NOVEL SPECTRAL SIMILARITY MEASURE FOR ... - Semantic Scholar
NOVEL SPECTRAL SIMILARITY MEASURE FOR HIGH RESOLUTION URBAN SCENES Bin Chen, Anthony Vodacek∗
Nathan D. Cahill
Rochester Institute of Technology Chester F. Carlson Center for Imaging Science Rochester, NY, USA 14623
Rochester Institute of Technology School of Mathematical Sciences Rochester, NY, USA 14623
ABSTRACT Spectral similarity measures can differentiate subtle differences between spectral footprints extracted from remotely sensed images. Prior research on urban scenes examined lowto-medium resolution images where many details of an urban scene are lost. For high spatial resolution imagers, a number of problems arise in the analysis of urban scenes since virtually all objects are resolved. Issues of spatially varying scene illumination and unavailability of surface reflectance data prevent classical spectral similarity measures from being extensively applied. Commonly used measures have limitations from a feature space perspective such as favoring either spectral direction or spectral magnitude. In this paper, a novel spectral similarity measure based on Mahalanobis distance is proposed to take into account the unique properties of high resolution urban scenes. A simplified radiometric transfer model is also incorporated. The results confirm the advantage of the new spectral similarity measure when applied to complex urban images.
to the spatial domain and can be addressed by imposing contextual constraints. The illumination issue can be solved via brightness normalization. Yet most spectral similarity measures (e.g., SAM and SID) require surface reflectance input which is not easily obtainable. The pixel spectrum is mainly available as digital number (DN) or calibrated spectral radiance data. The conversion of either to reflectance is not easy. Therefore, it is desirable to find a way whereby the spectral comparison can be made with radiance data. Another characteristic of high resolution images is that there are fewer spectral mixing issues because each single pixel basically records a pure type of material. But the conventional spectral similarity measures are initially designed to provide the fractional abundance of mixed materials by matching with the pixel spectra[3], while it is prohibitive to find the representative endmember for each class in a complex urban scene. We describe a novel spectral similarity measure, evolved from the Mahalanobis distance for the analysis of high resolution urban images in an attempt to address the problems facing other similarity measures.
1. INTRODUCTION 2. METHODOLOGY Spectral similarity measures are criteria that are widely used to provide quantitative descriptions of the extent that two reflected/emitted spectra resemble each other. These measures are important in remote sensing imagery analysis, and among them the most commonly used are: spectral angle mapper (SAM)[5], spectral correlation mapper (SCM)[8], spectral information divergence (SID)[1], Euclidean distance (ED), Mahalanobis distance (MD), and spectral similarity value (SSV)[4]. These spectral similarity measures bear some common limitations when applied to high resolution urban images. Many details on the surface of the earth are visible in well-resolved urban regions. The improved spatial discriminability tremendously benefits the recognition of small isolated objects. However, the advanced resolving capability also gives excessive details, e.g., the small structures of a single object and local spectral variation caused by non-uniform illumination due to scene geometry. The former is confined ∗ This work was supported by NSF grant IIP-0917839 and AFOSR grant FA9550-11-1-0348.
978-1-4673-1159-5/12/$31.00 ©2012 IEEE
The proposed approach is very similar to Mahalanobis distance but obtains an appropriately adjustable covariance matrix based only on the chosen reference pixel spectrum, x. The new spectral similarity measure incorporates spectral direction and spectral magnitude naturally and offers freedom in designing a distance isosurface in feature space. 2.1. Proposed Spectral Similarity Measure Recalling the formula for computing Mahalanobis distance, MD (x, x ) = (x − x )T Σ−1 (x − x ), where x is a spectral vector to be compared to x, the expression of the new measure is the same. It is known that the covariance matrix Σ can be diagonalized as Σ = ΦΛΦT , where the orthonormal matrix Φ has as its columns the corresponding eigenvectors of Σ. Λ is a diagonal matrix whose diagonal elements are variances of the corresponding components. Its first element is intentionally made w2 times of the rest such that
6637
IGARSS 2012
⎡ ⎢ ⎢ Λ=⎢ ⎣
w2 0 .. . 0
0 ··· 1 ··· .. . . . . 0 ···
0 0 .. .
