Bin Luo
email: bin.luo@gipsa-lab.inpg.fr

Muhammad Murtaza Khan
email: muhammad-murtaza.khan@gipsa-lab.inpg.fr

Thibaut Bienvenu
email: thibaut.bienvenu@gmail.com

Jocelyn Chanussot
email: jocelyn.chanussot@gipsa-lab.inpg.fr

PANSHARPENING WITH A DECISION FUSION BASED ON THE LOCAL SIZE INFORMATION

Pansharpening may be defined as the process of synthesizing multispectral images at a higher spatial resolution. Different pansharpening methods produce images with different characteristics. In the 2006 IEEE Data Fusion Contest, À-trous Wavelet Transform based pansharpening (AWLP) and Context Adaptive (CBD) pansharpening methods were declared as joint winners. While assessing the quantitative quality of the pansharpened images, it was observed that the two methods outperform each other depending upon the local content of the scene. Hence, it is interesting to develop a method which could produce results locally approximately similar to the best method, among the two pansharpening methods. In this paper we propose a method which selects either of the two methods for performing pansharpening on local regions, based upon the size of the objects. The results obtained demonstrate that the proposed method produces images with quantitative results approximately similar to the method which is better among the AWLP and CBD pansharpening methods.

INTRODUCTION

Satellites provide both multispectral (MS) and panchromatic (Pan) images. The Pan and MS images captured by satellites are not at the same resolution. The MS images have a high spectral and low spatial resolution while the Pan image has a high spatial resolution, but no spectral diversity. Images with both high spectral and spatial resolution cannot be captured by satellites due to the fact that in order to maintain the same signal to noise ratio for the narrow spectral band MS images the aperture of the sensor has to be opened for a longer duration. Since the satellite moves during this duration the image captured is blurred, thus resulting in low resolution MS images. Generally, satellite images with both high spectral and spatial resolution are required for improving the results of automated tasks like classification, segmentation and object detection [START_REF] Sirguey | Improving MODIS Spatial Resolution for Snow Mapping Using Wavelet Fusion and ARSIS Concept[END_REF]. The process of pansharpening helps in synthesizing MS images having both high spectral and spatial resolution. For example, the Pan image captured by the IKONOS sensor has a spatial resolution of 1 m where as the MS image has a resolution of 4m. The pansharpening process synthesizes a MS image at the spatial resolution of 1m.

Pansharpening methods can be divided into Multiresolution Analysis (MRA) based methods and Component Substitution (CS) based methods. CS methods are based on the spectral transformation of MS data and the inverse transformation using the high resolution Pan image, which results in the high resolution MS image [START_REF] Chavez | Comparison of three different methods to merge multiresolution and multispectral data: Landsat TM and SPOT panchromatic[END_REF]. MRA based methods employ filters to obtain multi level projections of the MS & Pan images and use the high frequency information extracted from the Pan image during the pansharpening process [START_REF] Núnez | Multiresolution-based image fusion with additive wavelet decomposition[END_REF][START_REF] Wald | Data Fusion: Definitions and Architectures -Fusion of images of different spatial resolutions[END_REF][START_REF] Aiazzi | Context-Driven Fusion of High Spatial and Spectral Resolution Images Based on Oversampled Multiresolution Analysis[END_REF][START_REF] Otazu | Introduction of Sensor Spectral Response Into Image Fusion Methods. Application to Wavelet-Based Methods[END_REF][START_REF] Khan | Pansharpening of hyperspectral images using spatial distortion optimization[END_REF]. In the recently conducted Data Fusion Contest [START_REF] Alparone | Comparison of Pansharpening Algorithms: Outcome of the 2006 GRS-S Data Fusion Contest[END_REF] it was observed that the MRA methods quantitatively outperformed the CS methods. Among the different MRA methods tested, the AWLP [START_REF] Núnez | Multiresolution-based image fusion with additive wavelet decomposition[END_REF][START_REF] Otazu | Introduction of Sensor Spectral Response Into Image Fusion Methods. Application to Wavelet-Based Methods[END_REF] and ATWT-CBD [START_REF] Aiazzi | Context-Driven Fusion of High Spatial and Spectral Resolution Images Based on Oversampled Multiresolution Analysis[END_REF] pansharpening methods resulted in images having good quantitative characteristics. It was observed that the results obtained by the AWLP and ATWT-CBD methods are either the best or second best. Thus an attempt can be made to use the size of the objects for determining which of the two methods should be locally used for pansharpening. It was observed while testing on local regions, i.e. context adaptivity, that AWLP method generally works better for objects of small size in the image i.e. roads, cars etc. where as the ATWT-CBD method is better for pansharpening larger objects in the image, such as industrial buildings, fields, etc..

