variables and spatial correlation, while kriging with

external drift (KED) is more often used when a spatial trend (i.e. a nonstationary

mean) is caused and described through some auxiliary variables (Wackernagel 1998,

Chiles and Delfiner 1999). Goovaerts (1999) used the term kriging after detrending todefine this kriging process. The advantages of RK are that it does not suffer from

instability in practice (Goovaerts 1997) and RK can be easily integrated with statis-

tical computations such as general additive modelling or regression trees (McBratney

et al. 2000). The RK terminology is used in this research since it directly indicates that

regression is combined with kriging.

In an image fusion process, let the pixel values of a given band to be fused be indicated

as z(s1), z(s2), z(s3), . . ., z(sn), where si ¼ (xlat, ylong) is a location i with the coordinates of

xlat and ylong, respectively, for latitude and longitude, and pixels i ¼ 1, 2, 3,. . ., n. Thepixel values at a new and unrecorded location (s0) can be predicted using RK by adding

the predicted trend and residuals with the equation defined by Odeh et al. (1994).

zðs0Þ ¼ mðs0Þ þ eðs0Þ (1)

where the residuals e are interpolated using ordinary kriging, and the trend is fitted

using linear regression as follows:

zðs0Þ ¼Xp

k¼0

bk � qkðs0Þ þXn

i

wiðs0Þ � eðsiÞ (2)

where bk is the kth estimated regression coefficient, qk is the kth external auxiliary

variable or predictor at location s0 (e.g., the image bands with higher spatial resolu-

tion), and q0ðs0Þ ¼ 1, p is the number of auxiliary variables, wiðs0Þ are the weights

determined by the covariance function and eðsiÞ are the regression residuals. Rewrite

the RK model of equation (1) in matrix notation using the following equations:

z ¼ qT � bþ e (3)

zðs0Þ ¼ qT0 � bþ lT

0 � e (4)

where e is regression residuals, q0 is the vector of p auxiliary variables at s0, b is thevector of p þ 1 estimated model coefficients, l0 is the vector of n kriging weights and e

is the vector of n residuals. To take into account the spatial correlation of residuals,

the regression model coefficients are solved by the following generalized least squares

estimation (Cressie 1993).

b ¼ ðqT � C�1 � qÞ�1 � qT � C�1 � z (5)

where q is the matrix of auxiliary variables at all the observed locations, z is the vector of

sampled observations, and C is the n � n covariance matrix of residuals as follows:

Satellite image fusion using regression kriging 1859

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C ¼Cðs1; s2Þ � � � Cðs1; snÞ

..

. . ..

..

.

Cðsn; s2Þ � � � Cðsn; snÞ

264

375 (6)

The covariance matrix between pixel pairs Cðsi; sjÞ is estimated through semivario-

gram modelling C(h). An example using a typical Spherical semivariogram model is

below:

�ðhÞ ¼ C032

ha0� 1

2ha0

� �3� �

; for h � a0

C0; for h � a0

8<: (7)

where �ðhÞ is the function of Spherical semivariogram that has the relationship of �ðhÞ¼C0 – C(h) with the above covariance function. C0, a0, h are, respectively, the

semivariogram parameters of sill, range and lag distance. To summarize the mathe-

matical steps in regression kriging, we can write RK in matrix notation in equation (8)

as Christensen (1990) with all the variables defined the same as in the above equations.

zðs0Þ ¼ qT0 � bþ lT

0 � ðz� q � bÞ (8)

The regression kriging for image fusion can be summarized using a flow chart

(figure 1) once the mathematic conceptions of regression kriging are understood.

Figure 1 indicates that three main steps are included in the image fusion process.

First, image data preprocessing includes georegistration, co-registration, and ASCII

file transformation of both lower spatial resolution images (i.e., the response vari-

ables) and higher spatial resolution images (i.e., the predictor variables). Second,regression kriging of image fusion includes computation of empirical semivariogram,

modelling of theoretical semivariogram, and regression kriging for image fusion. The

Spherical semivariogram model is often tried to fit first since it meets the spatial

characteristics of many geographic phenomena and other mathematical semivario-

gram equations need to be fitted for selecting the best theoretical model. The last step

includes image transformation of ASCII files obtained through regression kriging,

georegistration of the fused images and evaluation of the results.