⎤ ⎥ ⎥ ⎥, ⎦
(1)
1
where w 1 and thus it is guaranteed that the first element w2 , corresponding to x, is reasonably larger than the other elements. It is implicitly assumed that there is no preference for spectral direction other than that determined by the reference vector. The customized covariance matrix can then be easily constructed because only the subspace spanned by the axial vectors normal to the reference matters: Σ = I + w2 − 1 xxT / x2 ,
(2)
where I is a N × N identity matrix. The construction only depends on reference vector x and multiple w. According to (2), the proposed measure is named as anisotropy-tunable distance (ATD), because it uses a userdefined covariance matrix Σ, rather than the fixed one generated from training data. Since only linear operations are involved, the generation of the covariance matrix is not computationally expensive. The spectral separation benefits from the user-selectable w because the measure is adaptive to various urban scenes. Larger w means spectral direction dominates more than magnitude. The hyper-ellipsoidal isosurface of ATD is elongated along the principal axis, i.e. the direction of x, which indicates its prior sensitivity to the change of spectral direction than spectral magnitude. The shape of the isosurface can be tuned by adjusting the first eigenvalues of Σ, based upon which the extent of sensitivity to varying spectral angles is strengthened or weakened. It is also straightforward to see that ATD is equivalent to ED when w = 1. 2.2. Simplified Radiometric Model In feature space, reflectance data from the same class must lie along a ray passing through the origin. Image data, however, are normally recorded as radiance or DN, neither of which has this nature. SAM and SID require reflectance input because only spectral direction is captured in measurements. The collinearity demand for ATD is weaker but highly probable, which is not unreasonable because of its preference for spectral direction over magnitude. Problems arise when no high-fidelity radiometric conversion is possible for high resolution images. To tackle the problem, a simple radiometric model is proposed. If a target sits on a slope or its sky dome is obstructed by adjacent objects, the downwelled radiance onto the target will be reduced. The fraction of the sky hemisphere above the target F is defined in accordance with scene geometry as 1 F = 1 − sinσF , 2
(3)
where σF is the target slope angle. Moreover we have σ = σs − σF , where σ is the solar zenith angle relative to the normal of target and σs is the solar zenith angle. The topof-atmosphere (TOA) radiance at the sensor, Lλ ,T OA , is then effectively approximated as[6] r
τ2 (λ ) + Lusλ , (4) Lλ ,T OA = Esλ cosσ τ1 (λ ) + F Edsλ π
where Esλ is the exoatmospheric solar irradiance onto a surface perpendicular to the incident beam, τ1 (λ ) is the atmospheric transmission along the sun-target path, Edsλ is the total downwelled spectral irradiance, r is the bidirectional reflectance factor, τ2 (λ ) is the atmospheric transmission along the target-sensor path, Lusλ is the radiance from the sun scattered upward into the sensor’s line of site along the sensortarget path. r should have been considered as a bidirectional reflectance distribution function (BRDF). To simplify the discussion, r is assumed wavelength-independent and capable of characterizing the less idealized (diffuse) surface which only has strong forward and backward scattering. Further assumptions are made that the imaging sensor does not happen to locate in either direction which means the target surface can be approximately treated as Lambertion (perfectly diffuse). If we assume that the downwelled irradiance is some fraction α (λ ) of the total direct solar irradiance incident on the target Edsλ = α (λ ) Esλ τ1 (λ ), (4) can be simplified:
r Lλ ,T OA = cosσ + F α (λ ) Esλ τ1 (λ ) τ2 (λ ) + Lusλ π ≡ fλ (σF ) · Aλ + Bλ .
The following substitutions are made: fλ (σF ) = cosσ + F α (λ ), Aλ = Esλ τ1 (λ ) πr τ2 (λ ), and Bλ = Lusλ . Another assumption is made that Aλ and Bλ only depend on the wavelength λ ; i.e., they are spatially invariant (not scenedependent). This would be a valid assumption as long as the scene does not cover large patches. The ground-leaving radiance of the target is then expressed as the difference between TOA radiance and upwelled radiance: Lλ ,ground ≡ Lλ ,T OA − Bλ = fλ (σF ) Aλ .