The second section of this paper presents a brief introduction to AWLP and CBD pansharpening methods. The third section describes briefly how to determine the relative size of objects in a satellite image. The fourth section presents the proposed pansharpening method in the context of using different pansharpening methods based upon the relative size of objects. The fifth section presents the results obtained for the simulated Pléiades sensor images and the final section presents the conclusions and perspectives.

EXISTING PANSHARPENING METHODS

À-trous Wavelet Pansharpening (AWLP)

This pansharpening method is based on the use of undecimated Wavelet transform to extract the high frequency details from the Pan image and the addition of these details in the upscaled low resolution MS image. A version of this method, for three MS bands, was proposed in [START_REF] Núnez | Multiresolution-based image fusion with additive wavelet decomposition[END_REF] and its extension to multi-band MS images, having more than three bands, was proposed in [START_REF] Otazu | Introduction of Sensor Spectral Response Into Image Fusion Methods. Application to Wavelet-Based Methods[END_REF]. The method does not incorporate any specific detail injection model. All the details extracted from the high resolution Pan image are added to the MS image with a detail injection gain value of 1. i.e. the fused method can be simply defined as:

MAW LP (x) = M (x) + P (x). ( 1
)
Where M is the upscaled low resolution MS image, P represents the high frequency details extracted from the high resolution Pan image and M represents the pansharpened image.

Context adaptive À-trous Wavelet Pansharpening (ATWT-CBD)

In [START_REF] Aiazzi | Context-Driven Fusion of High Spatial and Spectral Resolution Images Based on Oversampled Multiresolution Analysis[END_REF], Aiazzi et al. proposed a context adaptive multiresolution fusion process called Context Driven Fusion. The authors proposed the use of a Laplacian pyramid for the extraction of high frequency component from the Pan image and the addition of these details to the low resolution MS images using a context driven model. However, instead of using the Laplacian pyramid, the À-trous Wavelet transform can be used during the pansharpening process. The Fusion process can be described as follows: i) Upscale the low resolution MS images (noted as M (x)); ii) Extract the high frequency details from the high resolution Pan image (noted as P (x)); iii) The fused image MAW CBD (x) is defined by:

MAW CBD (x) = M (x)+1 L M, P ;B >θ (x)×G M, P ;B (x)× P (x), (2)
where P (x) is the upscaled low resolution Pan image. The indicator function 1 L M, P ;B >θ is defined as:

1 L M, P ;B >θ (x) = 1, L M, P ;B (x) > θ 0, L M, P ;B (x) ≤ θ (3)
θ is a constant threshold. L M, P ;B (x) is the local correlation computed on the neighborhood B x defined around the pixel x between M and P , i.e.

L M, P ;B (x) = y∈Bx ( M (y) -μ M ;B (x))( P (y) -μ P ;B (x)) |B x |σ M ;B (x)σ P ;B (x) , μ M ;B (x) (resp. μ P ;B (x)), σ M ;B (x) (resp. σ P ;B (x)
) are the mean and standard deviation computed on B x of the image M (resp. P ). In practice, B x is a square window of size s centred at x. G M, P ;B (x) is the local gain computed on B x and is defined as:

G M, P ;B (x) = σ M ;B (x) σ P ;B (x)
The method estimates a detail injection gain for the addition of the details if the local correlation between the low resolution MS and Pan images is high. In addition, one should notice that the detail injection model is local and hence context adaptive. However, the model can be determined on fixed size windows or local regions. Thus, the proposed method does not take into account the size of the object or region while determining the model.