Low spatial resolution images High spatial resolution images

Georegistration and co-registration

ASCII file transformation

Response Predictor variablesEmpirical Semivariogram

Theoretical Semivariogram

Fused high spatial resolution images Georegistration and evaluation

Figure 1. The flow chart of image fusion using regression kriging.

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2.2 Evaluation methods

Visual analyses are usually conducted to compare the quality of fused images with the

original image. While the visual analysis is subjective, several statistical methods are

performed to objectively quantify the colour preservation: (1) basic statistics includ-

ing mean, median, range, minimum, maximum, mode, standard deviation (SD),

Geary’s kurtosis, and histogram are used to describe the distribution of the fused

images and the original image and to compare their differences; (2) root mean square

error (RMSE) is used to compare the difference between the original band and the

fused band, while mean absolute error (MAE) and relative MAE (RMAE) are used tocompare the differences at per-pixel level; (3) Pearson correlation coefficient and

Kolmogorov–Smirnov (KS) test are applied to further check the similarity of the

distributions between the original and fused images; (4) after the original image and

fused images are processed using two morphological functions of dilate and erode,

spatial assessment then can be conducted using Pearson’s correlation index.

The basic statistics, SD, Geary’s kurtosis (G(k)), RMSE, MAE, RMAE, and

Pearson correlation coefficient (r) are calculated using equations (9), (10), (11), (12),

(13) and (14), respectively.

SD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

ðzi � �zÞ2

n� 1

vuuut(9)

GðkÞ ¼

1

n

Xn

i¼1

zi � �zj j2

SD2(10)

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�zoriginal � �zfusedÞ2 þ ðSDoriginal � SDfusedÞ2

q(11)

MAE ¼ 1

n

Xn

i¼1

zoriginali � z

fusedi

��� ��� (12)

Rmae ¼MAE=�zoriginal (13)

rzoriginsl zfused¼Pðzoriginal

i � �zoriginalÞðzfusedi � �zfusedÞ

ðn� 1Þ � SDoriginal � SDfused

(14)

The KS test is used to determine whether the fused image and the initial image come

from the same population. The KS test, being non-parametric and distribution-free,

has the advantage of making no assumption about the distribution of data. To

perform a KS test, the two experimental cumulative distributions (W(zoriginal) and

W(zfused)) that both contain n pixels are computed for the two data sets of interest(e.g., zoriginal and zfused). The KS test uses the maximum vertical deviation between the

two curves as the statistic D shows:

D ¼ max jWðzoriginali Þ �Wðzfused

i Þji ¼ 1; 2; . . . ; n (15)

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3. Examples

3.1 Example 1: fusion of Ikonos and QuickBird images

A coastal land with area of 0.15 km2 located in Camp Lejeune (34.57� N latitude,

77.28� W longitude, Figure 2(a)), southeastern North Carolina, is used first as the testarea, in a coastal plain with relative flat terrain (Onslow County 2005). Different

landscapes including hardwoods, mixed softwoods, grasslands, and vegetated wet-

lands cover this region.

Ikonos images and QuickBird images (table 1) covering this area were used for

image fusion. We selected their panchromatic band and three multi-spectral bands

(bands 2, 3, and 4) (table 1) to conduct the image fusion process. These images were

spatially registered to the Universal Transverse Mercator (UTM) coordinate system

on the WGS 84 datum and then co-registered with each other (figure 3).