(5)
Suppose that two targets T1 and T2 consisting of the same material are on a surface with varied surface normals. From (5), the ratio of ground radiance of the targets is Lλ ,ground,T1 Lλ ,T OA,T1 − Bλ fλ σF1 . = = (6) Lλ ,ground,T2 Lλ ,T OA,T2 − Bλ fλ σF2 If α (λ ) = α; i.e., the scaling factor is no longer dependent on wavelength λ , then fλ (σF ) = f (σF ). The ratio of ground-leaving radiance is no longer the function of the wavelength such that
6638
Lλ ,T OA,T1 − Bλ cosσ1 + F1 α . = Lλ ,T OA,T2 − Bλ cosσ2 + F2 α
(7)
This ratio is a constant across all spectral bands for any two arbitrarily given target pixels representing exactly the same material. This observation is applied to address the problem of radiometric conversion. If all of the above assumptions were satisfied, the spectral similarity measures could be applied on ground-leaving radiance data, rather than reflectance. Several methods are available to determine upwelled radiance. Given the fact that there are no ground truth data available, one of the simplest yet most widely used image-based absolute atmospheric correction approach - dark object subtraction (DOS)[7] - is implemented to correct atmospheric effect. DOS assumes the existence of a dark object with zero or small surface reflectance throughout a scene and a horizontally homogeneous atmosphere. The minimum value in the histogram from the entire scene is thus attributed to atmospheric effect and is subtracted from all the pixels[2]. Considering the difficulty of finding an urban object with negligible reflectance, this approach is applied in a slightly different way. We simply seek to find one of the darkest pixel in the scene which is defined as the one without direct solar illumination and receiving negligible sky light illumination. As a result, a shadow pixel surrounded by high-rise buildings and/or tall trees is a potentially good candidate for an urban dark object. Hence, TOA radiance from the dark pixel is contributed solely by the upwelled radiance and (4) reduces to a much simpler form: Lλ ,T OA,dark = Lusλ = Bλ .
(8)
The TOA radiance data of two targets T1 and T2 are not collinear due to the offset caused by upwelled radiance. Only after Bλ is subtracted from the TOA radiance, ATD, as well as SAM and SID, can be applied to radiance data. 3. RESULTS & DISCUSSION The proposed method is applied to a WorldView-2 pansharpened 6-band1 image (c.f. Fig. 1 image shown in natural color) . The yellow circle indicates the location of the chosen dark pixel. To test the effectiveness of the proposed model, four regions of interest (ROIs) are selected from Fig. 1b such that each region represents different illumination conditions on the same rooftop. The data within those regions before and after DOS are depicted in Fig. 2. Colors of the data points correspond to those of ROIs in Fig. 1b. Two separate lines passing through the origin are fitted to two sets of image data in the least-squared sense. R2 of two fitting lines with respect to corresponding data sets are 0.947 for original TOA radiance and 0.986 for ground radiance. The proposed radiometric conversion is justified because the collinearity of the 1 Two
Near IR bands were not used.
0
¨¹
25 50
100
150
200 Meters
(a)
(b)
Fig. 1: Downtown Rome. (image courtesy of DigitalGlobe)
(a) Red vs. green
(b) Red vs. blue
Fig. 2: Dark object subtraction with respect to color bands.
original radiance data of a homogeneous object with respect to the origin is greatly improved after DOS. Two cropped image portions (blue rectangles in Fig. 1) and the corresponding grayscale maps of measured similarity are shown in Figs. 3 and 4. The blue squares in Figs. 3a and 4a indicate the locations of the chosen reference pixel points. The remaining sub-figures show similarity maps using different spectral similarity measures. Lower intensity values represent higher spectral resemblance. Note that similarity maps of SCM are subject to NOT operation. ATD with higher w performs better than the other similarity measures. The north slope of the red roof (Fig. 3a) receives little direct sun light and it is the most difficult part for most measures. Neither the ATD with smaller w values (less than 5) nor ED give consistent similarity measurement in both north and south slopes. As w is greater than 7, no obviously visible change is observed in similarity maps. Thus a simple thresholding can outline the rooftop, although the northern boundary of the roof is somewhat fuzzy in Figs. 3i and 3j. The similarity maps for a typical urban street are given in Fig. 4. This image portion has three different types of objects within the same scene: road (center), roof (right), and tree (left). Road and roof are similar in spectral direction, while road and tree are similar in spectral magnitude. Referring to Figs. 4b and 4c, both maps show spectral separations of road versus roof and tree for SAM and SID. However, the contrast between road and roof is weaker when compared to ATD in Fig. 4. In regard to SCM, it basically provides little separation information. Overall ATD measures differentiate road from roof and tree, as well as roof from tree. Note that ED failed to identify the southern end of tree leaves because the spectral magnitude is similar to that of the road.