It was observed while testing both the AWLP and ATWT-CBD methods that the quantitative performances varied with regions, based upon the size of the object in the region of interest. Hence, an attempt was made to locally select which of the two methods should be used based upon the size of objects.

RELATIVE SIZE OF OBJECTS

In [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF], the authors propose a method based on the topographic map of the image to estimate the local scale of each pixel in the case of gray scale remote sensing images. The idea is that, for each pixel, the most contrasted shape containing it is extracted, and the scale of this shape defines the characteristic scale of this pixel. The topographic map [START_REF] Caselles | Topographic maps and local contrast changes in natural images[END_REF], which can be obtained by Fast Level Set Transformation (FLST) [START_REF] Monasse | Fast computation of a contrastinvariant image representation[END_REF], represents an image by an inclusion tree of the shapes (which are defined as the connected components of level sets). For each pixel x, there is a branch of shapes f i (x) (f i-1 ⊂ f i ) containing it. Note I(f i ) the gray level of the shape f i (x), S(f i ) its area and P (f i ) its perimeter. The contrast of the shape f i (x) is defined as

C(f i ) = |I(f i+1 ) -I(f i )|.
The most contrasted shape f î(x) of a given pixel is defined as the shape containing this pixel, of which the contrast is the most important, i.e.

f î(x) = arg max j {C(f j (x)} (4)
Since the optical instruments always blur remote sensing images, several shapes with very low contrasts can belong to the same structure. In order to deal with the blur, the authors of [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] propose a geometrical criterion to cumulate the contrasts of the shapes corresponding to one given structure. The idea is that the difference of the areas of two succesive shapes (for example f i and f i+1 ) corresponding to one given structure is proportional to the perimeter of the smaller shape, i.e. S(f i+1 ) -S(f i ) ∼ λP (f i ), where λ is a constant. It is shown in [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] that the most contrasted shapes extracted in an image form a segmentation of this image. For a given pixel, the segmented region obtained by the method of [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] is actually the most significant structure containing this pixel. The scale E(x) of the pixel x is then defined as the size of the segmented region containing it, i.e.

E(x) = S(f î(x)).

(5)

PROPOSED PANSHARPENING METHOD

It was observed that the performance of AWLP approach is better than ATWT-CBD on an image containing mainly small scale regions (for example images of dense urban area, etc.), because AWLP better preserves small details. However, the details P extracted from images containing mainly large scale homogeneous regions (for example images with fields, etc.) could be noisy. Therefore, on such images, the pansharpened results obtained by AWLP will be noisy. Contrarily, ATWT-CBD adds the details only in those regions where the upscaled MS image is well correlated with the upscaled low resolution Pan image. This ensures better performances of ATWT-CBD on the images containing large scale regions. This observation will be confirmed by the experiments presented in Section 5.

A natural solution for taking advantage of these two approaches is to select for each pixel the optimal approach according to the scale of the most significant structure containing this pixel, which is the main idea of our proposed approach. We associate to each pixel the scale of the most significant region containing it computed on the high resolution Pan image P according to Equation ( 5), noted as E P (x). With the help of this value, we incorporate Equations ( 2) and ( 1). If the scale E P (x) of the most significant region containing x is large (i.e. larger than a threshold γ), we use ATWT-CBD to fuse M (x) and P (x) on this pixel in order to reduce the noise introduced by AWLP. Otherwise if the scale of the region is small (i.e. E P (x) ≤ γ), we use AWLP in order to preserve the details. Therefore the fused image is finally defined by:

M (x) = 1 EP (x)≤γ MAW LP (x)+(1-1 EP (x)≤γ ) MAW CBD (x), (6)
where MAW LP (x) and MAW CBD (x) are computed respectively by Equation (1) and Equation (2).