3.2 Example 2: fusion of QuickBird and OrbView-3 images

QuickBird images cover 1.104 km2 of Chilika Lake region, India (85.35� E latitude

and 19.65� N longitude, figure 2(b)) and then are used to further check the efficiency

of image fusion through regression kriging. These QuickBird images (DigitalGlobe

2004) with four bands and 2.79 m pixel size and the OrbView-3 panchromatic image

(OrbImage 2005) with 0.45 m pixel size covering the same area were downloaded fromthe Global Land Cover Facility at the University of Maryland and they are georegis-

tered to UTM coordinate system on the GRS 83 datum and then co-registered. The

OrbView-3 image then is resampled to 1-m resolution for image fusion.

3.3 Example 3: a wavelet-based fusion approach

Wavelet transformation has been proposed for image data fusion and compared with

other techniques, which indicated that a wavelet-based approach usually improves thespatial resolution with relatively little distortion of spectral values of the original

image (Zhou et al. 1998, Ranchin and Wald 2000). In this example, we use the Wavelet

Resolution Merge tool (ERDAS 2005, p. 169) to merge the Ikonos images with the

QuickBird panchromatic band (i.e., the image data in example 1) and to fuse the

QuickBird images with the OrbView-3 panchromatic image (figure 6). The results are

compared with that from RK in terms of the diagnostics tests of RMSE, bias, MAE,

RMAE, kurtosis, correlation coefficient, and KS test.

Table 1. Ikonos and QuickBird images.

Image data Spectral bandwidth(mm) Spatial resolution(m) Acquisition date

Ikonos pan 0.45–0.90 1 5 February 2000Ikonos XS Band 2: 0.51–0.60 4 27 August 2001(Band 2, 3, 4) Band 3: 0.63–0.70

Band 4: 0.51–0.60QuickBird Pan 0.45–0.90 0.61 3 March 2003QuickBird XS Band 2: 0.52–0.60 2.44 24 March 2003(Band 2, 3, 4) Band 3: 0.63–0.69

Band 4: 0.76–0.85

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4. Results

4.1 Fusion of Ikonos and QuickBird images

The fused images are compared with the original images visually, spectrally, and

spatially to make a comprehensive evaluation of the new image fusion technique. The

fused images and original Ikonos and QuickBird images are portrayed in figure 3.Visually compared with the original Ikonos multi-spectral image, the fused Ikonos

image using Ikonos panchromatic image as predictor (FIIKP), fused Ikonos image

using QuickBird panchromatic image as predictor (FIQBP), fused Ikonos image

using QuickBird 3-band as predictor (FIQB3x), and fused Ikonos image using

QuickBird 1-band as predictor (FIQB1x) do not have any apparent colour distortion.

Compared with QuickBird multi-spectral images, the fused QuickBird image using

QuickBird panchromatic image (FQBP) as predictor does not have any apparent

colour distortion either. The visual comparison indicated that high spectral andspatial information is inherited in all the fused images from the original images

through regression kriging. The resulting image resampled to its original resolution

should be close to the original image (Wald et al. 1997). The histogram of the original

bands and the fused bands are depicted using figures 4 and 5. For Ikonos band 2, 3,

Figure 2. Study area of the examples. (a) is located in North Carolina, US; (b) is in the area ofChilika Lake, India.

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and 4, the fused images using Ikonos pan, QuickBird pan, QuickBird three bands orone band as predictor(s) all contain similar distributions to the original bands. The

comparison between the histograms of the fused images and those of the original

multi-spectral images indicates that the fused images almost maintain the same

spectral distribution as the initial ones.

To indicate detailed characteristics of the distributions, basic statistics including

mean, median, range, minimum, maximum, mode, SD, Geary’s kurtosis of the fused

bands and the original bands are listed in tables 2 and 3, which also show that the

fused bands keep almost the same values in mean, median, mode, standard deviation,and Geary’s kurtosis as the original bands, although there are some differences in

maximum or minimum pixel values.