6639
(a) Portion 1
(f) ED
(b) SAM
(c) SID
(d) SCM
object subtraction method using deeply shadowed areas is used to simplify the radiometric conversion and reinforce the data collinearity. The approach is applicable when the conversion of DN or radiance to surface reflectance is difficult or inaccurate. Given that the traditional spectral similarity measures are not initially designed for the high-resolution urban images, they are not sufficiently robust because only a piece of spectral content is used. ATD, however, exploits relatively complete information, and thus is able to provide accurate and desired measurement of the spectral similarity in high-resolution scenes where the traditional measures may not work well. Future work with ATD will assess its application beyond the illumination problem.
(e) SSV
(g) ATD,w=3 (h) ATD,w=5 (i) ATD,w=7 (j) ATD,w=9
Fig. 3: Spectral similarity measurement for roof. Intensity of similarity maps are subject to 2% linear stretch for better visualization. (same below)
5. REFERENCES [1] C. I. Chang, “An information-theoretic approach to spectral variability, similarity, and discrimination for hyperspectral image analysis,” IEEE Trans. Information Theory, vol. 46, pp. 1927–1932, 2000.
(a) Portion 2
(b) SAM
(c) SID
(d) SCM
[2] P. S. Chavez, “Radiometric calibration of Landsat Thematic Mapper multispectral images,” Photogr. Eng. Remote Sens., vol. 55, pp. 1285–1294, 1989.
(e) SSV
[3] J. Farifteh, F. Van Der Meer, and E. J. M. Carranza, “Similarity measures for spectral discrimination of saltaffected soils,” Int. J. Remote Sens., vol. 28, pp. 5273– 5293, 2007. (f) ED
(g) ATD,w=3 (h) ATD,w=5 (i) ATD,w=7 (j) ATD,w=9
[4] P. Keranen, A. Kaarna, and P. Toivanen, “Spectral similarity measures for classification in lossy compression of hyperspectral images,” vol. 4885. SPIE, 2003, pp. 285– 296.
Fig. 4: Spectral similarity measurement for road.
In summary, SAM and SID bear a resemblance in measuring similarity due to the common magnitude normalization process. SCM is not robust, sometimes weak, in detecting urban structures. If treating the non-uniformly illuminated object as a whole is not a concern, ED is good for spectral separation (c.f. Fig. 3f). However, ED is unable to exploit spectral direction information. SSV is computed as a combination of ED and SCM; thus it sometimes takes disadvantages from both measures and its performance is unpredictable. ATD has the unique advantage of flexibility and can be adapted to complex scenes by compensating for nonuniform illumination(Fig. 3) and spectrally different objects are identified (Fig. 4). Considering that larger w (>7) yields basically identical results and smaller w (< 5) does not provide enough discrimination, w between 5 and 7 is a good choice. 4. CONCLUSIONS
[5] F. A. Kruse, A. B. Lefkoff, J. W. Boardman, K. B. Heidebrecht, A. T. Shapiro, P. J. Barloon, and A. F. H. Goetz, “The spectral image-processing system (SIPS) - interactive visualization and analysis of imaging spectrometer data,” Remote Sens. Environ., vol. 44, pp. 145–163, 1993. [6] J. Schott, Remote Sensing: The Image Chain Approach, 2nd Ed. Oxford University Press, USA, 2007. [7] C. Song, C. E. Woodcock, K. C. Seto, M. P. Lenney, and S. A. Macomber, “Classification and change detection using Landsat TM data: When and how to correct atmospheric effects?” Remote Sens. of Environ., vol. 75, pp. 230–244, 2001. [8] F. van der Meer and W. Bakker, “Cross correlogram spectral matching: Application to surface mineralogical mapping by using AVIRIS data from Cuprite, Nevada,” Remote Sens. Environ., vol. 61, pp. 371–382, 1997.
