Subpixel Mapping Using Markov Random Field With ... - IEEE Xplore

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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 3, MAY 2013

Subpixel Mapping Using Markov Random Field With Multiple Spectral Constraints From Subpixel Shifted Remote Sensing Images Liguo Wang and Qunming Wang

Abstract—Subpixel mapping (SPM) is a promising technique to increase the spatial resolution of land cover maps. Markov random field (MRF)-based SPM has the advantages of considering spatial and spectral constraints simultaneously. In the conventional MRF, only the spectral information of one observed coarse spatial resolution image is utilized, which limits the SPM accuracy. In this letter, supplementary information from subpixel shifted remote sensing images (SSRSI) is used with MRF to produce more accurate SPM results. That is, spectral information from SSRSI is incorporated into the likelihood energy function of MRF to provide multiple spectral constraints. Simulated and real images were tested with the subpixel/pixel spatial attraction model, Hopfield neural networks (HNNs), HNN with SSRSI, image interpolation then hard classification, conventional MRF, and proposed MRF with SSRSI based SPM methods. Results showed that the proposed method can generate the most accurate SPM results among these methods. Index Terms—Land cover, Markov random field (MRF), multiple spectral constraints, subpixel mapping (SPM).

I. I NTRODUCTION

M

IXED pixels widely exist in remote sensing images, which contain more than one land cover class. Soft classification can estimate the proportion of each land class in mixed pixels by exploiting the spectral information of remote sensing images. However, it fails to predict the spatial location of land classes. Subpixel mapping (SPM) has been developed to predict the spatial location of land classes at the subpixel level. By SPM, each coarse spatial resolution pixel is divided into subpixels, and the ultimate task is to predict the class label of each subpixel. Atkinson introduced the concept of SPM based on the assumption of spatial dependence [1]. This assumption underpins some approaches to SPM, namely, linear optimization techniques [2], subpixel/pixel spatial attraction

Manuscript received April 9, 2012; revised June 15, 2012; accepted July 21, 2012. Date of publication October 15, 2012; date of current version November 24, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 60802059 and Grant 61275010 and in part by the Foundation for the Doctoral Program of Higher Education of China under Grant 200802171003. L. Wang is with the College of Information and Communications Engineering, Harbin Engineering University, Harbin 150001, China. Q. Wang is with the College of Information and Communications Engineering, Harbin Engineering University, Harbin 150001, China, and also with the Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2012.2215573

model (SPSAM) [3], Hopfield neural networks (HNNs) [4], genetic algorithm [5], and pixel swapping algorithm [6]. These approaches take the fraction of each land cover class as a constraint. According to the fractions, the number of subpixels belonging to each class is strictly maintained (or to a large extent, such as HNN). In the class allocation process at the subpixel level, only spatial dependence is used, and the spectral information is not taken into account. As a preprocessing step, the soft classification has a direct influence on the SPM, and errors from the former can be transmitted to the latter. For this reason, there may be many isolated pixels in the resulting land cover maps. Some methods, however, are not bound by the exact value of the soft classification. For example, techniques in the fields of digital image processing have been employed to remove isolated pixels in subpixel maps, such as mathematical morphology [7], notch filter [8], and modal filter [2]. In addition, Foody et al. [9] fitted a class membership contour through the pixels after soft classification to determine the waterline or shoreline at the subpixel level. However, during the process, they still only consider spatial information in nature and may fail to map the land cover objects smaller than a coarse pixel. The predicted subpixel boundaries of land cover objects are also likely to be destroyed while removing isolated pixels near the boundaries. Unlike the approaches in [2]–[6], the Markov random field (MRF)-based SPM method considers spatial (prior energy) and spectral (likelihood energy) constraints simultaneously through the posterior probability model [10]. It may utilize the soft classification result to obtain the initial spatial distribution of land classes at the subpixel scale but does not strictly rely on this input, which may be modified in the SPM process. In addition, it exploits the intraclass spectral variability through a covariance matrix in a likelihood energy term and utilizes spectral information to a large degree. MRF-based SPM can be directly applied to a remote sensing image in units of reflectance [11]. The spectral constraint in conventional MRF is based on one observed coarse spatial resolution image used as an input for SPM. The SPM problem is underdetermined, in which it has multiple plausible solutions, and many fine spatial resolution class maps can lead to an equally good reproduction of the available coarse image [12]. The additional information has been used to address the underdetermined problem, such as intermediate spatial resolution fused images [13], subpixel shifted remote sensing images (SSRSI) [14] (also termed time series images in [8]), and panchromatic images [15]. Data in