EXPERIMENTS AND RESULTS

Data sets

For comparing the results of the proposed method with existing pansharpening methods (AWLP and ATWT-CBD) both qualitative and quantitative analysis were performed on simulated Pléiades data. Since, the Pléiades data sets used is simulated the high resolution MS image is available and hence the quantitative analysis can be performed at the desired high resolution. The high resolution MS and Pan images along with the low resolution MS image are provided by CNES (the French Space Agency) at 0.8m and 3.2m resolutions, respectively. The 1024 x 1024 pixel test images were selected from the images taken over the cities of Toulouse and Strasbourg in France. The Toulouse image represents an urban region having small objects and fine details whereas, the Strasbourg image represents countryside having large objects. Two such regions for each image were used for testing and verification however, due to constraints of space only one result for each region is presented in this paper.

Quantitative evaluations

Quantitative quality assessment is generally performed using a reference MS image to assess the quality of the pansharpened images, as defined by Wald's synthesis property [START_REF] Wald | Data Fusion: Definitions and Architectures -Fusion of images of different spatial resolutions[END_REF]. Relative Global Error of Synthesis (ERGAS), Spectral Angle Mapper (SAM) and Quaternion Index (Q4) [START_REF] Alparone | Comparison of Pansharpening Algorithms: Outcome of the 2006 GRS-S Data Fusion Contest[END_REF] can be used to determine global quality of pansharpened images. We have used these indexes for quality assessment as they were used in the 2006 IEEE Data Fusion Contest [START_REF] Alparone | Comparison of Pansharpening Algorithms: Outcome of the 2006 GRS-S Data Fusion Contest[END_REF].

Parameters and Results

For comparison, we performed AWLP, ATWT-CBD and the proposed method for the pansharpening of the data sets. The threshold of the local correlations for the ATWT-CBD method is θ = 0.5 (see Equation ( 2)). According to [START_REF] Aiazzi | Context-Driven Fusion of High Spatial and Spectral Resolution Images Based on Oversampled Multiresolution Analysis[END_REF], the result of pansharpening is not very sensitive to the choice of θ if it is less than 0.8. The neighbourhood B x (see Equation ( 3)) for computing the local correlations and variances is defined by a 16 × 16 analysis window. This choice is based on the experiments using the ATWT-CBD method with different window sizes varying from 12 to 32. When the window size is too small, the considered context is not enough for reducing the noise. If the window size is too large, the improvement of the result is not significant, while the computation time increases a lot. The threshold of the local object size is γ = 256 (see Equation ( 6)). If the object is smaller than the neighbourhood B x , it is considered as a small object. We therefore use AWLP method on this object in order to preserve more details. Otherwise, we use ATWT-CBD method in order to reduce the noise.

The results obtained for the Strasbourg and Toulouse Pléiades images are presented in Tables 1& 2, respectively. From Table 1 it can be seen that for all the indexes determined the ATWT-CBD produces the best results. However from Table 2, it is observed that, for SAM and Q4 indexes, AWLP produces better results, while for ERGAS, ATWT-CBD produces better results. This indicates that AWLP produces better results for urban images with fine details whereas ATWT-CBD performs better for objects with a large size. The proposed method which selects among the two methods produces results that are closer to the best results among the two methods. This means that the results of SAM, ERGAS and Q4 are closer to the ATWT-CBD method for the Strasbourg image and are closer to the AWLP method for the Toulouse image. The ERGAS value of the proposed method is even better than both the ATWT-CBD and AWLP methods. Thus, it can be concluded that the proposed selection method results in images which are quantitatively closer to the best result among the two methods. In Figure 1, we zoom at two regions of the pansharpening results obtained by AWLP, ATWT-CBD and the proposed method on the two simulated Pléiades images. For the visualization, only the red, green and blue channels are shown. It can be observed that, for the Strasbourg image, the AWLP method introduces spectral distortions on the building roofs and the fields, since their sizes are large. While the results of the ATWT-CBD method and the proposed method are quite consistent to the reference. On the Toulouse image, the results of the AWLP and the proposed method well preserve the details of the small objects, such as the cars. While the result of the ATWT-CBD method is more blurred. Filter can be used along with the CBD method thus producing a variant called ATWT-CBD. We observed that generally AWLP method produced better quantitative results for urban areas with fine details whereas the ATWT-CBD method quantitatively performed better when tested with country side scenes with large objects. Hence, a merging of the two algorithms was proposed in this paper to make use of AWLP method for pansharpening of finer details whereas ATWT-CBD for large objects in the same scene. The proposed algorithm was tested on Pléiades simulated data sets and the results obtained for the two different scenarios were closer to the best results from either of the two methods. The extension of this work to IKONOS and QuickBird images is actually investigated, as well as other decision fusion rules on the object size.