The ATD spectral similarity measure is especially designed to account for high resolution urban scenes. A revised dark
6640
Nathan D. Cahill
Rochester Institute of Technology Chester F. Carlson Center for Imaging Science Rochester, NY, USA 14623
Rochester Institute of Technology School of Mathematical Sciences Rochester, NY, USA 14623
ABSTRACT Spectral similarity measures can differentiate subtle differences between spectral footprints extracted from remotely sensed images. Prior research on urban scenes examined lowto-medium resolution images where many details of an urban scene are lost. For high spatial resolution imagers, a number of problems arise in the analysis of urban scenes since virtually all objects are resolved. Issues of spatially varying scene illumination and unavailability of surface reflectance data prevent classical spectral similarity measures from being extensively applied. Commonly used measures have limitations from a feature space perspective such as favoring either spectral direction or spectral magnitude. In this paper, a novel spectral similarity measure based on Mahalanobis distance is proposed to take into account the unique properties of high resolution urban scenes. A simplified radiometric transfer model is also incorporated. The results confirm the advantage of the new spectral similarity measure when applied to complex urban images.
to the spatial domain and can be addressed by imposing contextual constraints. The illumination issue can be solved via brightness normalization. Yet most spectral similarity measures (e.g., SAM and SID) require surface reflectance input which is not easily obtainable. The pixel spectrum is mainly available as digital number (DN) or calibrated spectral radiance data. The conversion of either to reflectance is not easy. Therefore, it is desirable to find a way whereby the spectral comparison can be made with radiance data. Another characteristic of high resolution images is that there are fewer spectral mixing issues because each single pixel basically records a pure type of material. But the conventional spectral similarity measures are initially designed to provide the fractional abundance of mixed materials by matching with the pixel spectra[3], while it is prohibitive to find the representative endmember for each class in a complex urban scene. We describe a novel spectral similarity measure, evolved from the Mahalanobis distance for the analysis of high resolution urban images in an attempt to address the problems facing other similarity measures.
1. INTRODUCTION 2. METHODOLOGY Spectral similarity measures are criteria that are widely used to provide quantitative descriptions of the extent that two reflected/emitted spectra resemble each other. These measures are important in remote sensing imagery analysis, and among them the most commonly used are: spectral angle mapper (SAM)[5], spectral correlation mapper (SCM)[8], spectral information divergence (SID)[1], Euclidean distance (ED), Mahalanobis distance (MD), and spectral similarity value (SSV)[4]. These spectral similarity measures bear some common limitations when applied to high resolution urban images. Many details on the surface of the earth are visible in well-resolved urban regions. The improved spatial discriminability tremendously benefits the recognition of small isolated objects. However, the advanced resolving capability also gives excessive details, e.g., the small structures of a single object and local spectral variation caused by non-uniform illumination due to scene geometry. The former is confined ∗ This work was supported by NSF grant IIP-0917839 and AFOSR grant FA9550-11-1-0348.
978-1-4673-1159-5/12/$31.00 ©2012 IEEE
The proposed approach is very similar to Mahalanobis distance but obtains an appropriately adjustable covariance matrix based only on the chosen reference pixel spectrum, x. The new spectral similarity measure incorporates spectral direction and spectral magnitude naturally and offers freedom in designing a distance isosurface in feature space. 2.1. Proposed Spectral Similarity Measure Recalling the formula for computing Mahalanobis distance, MD (x, x ) = (x − x )T Σ−1 (x − x ), where x is a spectral vector to be compared to x, the expression of the new measure is the same. It is known that the covariance matrix Σ can be diagonalized as Σ = ΦΛΦT , where the orthonormal matrix Φ has as its columns the corresponding eigenvectors of Σ. Λ is a diagonal matrix whose diagonal elements are variances of the corresponding components. Its first element is intentionally made w2 times of the rest such that
6637
IGARSS 2012
⎡ ⎢ ⎢ Λ=⎢ ⎣
w2 0 .. . 0
0 ··· 1 ··· .. . . . . 0 ···
0 0 .. .