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WANG AND WANG: SUBPIXEL MAPPING USING MRF WITH MULTIPLE SPECTRAL CONSTRAINTS

other forms have been also used, such as vector boundaries [16], LIDAR data [17], and digital elevation models [18]. This letter presents a novel SPM method utilizing supplementary information from SSRSI using MRF. SSRSI are taken by observation satellites taking images over the same area at different times. Due to the slight orbit translation, SSRSI are usually shifted at the subpixel level [14]. The spectral information from these images is incorporated in the likelihood energy term of the MRF’s energy function to provide multiple spectral constraints. The difference between the proposed method and the techniques proposed in [8], [13], and [14] is that the former makes use directly of the spectral information from SSRSI in units of reflectance rather than the land cover fractions from the soft classification of additional images (as in [8], [13], and [14]). The rest of this letter is organized as follows. Section II presents details of MRF and MRF with SSRSI. Experimental results are discussed in Section III and then followed by conclusions in Section IV. II. M ETHODS A. MRF-Based SPM Suppose Y is the observed coarse spatial resolution image with M × N pixels and X is the fine spatial resolution SPM result with SM × SN pixels, where S is the scale factor (i.e., each coarse pixel is divided into S 2 subpixels of X). Let P (X) be the prior probability; P (Y |X) be the conditional probability that Y is observed, given X; and P (X|Y ) be the posterior probability for X, given Y . These probabilities can be expressed as follows: P (X) =

1 exp (−U (X)/M ) Zp

1 exp (−U (Y |X)/M ) Zl 1 P (X|Y ) = exp (−U (X|Y )/M ) Z P (Y |X) =

B S n α δ(qij , qk ) A0 dk j=1 i=1

μj =

L

elj μl

(6)

L 1 elj C l S2

(7)

l=1

Cj =

l=1

where elj is the proportion of class l in pixel Qj ; L is the total number of classes; μl and C l are the mean vector and the covariance matrix of class l, respectively. U (Y |X) is expressed as B 1 1 (y j −μj ) C −1 ln |C U (Y |X) = (y −μ )+ | . (8) j j j j 2 2 j=1 In (8), the intraclass spectral variability is described by a covariance matrix, which is of great significance to express the spectral information in a remote sensing image. Based on the maximum a posteriori theory, the most likely SPM results X opt correspond to the maximum posterior probability P (X|Y ) (i.e., the minimum posterior energy function U (X|Y )), i.e., X opt = arg max [P (X|Y )] = arg min [U (X|Y )] X X (9) = arg min [βUspatial + (1 − β)Uspectral ] X

(2) (3)

B. MRF With SSRSI

(1)

(4)

Assume that Qj is a coarse pixel of Y that corresponds to the area covered by S 2 subpixel qij (i = 1, 2, . . . , S 2 ; and j = 1, 2, . . . , B, where B is the total number of coarse pixels of Y ). Then, U (X) can be given by 2

U (X) =

Each pixel Qj in Y corresponds to vector y j composed of spectral values in each band of the remote sensing image. The vector is assumed to be normally distributed with mean vector μj and covariance matrix C j , which are given by

where β = (α/1 + α) ∈ (0, 1) is a weight coefficient that controls the balance between the prior energy function and the likelihood energy function. The former exploits the spatial information and provides a spatial constraint Uspatial [see (5)], whereas the latter exploits the spectral information and provides a spectral constraint Uspectral [see (8)]. The two constraints are simultaneously considered to determine X opt .

where Zp , Zl , and Z are normalizing constants; M is a constant; U (X) is the prior energy function of X; U (Y |X) is the likelihood energy function of Y , given X; and U (X|Y ) is the posterior energy function of X, given Y . For these energy functions, (4) holds on, i.e., U (X|Y ) = U (X) + U (Y |X).