  (a) Reference c CNES (b) AWLP (c) ATWT-CBD (d)Proposed method (e) Reference c CNES (f) AWLP (g) ATWT-CBD (h)Proposed method Fig. 1. Zoom (128 × 128) at the reference MS images and the pansharpening results on (a)-(d) the Strasbourg image and (e)-(h) the Toulouse image.

  Wavelet Fusion (AWLP) along with Laplacian Pyramid based Context Decision Fusion(CBD) were rated as the joint winners of the 2006 IEEE Data Fusion Contest . It was observed that instead of the Laplacian Pyramid the À-trous

Table 1 .

 1 Quantitative evaluation of the results obtained on the Strasbourg image by the three pansharpening approaches.

	Approach AWLP ATWT-CBD proposed method
	ERGAS	2.2853	1.8735	2.0283
	SAM	2.6383	2.2914	2.3573
	Q4	0.8771	0.8968	0.8929
	Approach AWLP ATWT-CBD proposed method
	ERGAS	3.4911	3.5539	3.4762
	SAM	5.3153	4.4998	4.768
	Q4	0.9378	0.9301	0.9353

Table 2 .

 2 Quantitative evaluation of the results obtained on the Toulouse image by the three pansharpening approaches.


Bin Luo
email: bin.luo@gipsa-lab.inpg.fr

Jocelyn Chanussot
email: jocelyn.chanussot@gipsa-lab.inpg.fr

GEOMETRICAL FEATURES FOR THE CLASSIFICATION OF VERY HIGH RESOLUTION MULTISPECTRAL REMOTE-SENSING IMAGES

In order to extract geometrical features from a multispectral image and derive a classification, an approach based on the topographic map of the image is proposed. For each pixel, the most significant structure containing it is extracted. The classification of this pixel is based on its spectral information and the geometrical features of the corresponding structure (its area and perimeter). The results obtained on multispectral remote sensing images taken by two different sensors show the efficiency of the extracted geometrical features for separating some classes with very similar spectral attributes but of different semantic meanings.

INTRODUCTION

Recently, very high resolution multispectral remote sensing images are increasingly used for Earth observation. These images, providing rich spatial and spectral information on the observed scenes, better describe the material attributes than panchromatic images. However, by using only the spectral bands, it is difficult to distinguish some classes with similar spectral attributes, such as water and shadow. Moreover, it is impossible to separate objects made by the same material but with different semantic meanings (such as the roads and the roofs of some buildings). Therefore, contextual information, in particular geometrical features, become important for the classification of very high resolution multispectral images.