⎤ ⎥ ⎥ ⎥, ⎦
(1)
1
where w 1 and thus it is guaranteed that the first element w2 , corresponding to x, is reasonably larger than the other elements. It is implicitly assumed that there is no preference for spectral direction other than that determined by the reference vector. The customized covariance matrix can then be easily constructed because only the subspace spanned by the axial vectors normal to the reference matters: Σ = I + w2 − 1 xxT / x2 ,
(2)
where I is a N × N identity matrix. The construction only depends on reference vector x and multiple w. According to (2), the proposed measure is named as anisotropy-tunable distance (ATD), because it uses a userdefined covariance matrix Σ, rather than the fixed one generated from training data. Since only linear operations are involved, the generation of the covariance matrix is not computationally expensive. The spectral separation benefits from the user-selectable w because the measure is adaptive to various urban scenes. Larger w means spectral direction dominates more than magnitude. The hyper-ellipsoidal isosurface of ATD is elongated along the principal axis, i.e. the direction of x, which indicates its prior sensitivity to the change of spectral direction than spectral magnitude. The shape of the isosurface can be tuned by adjusting the first eigenvalues of Σ, based upon which the extent of sensitivity to varying spectral angles is strengthened or weakened. It is also straightforward to see that ATD is equivalent to ED when w = 1. 2.2. Simplified Radiometric Model In feature space, reflectance data from the same class must lie along a ray passing through the origin. Image data, however, are normally recorded as radiance or DN, neither of which has this nature. SAM and SID require reflectance input because only spectral direction is captured in measurements. The collinearity demand for ATD is weaker but highly probable, which is not unreasonable because of its preference for spectral direction over magnitude. Problems arise when no high-fidelity radiometric conversion is possible for high resolution images. To tackle the problem, a simple radiometric model is proposed. If a target sits on a slope or its sky dome is obstructed by adjacent objects, the downwelled radiance onto the target will be reduced. The fraction of the sky hemisphere above the target F is defined in accordance with scene geometry as 1 F = 1 − sinσF , 2
(3)
where σF is the target slope angle. Moreover we have σ = σs − σF , where σ is the solar zenith angle relative to the normal of target and σs is the solar zenith angle. The topof-atmosphere (TOA) radiance at the sensor, Lλ ,T OA , is then effectively approximated as[6] r
τ2 (λ ) + Lusλ , (4) Lλ ,T OA = Esλ cosσ τ1 (λ ) + F Edsλ π
where Esλ is the exoatmospheric solar irradiance onto a surface perpendicular to the incident beam, τ1 (λ ) is the atmospheric transmission along the sun-target path, Edsλ is the total downwelled spectral irradiance, r is the bidirectional reflectance factor, τ2 (λ ) is the atmospheric transmission along the target-sensor path, Lusλ is the radiance from the sun scattered upward into the sensor’s line of site along the sensortarget path. r should have been considered as a bidirectional reflectance distribution function (BRDF). To simplify the discussion, r is assumed wavelength-independent and capable of characterizing the less idealized (diffuse) surface which only has strong forward and backward scattering. Further assumptions are made that the imaging sensor does not happen to locate in either direction which means the target surface can be approximately treated as Lambertion (perfectly diffuse). If we assume that the downwelled irradiance is some fraction α (λ ) of the total direct solar irradiance incident on the target Edsλ = α (λ ) Esλ τ1 (λ ), (4) can be simplified:
r Lλ ,T OA = cosσ + F α (λ ) Esλ τ1 (λ ) τ2 (λ ) + Lusλ π ≡ fλ (σF ) · Aλ + Bλ .
The following substitutions are made: fλ (σF ) = cosσ + F α (λ ), Aλ = Esλ τ1 (λ ) πr τ2 (λ ), and Bλ = Lusλ . Another assumption is made that Aλ and Bλ only depend on the wavelength λ ; i.e., they are spatially invariant (not scenedependent). This would be a valid assumption as long as the scene does not cover large patches. The ground-leaving radiance of the target is then expressed as the difference between TOA radiance and upwelled radiance: Lλ ,ground ≡ Lλ ,T OA − Bλ = fλ (σF ) Aλ .