599

(5)

k=1

where α ∈ (0, +∞) is a weighting coefficient, n is the number of neighbors, dk is the distance between geometric centers of subpixel qij and its neighbor qk , and A0 is a normalizing constant. Based on the Ising model, δ(qij , qk ) takes 0 if qij and qk belong to the same class; otherwise, it takes 1 [19].

To use the SSRSI for SPM by MRF, (9) was modified by incorporating multiple spectral constraints, i.e.,

T 1 t X opt = arg min βUspatial + (1 − β)Uspectral X T t=1 (10) B

t −1 t 1 1 t t t t t

Uspectral = y j −μj + ln C j y − μj C j 2 j 2 j=1 (11) where T is the number of SSRSI, including the input observed coarse spatial resolution image Y ; y tj is the vector of pixel Qtj in the tth SSRSI of Y , with mean vector μtj and covariance matrix C tj . To result in a faster convergence rate for the MRFbased SPM, during the iterations, each subpixel was updated to the class that minimizes the posterior energy function. Suppose (fc , fd ) is the coordinate of the subpixel in the fine spatial resolution map X, subpixel (fc , fd ) has to satisfy the spectral constraint from coarse pixel Qt(Fe ,Ff ) in the tth shifted image and not only the constraint from the single

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observed image in the conventional MRF model. Fe and Ff are calculated as Fe = floor ((fc − fat − 1)/S) + 1 Ff = floor ((fd − fbt − 1)/S) + 1

(12)

where floor(•) is a function that takes an integer nearest to • but not larger than it, and (fat , fbt ) is the subpixel shift of the tth image. The proposed approach has several properties. 1) It inherits the advantages of the MRF model. The supplementary information is directly used in the form of spectra (i.e., in units of reflectance). This way, the soft classification procedure, as done in [8], [13], and [14], is not needed to process the remote sensing image to exploit spectral information, which may introduce errors. Therefore, the supplementary information is conveniently used. 2) The SSRSI are also coarse spatial resolution images that have the same spatial resolution as the observed image. They are easy to acquire; and the fusion process, as done in [13], is not required to generate the high spectral and spatial resolution image that may lead to spectral distortion. 3) The ease by which any additional information in units of reflectance can be easily coded into the model as an extra spectral constraint to enhance the SPM accuracy. III. E XPERIMENTAL R ESULTS To objectively evaluate the performances of SPM methods, synthetic coarse spatial resolution images were studied. They were created by degrading the fine spatial resolution image band by band via a mean filter. Then, the known fine spatial resolution reference land cover map was used for supervised assessment. The misregistration is an active topic while multiple images are utilized. Since many algorithms can be used to deal with this problem, it is beyond the scope of this letter. To solely concentrate on the performance of the proposed SPM approach, the SSRSI were generated by shifting the fine spatial resolution image, and the subpixel shifts were assumed to be known. Here, experiments were conducted on a simulated AVIRIS image and a real Reflective Optics System Imaging Spectrometer (ROSIS) image. Four shifted images were used, and the subpixel shifts at the scale factor S were assumed to be (0, 0), (floor(S/2), 0), (0, floor(S/2)), and (floor(S/2), floor(S/2)). We compared the SPM results based on SPSAM, HNN, HNN with SSRSI, conventional MRF, and MRF with SSRSI. For SPSAM, the neighborhood window size was set to 3 × 3. For both HNN and HNN with SSRSI, all the weighting constants in the network energy function were set to 1, the steepness of the tanh function was set to 10, the time step was set to 0.001, and the number of iterations was set to 1000. In addition, another method was also tested, namely, image interpolation then hard classification (IITHC). Specifically, a linear interpolation of the T shifted images was carried out band by band for the desired scale at first. Then, the T interpolated images were averaged to generate an enhanced image. At last, a hard classification of the enhanced image was implemented using the classical spectral angle mapper.

Fig. 1.

Reference land cover map of the simulated AVIRIS image.