Figure 3. The fused images using regression kriging and the original Ikonos and QuickBirdimages. (a), Fused Ikonos image using Ikonos panchromatic image as predictor; (b), fusedIkonos image using QuickBird panchromatic image as predictor; (c), fused Ikonos image usingQuickBird 3-band as predictor; (d), fused Ikonos image using QuickBird 1-band as predictor;(e), fused QuickBird image using QuickBird panchromatic image as predictor; (f), Ikonosmultispectral image; (g), QuickBird multi-spectral image; (h), QuickBird panchromaticimage; (i), Ikonos panchromatic image.

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The Pearson correlation coefficient between the fused band and original band is

used to check spectral differences. The higher the correlation values, the better the

spectral consistency between the fused image and the original image. For the fused

Ikonos bands 2 and 3, the values of the correlation coefficients are between 0.95 and

0.98, while the correlation coefficients are from 0.91 to 0.94 for fused band 4 (table 4).

Fused Quickbird band 4 has a relatively low correlation coefficient 0.82 (table 4). TheKS test is applied to further check the distribution differences between the fused band

and the original band. Fused Ikonos images using panchromatic have significant

differences from the original images (i.e., p-values are smaller than 0.05), while the

fused Ikonos images using QuickBird one band, three bands, and panchromatic band

as predictors are not significantly different from the original images except for Ikonos

band 4 (table 5). The fused QuickBird images using its panchromatic band are not

Figure 4. Histograms of the original multi-spectral images and fused Ikonos images usingIkonos panchromatic image (FIIKP), QuickBird panchromatic image (FIQBP), QuickBird 3-band images (FIQB3x), and QuickBird 1-band image (FIQB1x). Column 1 is for band 2,column 2 is for band 3, and column 3 is for band 4.

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significantly different from its original images although fused band 4 has relatively

small p-value and does not have a large correlation coefficient (table 5).

The RMSE is used to check the difference between the original and fused bands.

The MAE and RMAE are used to calculate differences at per pixel level for fused

images and original images (table 6). For the fused Ikonos images, bands 2 and 3 have

smaller RMSE, MAE and RMAE, while band 4 has relatively large errors; the fusedimage using QuickBird three bands has relatively smaller errors compared with the

fusion approaches using one band or panchromatic band. The fused QuickBird image

also has relatively smaller errors.

Morphological processing is typically used to understand the structure of an image

and identify boundaries or objects within an image. To evaluate the spatial effects of

the fused images, morphological processes with a 7 � 7 neighbourhood including

erode and dilate are first applied to the fused and the original images. Here

Figure 5. Histograms of the original multi-spectral images (first row) and fused QuickBirdimages (second row) using QuickBird panchromatic image (FQQP). Column 1 is for band 2,column 2 is for band 3, and column 3 is for band 4.

Figure 6. The OrbView-3 panchromatic image (a) and the QuickBird images (b).

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morphological techniques are used to indicate the potential spatial structure of

spectral values within an image. Dilate is used as a maximum operator to select the

greatest values in the neighbourhood, while erode is used as a minimum operator to

select the smallest values in the neighbourhood. Then based on the processed original

band and the fused band, Pearson correlation coefficients are calculated (tables 7 and

8). The higher the correlation, the more similar are the spatial structures between thebands. Tables 7 and 8 show that the fused images are highly correlated with the initial

images after dilate processing. The correlation coefficients are also large after the

fused and original images are processed using the erode function.

Table 3. Basic statistics of QuickBird original image and fused images.

Minimum Maximum Range Mean Median Mode SD* Kurtosis

Original QuickBirdimages

Band 2 127 531 404 205 197 166 51 0.62Band 3 55 432 377 135 130 152 57 0.71Band 4 52 680 628 288 282 271 75 0.75

Fused QuickBirdimages usingits panchromaticband

Band 2 1 819 818 205 196 212 50 0.63Band 3 1 838 837 135 128 82 57 0.71Band 4 2 999 997 288 281 281 68 0.76

*Standard deviation.

Table 4. Pearson correlation coefficients for the fused Ikonos images.