(5)
Suppose that two targets T1 and T2 consisting of the same material are on a surface with varied surface normals. From (5), the ratio of ground radiance of the targets is Lλ ,ground,T1 Lλ ,T OA,T1 − Bλ fλ σF1 . = = (6) Lλ ,ground,T2 Lλ ,T OA,T2 − Bλ fλ σF2 If α (λ ) = α; i.e., the scaling factor is no longer dependent on wavelength λ , then fλ (σF ) = f (σF ). The ratio of ground-leaving radiance is no longer the function of the wavelength such that
6638
Lλ ,T OA,T1 − Bλ cosσ1 + F1 α . = Lλ ,T OA,T2 − Bλ cosσ2 + F2 α
(7)
This ratio is a constant across all spectral bands for any two arbitrarily given target pixels representing exactly the same material. This observation is applied to address the problem of radiometric conversion. If all of the above assumptions were satisfied, the spectral similarity measures could be applied on ground-leaving radiance data, rather than reflectance. Several methods are available to determine upwelled radiance. Given the fact that there are no ground truth data available, one of the simplest yet most widely used image-based absolute atmospheric correction approach - dark object subtraction (DOS)[7] - is implemented to correct atmospheric effect. DOS assumes the existence of a dark object with zero or small surface reflectance throughout a scene and a horizontally homogeneous atmosphere. The minimum value in the histogram from the entire scene is thus attributed to atmospheric effect and is subtracted from all the pixels[2]. Considering the difficulty of finding an urban object with negligible reflectance, this approach is applied in a slightly different way. We simply seek to find one of the darkest pixel in the scene which is defined as the one without direct solar illumination and receiving negligible sky light illumination. As a result, a shadow pixel surrounded by high-rise buildings and/or tall trees is a potentially good candidate for an urban dark object. Hence, TOA radiance from the dark pixel is contributed solely by the upwelled radiance and (4) reduces to a much simpler form: Lλ ,T OA,dark = Lusλ = Bλ .
(8)
The TOA radiance data of two targets T1 and T2 are not collinear due to the offset caused by upwelled radiance. Only after Bλ is subtracted from the TOA radiance, ATD, as well as SAM and SID, can be applied to radiance data. 3. RESULTS & DISCUSSION The proposed method is applied to a WorldView-2 pansharpened 6-band1 image (c.f. Fig. 1 image shown in natural color) . The yellow circle indicates the location of the chosen dark pixel. To test the effectiveness of the proposed model, four regions of interest (ROIs) are selected from Fig. 1b such that each region represents different illumination conditions on the same rooftop. The data within those regions before and after DOS are depicted in Fig. 2. Colors of the data points correspond to those of ROIs in Fig. 1b. Two separate lines passing through the origin are fitted to two sets of image data in the least-squared sense. R2 of two fitting lines with respect to corresponding data sets are 0.947 for original TOA radiance and 0.986 for ground radiance. The proposed radiometric conversion is justified because the collinearity of the 1 Two
Near IR bands were not used.
0
¨¹
25 50
100
150
200 Meters
(a)
(b)
Fig. 1: Downtown Rome. (image courtesy of DigitalGlobe)
(a) Red vs. green
(b) Red vs. blue
Fig. 2: Dark object subtraction with respect to color bands.
original radiance data of a homogeneous object with respect to the origin is greatly improved after DOS. Two cropped image portions (blue rectangles in Fig. 1) and the corresponding grayscale maps of measured similarity are shown in Figs. 3 and 4. The blue squares in Figs. 3a and 4a indicate the locations of the chosen reference pixel points. The remaining sub-figures show similarity maps using different spectral similarity measures. Lower intensity values represent higher spectral resemblance. Note that similarity maps of SCM are subject to NOT operation. ATD with higher w performs better than the other similarity measures. The north slope of the red roof (Fig. 3a) receives little direct sun light and it is the most difficult part for most measures. Neither the ATD with smaller w values (less than 5) nor ED give consistent similarity measurement in both north and south slopes. As w is greater than 7, no obviously visible change is observed in similarity maps. Thus a simple thresholding can outline the rooftop, although the northern boundary of the roof is somewhat fuzzy in Figs. 3i and 3j. The similarity maps for a typical urban street are given in Fig. 4. This image portion has three different types of objects within the same scene: road (center), roof (right), and tree (left). Road and roof are similar in spectral direction, while road and tree are similar in spectral magnitude. Referring to Figs. 4b and 4c, both maps show spectral separations of road versus roof and tree for SAM and SID. However, the contrast between road and roof is weaker when compared to ATD in Fig. 4. In regard to SCM, it basically provides little separation information. Overall ATD measures differentiate road from roof and tree, as well as roof from tree. Note that ED failed to identify the southern end of tree leaves because the spectral magnitude is similar to that of the road.