In order to extract the contextual information from remote sensing images, one can use models based on Markov Random Field (MRF) [START_REF] Poggi | Supervised segmentation of remote sensing images based on a tree-structure MRF model[END_REF]. However, the MRF based methods provide only statistical information on the neighborhood of the considered pixel. Another family of methods for contextual information extraction is based on morphological operators, which allow to extract descriptive features, such as the geometrical features about the structures [START_REF] Pesaresi | A new approach for the morphological segmentation of high-resolution satellite imagery[END_REF]. In [START_REF] Fauvel | Kernel principal component analysis for the classification of hyperspectral remote-sensing data over urban areas[END_REF], it is proposed to extract the extended morphological profiles (EMP) of the principle components of hyperspectral images for the classification. However, since all morphological methods require a structuring element, the features extracted by such methods depend on the used structuring element. Moreover, in order This work is funded by French ANR project VAHINE.

to describe the contextual information of the structures with different scales, one has to use structuring elements with a large discrete scale range. This fact has two major drawbacks. Firstly, the scale range has to be chosen manually and is not necessarily adapted to the image. Secondly, it considerably increases the feature dimension. For example, in order to describe the geometrical information of one pixel on a principle component, the EMP method in [START_REF] Fauvel | Kernel principal component analysis for the classification of hyperspectral remote-sensing data over urban areas[END_REF] uses 8 values.

In [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF], for panchromatic remote-sensing images, it is proposed to define the pixel-wise characteristic scale of the data using the topographic map of gray scale images. The main idea is that for each pixel, the most contrasted structure containing it is extracted, and the scale of this structure defines the scale of this pixel. In [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF], it was shown that the structures extracted on all the pixels form a partition of the image and that the structure extracted on a pixel is in fact the most significant structure containing it. The geometrical features of these structures are very pertinent for describing the contextual information. In this paper, we try to extend the algorithm presented in [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] for extracting geometrical features (the areas and the perimeters of the structures) of multispectral images. The main advantages of our proposed method when compared to EMP are two-fold. Firstly, no structuring element is required. The proposed approach can extract the geometrical features of structures of any scale. In addition, only a very few values are required to describe the geometrical attributes of a spatial position when compared the EMP.

The outline of the paper is as follow: in Section 2, we briefly introduce and then extend the approach proposed in [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] for extracting geometrical features of remote-sensing images. In Section 3, the classification results obtained with the help of the geometrical features are presented. In Section 4, the conclusion is drawn and some perspectives are given.

EXTRACTION OF GEOMETRICAL FEATURES

Extraction of geometrical features on panchromatic remote sensing images

In [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF], the authors propose a method based on the topographic map of the image to estimate the local scale of each pixel in the case of gray scale remote sensing images. idea is that, for each pixel, the most contrasted shape containing it is extracted, and the scale of this shape defines the characteristic scale of this pixel. The topographic map [START_REF] Caselles | Topographic maps and local contrast changes in natural images[END_REF], which can be obtained by Fast Level Set Transformation (FLST) [START_REF] Monasse | Fast computation of a contrastinvariant image representation[END_REF], represents an image by an inclusion tree of the shapes (which are defined as the connected components of the level sets). For each pixel (x, y), there is a branch of shapes f i (x, y) (f i-1 ⊂ f i ) containing it. Note I(f i ) the gray level of the shape f i (x, y), S(f i ) its area and P (f i ) its perimeter. The contrast of the shape f i (x, y) is defined as

C(f i ) = |I(f i+1 ) -I(f i )|.
The most contrasted shape f î(x, y) of a given pixel is defined as the shape containing this pixel, of which the contrast is the most important, i.e.

f î(x, y) = arg max j {C(f j (x, y)} (1)
Since the optical instruments always blur remote sensing images, several shapes with very low contrasts can belong to the same structure. In order to deal with the blur, the authors of [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] propose a geometrical criterion to cumulate the contrasts of the shapes corresponding to one given structure. The idea is that the difference of the areas of two successive shapes (for example f i and f i+1 ) corresponding to one given structure is proportional to the perimeter of the smaller shape, i.e. S(f i+1 ) -S(f i ) ∼ λP (f i ), where λ is a constant. It is shown in [START_REF] Luo | Local scale measure from the topographic map and application to remote sensing images[END_REF] that the most contrasted shapes extracted in an image form a partition of this image. The area and the perimeter values of the most contrasted shape f î(x, y) are very pertinent geometrical features for the classification task.