All experiments were tested on an Intel Core 2 Processor (1.80-GHz Duo central processing unit, 2.00-GB random access memory) with MATLAB 7.1 version. It took less than 5 min to run the proposed method in all experiments. Note that the computing time of MRF-based SPM methods is closely related to the size of the studied scene and the number of the bands of the remote sensing image: Computing complexity linearly scales with the number of coarse pixels and quadratically scales with the number of bands of the studied image. A. Experiments on the Simulated AVIRIS Image In the first experiment, a simulated image was used to evaluate the performance of the proposed method. The land cover map for the simulated image was from the image in [20]. It covers an agricultural area in Flevoland, The Netherlands. For simplicity, the image was manually segmented to three classes, namely, C0, C1, and C2 (see Fig. 1). It contains 60 × 60 pixels; and C0, C1, and C2 cover an area of 1011, 1710, and 879 pixels, respectively. The spectral data for the three classes were from the AVIRIS data acquired over the Indian Pine test site [21]. We randomly selected 1011, 1710, and 879 spectra samples from Corn_notill (1434 samples in all), Soybeans_min (2468 samples in all), and Soybeans_notill (968 samples in all), of which the spectral data corresponded to C0, C1, and C2, respectively. To reduce the computing burden, the band selection method in [22] was applied; and five bands, i.e., 17, 29, 41, 97, and 200, were selected out for the experiment. The fine spatial resolution image was degraded with three scales, namely, S = 4, 6, and 10. First, the soft classification was processed; and linear spectral mixture analysis (LSMA) [23] was employed, appreciating its simple physical meaning and convenience in application. After the LSMA, SPM methods were applied to produce the fine spatial resolution land cover maps. Fig. 2 shows the resulting subpixel maps when scale 6 was considered. Referring to Fig. 1, we can see that the boundaries of three classes in the SPSAM result are nearly wrong, and there are many obvious incorrectly classified pixels appearing as pits and isolated pixels. This is because SPSAM heavily relied on soft classification results and errors from soft classification could not be eliminated in the SPM process. Different from SPSAM, HNN-based methods alleviated the aforementioned phenomenon slightly, and the boundaries are clearer. The HNN used for SPM is an optimization tool. In this model, the class proportions were used as constraint terms in the energy function to restrict the number of subpixels per class. The attribute value of each class (between 0 and 1) was changed, and the energy function was iteratively minimized to approach a solution. Finally, the attribute values were quantized to create a hardclassified subpixel map. In the resulting subpixel map, the class proportion information provided by the soft classification

WANG AND WANG: SUBPIXEL MAPPING USING MRF WITH MULTIPLE SPECTRAL CONSTRAINTS

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Fig. 3. Real ROSIS image. (a) 91th band of the scene. (b) Reference land cover map. (Gray) Asphalt. (Light green) Meadows. (Dark green) Trees. (Red) Bricks. (Black) Unlabeled.

Fig. 2. SPM results of the simulated AVIRIS image with different methods. (a) SPSAM. (b) HNN. (c) HNN+SSRSI. (d) IITHC. (e) MRF. (f) MRF+SSRSI. TABLE I PCC (%) OF THE SPM METHODS FOR THE SIMULATED AVIRIS IMAGE

would be maintained to a large extent but might not be perfectly so. As a result, some previous pits and isolated pixels may not appear in the HNN result. Comparing the maps in Fig. 2(b) and (c), we can conclude that with SSRSI, HNN generated more accurate mapping results than HNN without SSRSI, which suggests that SSRSI were helpful for HNN in this experiment. As for IITHC, the resulting map seems smooth, as the linear interpolation had a smooth effect and the average process on multiple interpolated images eliminated the noise. However, this also led to the fact that the boundaries between classes were distorted and quite different from the reference map in Fig. 1. Parameter β controls the contributions from the spatial and spectral constraints in the MRF model. Its influence was tested for conventional MRF and MRF with SSRSI at S = 6 (the neighborhood window size was set to the same number as in [20]) by the overall accuracy in terms of the percentage of correctly classified pixels (PCC). The PCC was on a parabola change when β ranged from 0.1 to 0.9 with a step of 0.1, peaking at β = 0.4 for both methods. Fig. 2(e) and (f) shows the subpixel maps of the two methods when β = 0.4. The conventional MRF-based method removed the majority of the isolated pixels. The map is more accurate than those produced by SPSAM, HNN, HNN with SSRSI, and IITHC, as the boundaries of three classes in Fig. 2(e) are much clearer. However, some isolated pixels and compact boundaries exist in the conventional MRF map, as the effect of the conventional MRF was limited for its single spectral constraint. Examining the resulting map of MRF with SSRSI, the improvement is pronounced. Compared with conventional MRF, the boundaries of three classes are more similar to the reference map. Among the six methods, the proposed method produced the most accurate SPM result. The advantage of the proposed method was also quantitatively confirmed by comparison of PCC. Table I displays the