FIIKP(1) FIQB3x(2) FIQBP(3) FIQB1x(4) FQQP(5)

Band 2 0.9567 0.9702 0.9696 0.9709 0.9622Band 3 0.9639 0.9727 0.9722 0.9741 0.9703Band 4 0.9128 0.9143 0.9383 0.9254 0.8181

(1)Fused Ikonos images using the Ikonos panchromatic image.(2)Fused Ikonos images using the QuickBird three bands.(3)Fused Ikonos images using the QuickBird panchromatic image.(4)Fused Ikonos images using the QuickBird one band.(5)Fused QuickBird images using QuickBird panchromatic.

Table 5. Kolmogorov–Smirnov (KS) test for the fused Ikonos images.

FIIKP(1) FIQB3x(2) FIQBP(3) FIQB1x(4) FQQP(5)

Band 2 0.0431a 0.016 0.0141 0.0301 0.0227(0.0176)b (0.8972) (0.9616) (0.0963) (0.4939)

Band 3 0.0472 0.0172 0.0196 0.0235 0.0216(0.0065) (0.8434) (0.7121) (0.4816) (0.5595)

Band 4 0.0728 0.0587 0.0556 0.0705 0.0349(0.0001) (0.0002) (0.0008) (0.0001) (0.07499)

aThe maximum vertical deviation D in KS test.bThe p-value of KS test.Other notes as table 4.

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4.2 Fusion of QuickBird and OrbView-3 images

The QuickBird images covering a part of Chilika Lake area, India then are fused using

1-m resolution OrbView-3 panchromatic images (figure 6) to check the robust of

regression kriging for image fusion. The fused images maintain the basic spectral and

spatial characteristics of the initial QuickBird data and no apparent difference exists

(figure 7). The basic statistics including the very similar values of mean, standard

deviation, minimum values, and maximum values, the very large values of Pearson

correlation coefficient, the smaller Kolmogorov–Smirnov D values, and very large

Table 7. Pearson correlation coefficient for fused Ikonos images after morphologicalprocessing.

After dilate processing After erode processing

FIIKP(1) FIQB3x(2) FIQBP(3) FIQB1x(4) FIIKP(1) FIQB3x(2) FIQBP(3) FIQB1x(4)

Band 2 0.9926 0.9963 0.9966 0.9964 0.9557 0.9637 0.913 0.9662

Band 3 0.9913 0.9958 0.9962 0.9959 0.9616 0.969 0.9558 0.9718

Band 4 0.9906 0.995 0.9956 0.9951 0.9217 0.9214 0.8478 0.9258

Notes as table 4.

Table 8. Pearson correlation coefficient for fused QuickBird images after morphologicalprocessing.

Band 2 Band 3 Band 4

Dilate operation 0.9716 0.9747 0.8198Erode operation 0.9708 0.9620 0.8689

Table 6. Errors of the fused Ikonos image.

Band 2 Band 3 Band 4

Fused Ikonos image using Ikonos panchromatic RMSEa 3.99 4.49 9.95MAEb 9.83 12.24 24.12RMAEc 0.04 0.06 0.07

Fused Ikonos image using QuickBird 3 bands RMSE 0.89 1.02 7.28MAE 8.56 10.83 20.35RMAE 0.03 0.05 0.06

Fused Ikonos image using QuickBird panchromatic RMSE 2.66 2.85 7.34MAE 8.17 10.45 23.52RMAE 0.03 0.05 0.07

Fused Ikonos image using QuickBird 1band RMSE 3.26 3.26 9.61MAE 8.08 10.09 22.44RMAE 0.03 0.05 0.07

Fused QuickBird using QuickBird Panchromatic RMSE 0.43 0.57 6.05MAE 8.18 8.42 29.27RMAE 0.04 0.06 0.1

aRoot mean square errors; bmean absolute error; crelative mean absolute error; other notes astable 2.

Satellite image fusion using regression kriging 1869

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Figure 7. Fused QuickBird images using OrbView-3 panchromatic band. Column I, theQuickBird images; Column II, the fused QuickBird images; Row 1, band 1; Row 2, band 2;Row 3, band 3; Row 4, band 4.