Extension to multispectral images

We note Y n (x, y), (n = 1, . . . , N c ) the nth channel of the multispectral image and N c the number of spectral channels. In order to extend the estimation of local scale to multispectral images, we firstly extract the most contrasted shapes for each channel. Therefore, for each spatial position (x, y) in a multispectral image, N c shapes are extracted. We note f î,n (x, y) the most contrasted shape extracted by Equation (1) for the pixel (x, y) on the nth channel. The simplest way to use the scale features of a pixel is to use all the scale values of all the channels as features for the classification. The major drawback is that for a given object, if the nth channel is dominant, the area and perimeter values computed on the other channels may introduce classification errors. Therefore, we try to define an unique area value and an unique perimeter value for each pixel by the most significant channel. For a given pixel (x, y), we try to select the most significant shape f î,n (x, y) from the most contrasted shapes f î,n (x, y) extracted from all the channels of the image. The criterion of the selection is always based on the contrasts of these shapes C(f î,n (x, y)). Remark that, since the dynamic of each channel may be quite different, we normalize the contrast value by the total variation

T V n = ∂ x Y 2 n + ∂ y Y 2 n of the corresponding channel.
Therefore, the most significant shape of the pixel (x, y) is defined as:

f î,n (x, y) = arg max n C(f î,n (x, y)) T V n . (2)
Since the area and the perimeter values of the structures in an image may significantly vary (from several pixels to tens of thousands of pixels), we associate to the pixel (x, y) the logarithmic values of the area and the perimeter of the shape f î,n (x, y).

Therefore, the feature vector Θ(x, y) of a given pixel (x, y) containing spectral and geometrical features for classification is defined as:

Θ(x, y) = {Y 1 (x, y), . . . Y Nc (x, y), log S(f î,n (x, y)), log P (f î,n (x, y))}. ( 3
)

EXPERIMENTS AND RESULTS

Geometrical features are more pertinent on very high resolution remote-sensing images, since fine structures (such as buildings, roads, etc.) of urban areas become individually visible. Therefore, in this section, we validate the efficiency of the geometrical features on classification by the experiments carried out on two multispectral remote sensing images taken by two different high resolution sensors: Quickbird (2.4m) and simulated Pleïade (1m).

Experiments on QuickBird image

The first experiment is carried out on a Quickbird multispectral image of Toulouse (see Figure 1 The overall accuracy and the class accuracies obtained by using the two feature sets on the Quickbird image are shown in Table 1. Figure 3 shows the classification results on the whole image with the help of the SVM trained by 10% of the labelled pixels by using the two feature sets. It can be seen that the addition of geometrical features improves the classification results, especially for the classes of Shadow and Water. It is very difficult to separate these two classes when we use only the spectral channels (there isn't even any pixel which is classified as Water). The geometrical features can separate these classes, since the area and the perimeter of the water region in this image (which is in fact a lake) are much larger than for the shadows. Moreover, the geometrical features considerably improve the classification results of the Road and Building classes. The color of the roads and the color of some building roofs are exactly the same, since they are all made of asphalt. Their geometrical attributes are different, since the ratio between the area and perimeter of the buildings is larger than for the roads. The same phenomenon can be observed for the Parkings. By using only spectral channels, no pixel is classified to this class, because the color of the parkings is the same as for the roads.

Experiments on simulated Pleïade data

The second experiment is carried out on a simulated Pleïade image (see Figure 4(a)) taken on Strasbourg with four spectral channels (near infra red, red, green and blue). The spatial resolution is 1m and the image size is 1024 × 1024 pixels. 38.9% of the pixels are manually labelled into 7 classes serving as ground truth. The labelled pixels and the definitions of SVM with Gaussian kernel is used, its optimal scale parameter is selected by 5-fold cross validation. 10% of the pixels of each class are used for training the SVM. For comparison, the image is classified by two feature sets: the original channels {Y n , n = 1, . . . , 4} and the feature set Θ (see Equation (3)). Figure 5 shows the computed logarithmic values of the areas and perimeters computed.