Fig. 4. SPM results of the real ROSIS image with different methods. (a) SPSAM. (b) HNN. (c) HNN+SSRSI. (d) IITHC. (e) MRF. (f) MRF+SSRSI. TABLE II PCC (%) OF THE SPM METHODS FOR REAL ROSIS IMAGE (S = 5)

accuracy of the six methods at scales 4, 6, and 10. As for SPSAM, HNN, and HNN with SSRSI, the PCC was smaller than 80.0% at all three scales, much lower than for the two MRF-based methods. Compared with conventional MRF and IITHC, the proposed MRF with SSRSI had a higher PCC. B. Experiments on the Real ROSIS Image To further evaluate the proposed algorithm, another experiment on a real image was conducted. The image was acquired by the ROSIS sensor during a flight campaign over Pavia, northern Italy, on July 8, 2002. It covers an area around the Engineering School at the University of Pavia. The ROSIS data set has 103 bands, ranging from 0.43 to 0.86 μm, with a spatial resolution of 1.3 m. Here, a region with 100 × 100 pixels was used for the test, which contains four classes of interest, namely, asphalt, meadows, trees, and bricks. Likewise, to reduce the computational burden of the MRF-based SPM methods, bands 1, 3, 13, and 91 were selected out by the method in [22]. The 91th band of the region studied is shown in Fig. 3(a). The corresponding reference map is shown in Fig. 3(b). The ROSIS image was degraded with S = 5. Fig. 4 shows the SPM results produced by the six methods. β was set to 0.9, and the neighborhood window size was set to 3 × 3 for both MRF and MRF with SSRSI.

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As shown in Fig. 4, there are many isolated pixels in the map obtained with SPSAM, and they appear as obvious noise due to the errors from the LSMA. Although the HNN alleviated this phenomenon, as the proportion constraint was not perfectly maintained, it is easy to see that many places for trees were wrongly classified. Unlike the HNN result, in Fig. 4(c), the brick class appears in many places, particularly where the meadow class is appropriate. This is because SSRSI used in the experiments of this group provide inaccurate multiple proportion constraints in the HNN model that were generated by the LSMA. This adversely affects the overall performance of the HNN with SSRSI method. Although there are little isolated pixels in the IITHC result due to the smooth effect of the linear interpolation and the average process on multiple interpolated images, many places were incorrectly classified, particularly for trees and asphalt. As for the two MRF-based methods, the results seem more reasonable than the other four methods. Moreover, compared with the conventional MRF model, when SSRSI were used, more trees and meadows were correctly labeled than by the conventional MRF-based method. The labeled test samples in Fig. 3(b) were used for supervised quantitative assessment (see Table II). The PCC of SPSAM was 76.5%, which is 8.5% less than that of HNN. However, with SSRSI, the PCC of HNN did not increase as expected because many pixels belong to meadow classes and brick classes were misclassified. The PCC of conventional MRF was nearly the same as that of IITHC but of much higher accuracy than that of SPSAM, HNN, and HNN with SSRSI. With SSRSI, the PCC of the MRF-based model increased by 1.8%. Among these methods, the proposed method had the highest accuracy. IV. C ONCLUSION This letter presented a novel SPM method, i.e., MRF with SSRSI, making use of supplementary spectral information from SSRSI to provide multiple spectral constraints by incorporating them in the likelihood energy function of the MRF classifier. The proposed method’s effectiveness was tested by experiments on simulated AVIRIS and real ROSIS data. The results demonstrated that the proposed SPM method can produce the most accurate subpixel land cover maps in comparison with SPSAM, HNN, HNN with SSRSI, IITHC, and conventional MRF. The additional information used in this letter was the spectral information from SSRSI. How to exploit spectral information from data from other sources, such as fused images, is worthy of consideration. In addition, additional spatial structure information may be also applied to the MRF model. These seem to be promising, and our further work will focus on these issues. ACKNOWLEDGMENT The authors would like to thank Prof. P. Atkinson from the University of Southampton for his careful proofreading and Prof. P. Gamba from the University of Pavia for providing the ROSIS data.

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