6639
(a) Portion 1
(f) ED
(b) SAM
(c) SID
(d) SCM
object subtraction method using deeply shadowed areas is used to simplify the radiometric conversion and reinforce the data collinearity. The approach is applicable when the conversion of DN or radiance to surface reflectance is difficult or inaccurate. Given that the traditional spectral similarity measures are not initially designed for the high-resolution urban images, they are not sufficiently robust because only a piece of spectral content is used. ATD, however, exploits relatively complete information, and thus is able to provide accurate and desired measurement of the spectral similarity in high-resolution scenes where the traditional measures may not work well. Future work with ATD will assess its application beyond the illumination problem.
(e) SSV
(g) ATD,w=3 (h) ATD,w=5 (i) ATD,w=7 (j) ATD,w=9
Fig. 3: Spectral similarity measurement for roof. Intensity of similarity maps are subject to 2% linear stretch for better visualization. (same below)
5. REFERENCES [1] C. I. Chang, “An information-theoretic approach to spectral variability, similarity, and discrimination for hyperspectral image analysis,” IEEE Trans. Information Theory, vol. 46, pp. 1927–1932, 2000.
(a) Portion 2
(b) SAM
(c) SID
(d) SCM
[2] P. S. Chavez, “Radiometric calibration of Landsat Thematic Mapper multispectral images,” Photogr. Eng. Remote Sens., vol. 55, pp. 1285–1294, 1989.
(e) SSV
[3] J. Farifteh, F. Van Der Meer, and E. J. M. Carranza, “Similarity measures for spectral discrimination of saltaffected soils,” Int. J. Remote Sens., vol. 28, pp. 5273– 5293, 2007. (f) ED
(g) ATD,w=3 (h) ATD,w=5 (i) ATD,w=7 (j) ATD,w=9
[4] P. Keranen, A. Kaarna, and P. Toivanen, “Spectral similarity measures for classification in lossy compression of hyperspectral images,” vol. 4885. SPIE, 2003, pp. 285– 296.
Fig. 4: Spectral similarity measurement for road.
In summary, SAM and SID bear a resemblance in measuring similarity due to the common magnitude normalization process. SCM is not robust, sometimes weak, in detecting urban structures. If treating the non-uniformly illuminated object as a whole is not a concern, ED is good for spectral separation (c.f. Fig. 3f). However, ED is unable to exploit spectral direction information. SSV is computed as a combination of ED and SCM; thus it sometimes takes disadvantages from both measures and its performance is unpredictable. ATD has the unique advantage of flexibility and can be adapted to complex scenes by compensating for nonuniform illumination(Fig. 3) and spectrally different objects are identified (Fig. 4). Considering that larger w (>7) yields basically identical results and smaller w (< 5) does not provide enough discrimination, w between 5 and 7 is a good choice. 4. CONCLUSIONS
[5] F. A. Kruse, A. B. Lefkoff, J. W. Boardman, K. B. Heidebrecht, A. T. Shapiro, P. J. Barloon, and A. F. H. Goetz, “The spectral image-processing system (SIPS) - interactive visualization and analysis of imaging spectrometer data,” Remote Sens. Environ., vol. 44, pp. 145–163, 1993. [6] J. Schott, Remote Sensing: The Image Chain Approach, 2nd Ed. Oxford University Press, USA, 2007. [7] C. Song, C. E. Woodcock, K. C. Seto, M. P. Lenney, and S. A. Macomber, “Classification and change detection using Landsat TM data: When and how to correct atmospheric effects?” Remote Sens. of Environ., vol. 75, pp. 230–244, 2001. [8] F. van der Meer and W. Bakker, “Cross correlogram spectral matching: Application to surface mineralogical mapping by using AVIRIS data from Cuprite, Nevada,” Remote Sens. Environ., vol. 61, pp. 371–382, 1997.
The ATD spectral similarity measure is especially designed to account for high resolution urban scenes. A revised dark
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