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Kolmogorov–Smirnov P values show that there is no significant difference of spectral

values between the fused QuickBird images and the initial images. The very small values

of the bias (i.e., initial values minus fused values), absolute mean error, relative absolute

mean error, and RMSE indicate that the per-pixel level differences between the fused

and the initial images are too small to be considered significant errors (table 9).

4.3 Results of a wavelet-based approach

Using the ERDAS Wavelet Resolution Merge tool, we first fused the Ikonos images

with the QuickBird panchromatic band and the same procedure is processed forQuickBird image fusion with the OrbView-3 panchromatic band (figure 8).

Compared with the original images, there are some colour distortions in the fused

Ikonos and QuickBird images. This coincides with the indications of the large errors

of bias, RMSE, MAE, RMAE and the smaller correlation coefficient and the much

smaller p values of the KS test (table 10).

Tables 9 and 10 indicate that the values of bias, RMSE, MAE, and RMAE from the

QuickBird fusion using this wavelet-based approach are much larger than those

calculated based on RK; the correlation coefficients between the fused image andthe original image are much smaller than those measured for RK fusion. The very

small p values (0–0.0003) of the KS test also indicate that the fused QuickBird images

are significantly different from the original images, while the much larger p values of

the KS test in table 9 show that there is no significant difference between the fused

images achieved by RK and the original images. With regard to the fused Ikonos

images, comparisons of table 10 with table 6, table 10 with table 5, and table 10 with

table 4 show likewise the same conclusion as above.

5. Conclusions

The multi-date panchromatic sharpening using either Ikonos or QuickBird images

results in satisfying images, while the individual band-based image fusion (i.e. Ikonos

band fusion using one QuickBird band as predictor) achieves relatively better results

than the process of panchromatic sharpening and the image fusion using three

QuickBird bands as predictor. The QuickBird image fusion with the OrbView-3

panchromatic image as predictor resulting in very small errors also shows that

Table 9. Diagnostic check of fused QuickBird bands using OrbView-3.

Mean SD1 Minimum Maximum Bias RMSE2 MAE3 RMAE4 Coor5 KS test6

Band 1 242.45 11.35 219 329 0.022Fused 1 242.48 10.67 187 339 0.03 4.5457 2.52 0.011 0.92 (0.969)*Band 2 327.26 27.36 271 539 0.026Fused 2 327.22 24.41 268 514 –0.05 8.8814 5 0.015 0.95 (0.888)Band 3 197.83 31.21 135 442 0.026Fused 3 197.79 28.11 121 428 –0.05 9.0227 4.8 0.024 0.96 (0.888)Band 4 161.11 77.43 79 569 0.036Fused 4 161.07 73.08 27 540 –0.05 15.5410 7.38 0.045 0.98 (0.536)

1Standard deviation.2Root mean square error.3Mean absolute error.4Relative mean absolute error.5Pearson correlation coefficient.6Kolmogorov–Smirnov test.*p-value of the KS test.

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regression kriging is a robust and effective image fusion method. The multi-sensor

temporal data fusion (i.e. Ikonos image with QuickBird pan and QuickBird image

with OrbView pan) using a wavelet-based method again indicates that RK results in

less colour distortion and change of pixel values.

Regression kriging is designed and applied as a new image fusion technique that is

applied to fuse multi-date single-sensor and different-sensor images. One advantage

of this technique is that regression kriging can be directly applied both for

panchromatic-based image fusion and different-sensor band-based image fusionwithout any additional image processing such as normalization and transformation.

An ideal band-based image fusion approach can be selected by diagnostic checking of

predictor combinations of different bands.

The qualitative and quantitative diagnostic checks indicate that regression kriging

has a significant advantage of colour preservation that is a critical property in image

fusion. The visual appearance of the fused images using regression kriging does not

show any apparent colour distortion, which coincides with the quantitative analysis.