The overall accuracy and the class accuracies obtained by using the two feature sets are shown in Table 2. with the help of the SVM trained by 10% of the labelled pixels by using the two feature sets. It can be seen that, again, the classification results are considerably improved by adding the geometrical features. The overall accuracy increases by nearly 9.14% and the accuracy of each class increases too.

In particular, the accuracies are significantly improved for the Building, Field and Tree classes. For the class of Building, the roads and some building roofs are made by the same material, and hence have the same color. However, the ratio between the area and the perimeter of buildings is larger than for the roads. The colors of the fields and of the trees are quite similar. However, the fields have larger areas and much more regular contours. Therefore, geometrical features can better separate these two classes. Moreover, it can be observed from Figure 6 that when only the spectral channels are used, lots of errors occur on the top left corner of the image. Many pixels corresponding to buildings (mainly small houses) are classified as Bare soil. Again, by adding geometrical features, these pixels are better classified, because the Bare soil class in this image corresponds to larger area and perimeter.

CONCLUSION AND PERSPECTIVES

In this paper, we describe an approach for automatically extracting the geometrical features (the areas and the perime- ters of the structures) of multispectral remote sensing images based on topographic map. The results on both Quickbird and simulated Pleïade images show the efficiency of these features: by adding only two features, it is possible to separate some classes which are difficult to separate by using spectral bands only. However, it can be seen that, in this paper, the geometrical features are simply appended to the feature set. It will be interesting to apply a feature ponderation stradegy (such as the method proposed in [START_REF] Fauvel | Decision fusion for the classification of urban remote sensing images[END_REF]) in order to avoid potential classification errors introduced by the geometrical features.

Fig. 1 .

 1 Fig. 1. (a) Quickbird image on Toulouse. For the visualization, only the infra red, red and green channels are shown. (b) Manually labeled pixels. The definitions of the classes are: Grass, Tree, Shadow, Building, Road, Water, Bare soil, Parking.We use SVM (Support Vector Machines) with Gaussian kernel for classification. 10% of the pixels in each class are

Fig. 2 .

 2 Fig. 2. Logarithmic values of (a) the areas and (b) the perimeters of the most contrasted shapes computed on all the pixels of the Quickbird image.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Classification results obtained on the Quickbird image (a) by using the original 4 spectral bands only, and (b) by using feature vector Θ (see Equation (3)).

Fig. 5 .

 5 Fig. 5. Logarithmic values of (a) the areas and (b) the perimeters of the most contrasted shapes computed on all the pixels of the simulated Pleïade image.

Fig. 6 .

 6 Fig. 6. Classification results obtained on the simulated Pleïade image (a) by using the original 4 spectral bands only, and (b) by using feature vector Θ (see Equation (3)).

Table 1 .

 1 Feature set{Y n } Original channels Θ (see Eq. (3)) Overall accuracy and class accuracies obtained on the Quickbird image.

	Feature dimension	4	6
	Overall Accuracy	77.27%	84.84%
	1. Grass	96.94%	97.96%
	2. Tree	88.75%	88%
	3. Shadow	74.44%	75.07%
	4. Building	51.97%	74.82%
	5. Road	77.54%	87.68%
	6. Water	0%	71.5%
	7. Bare soil	94.05%	97.83%
	8. Parking	0%	22.86%

Table 2 .

 2 Overall accuracy and class accuracies obtained on the simulated Pleïade image.

	Feature set	{Y n } Original channels Θ (see Eq. (3))
	Feature dimension	4	6
	Overall Accuracy	83.87%	93.01%
	1. Road	91.84%	96.88%
	2. Building	77.37%	84.86%
	3. Water	94.14%	97.93%
	4. Field	63.23%	92.36%
	5. Bare soil	89.17%	90.78%
	6. Grass	88.39%	93.03%
	7. Tree	63.2%	89.96%