Besides histograms, Pearson correlation coefficients, and RMSE, other statisticsincluding mean absolute error, relative mean absolute error, Kolmogorov–Smirnov

Figure 8. Image fusion using a Wavelet Resolution Merge tool (ERDAS 2005, p. 169). (1)Original QuickBird images, (2) fused QuickBird images with OrbView-3 panchromatic band,(3) original Ikonos images, (4) fused Ikonos images with QuickBird panchromatic band.

1872 Q. Meng et al.

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test, and morphological analysis are applied to further assess the difference between

the fused and the original images. The fused images are highly correlated with the

original images, although the Kolmogorov–Smirnov test still indicates that very small

significant differences exist even when the correlation coefficients are around 0.9. This

may indicate that the Kolmogorov–Smirnov test can be more sensitive to differencesbetween the original images and the fused data.

6. Discussion

The results obtained using regression kriging are comparable with other image fusion

methods. We compared a wavelet-based approach with the regression kriging method

and the regression kriging results in better fusion images.

Per-pixel difference and Person correlation coefficients between fused and originalimages are usually applied to evaluate the fused images. We could use these two criteria

to make general comparisons between regression kriging and other fusion algorithms.

Klonus and Ehlers (2007) compared a number of sophisticated image fusion algorithms

in order to evaluate the problems of colour distortions in fused images; as for Quickbird

image fusion, they concluded that the best correlation coefficients were between 0.92

and 0.98 and the best per-pixel differences were between 12 and 26. In this image fusion

research, the correlation coefficients for the fused Quickbird images with OrbView-3 as

predictor are between 0.92 and 0.98 and per-pixel differences are between 2.5 and 7.4(table 9). Ling et al. (2007) fused the very similar Ikonos images with the same

QuickBird images, while it seemed that there were some apparent colour distortions

along the main road in the fused images using Fast Fourier Transform (FFT)-enhanced

IHS approach, although the correlation coefficients were between 0.92 and 0.98 and the

per-pixel differences were between 4 and 8. The fused Ikonos images using regression

kriging in this study result in good correlation coefficients from 0.93 to 0.98 and the best

per-pixel differences between 8 and 20.

Regression kriging for image fusion includes three procedures that are relativelysimple, easy to understand, and computation-efficient, while other effective methods

are complex and may not be reproducible – for example, image fusion through

wavelet transformation. Carr (2004) concluded: (1) wavelet transformation is not

only a complex process but also computation-consuming, which is required by the

wavelet component images that must be maintained at higher numerical precision in

fusion process to finally obtain a correct result; and (2) image fusion using wavelet

transformation is difficult or impossible to reproduce if the algorithms are not

discussed in detail.Regression kriging takes the advantages of the correlation between pixel values of

different images of multi-date single-sensor or multi-date multi-sensor and incorpo-

rates the spatial dependence of pixel values into residual kriging, in which regression

kriging improves spatial resolution while reducing colour distortion in the fused

images. However, regression kriging also has some limitations. Semivariogram mod-

elling could be very computation-intensive, especially when the full variogram surface

is generated. However, the full semivariogram surface is not really needed when the

study area is not a homogeneous region, while homogeneity is not a typical propertyof geographical phenomena. Different semivariogram models might be applied

because of analysts’ background and experience, which means that a non-optimal

or a relatively optimal semivariogram model could be applied to represent the spatial

dependence and variation in the data without an extensive data exploration. We could

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say the estimation of a semivariogram equation is more-or-less arbitrary decisions.

However and fortunately, ‘even a fairly crudely determined set of weights can give

excellent results when applied to data’ (Cressie 1993), (see also Chiles and Delfiner

1999). Additionally, kriging of residuals in regression kriging adjusts the predictions

of fused pixel values that are basically predicted by the linear correlation betweendifferent images. If there is little linear correlation between different images, regres-

sion kriging will result in poor image fusion. Fortunately, a relatively large correlation

between different images (e.g. correlation of band-to-band, band-to-pan, or band-to-

multiband) covering the same area typically exists.

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