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Invariant representation for spectral reflectance images and its application
EURASIP Journal on Image and Video Processing volume 2011, Article number: 2 (2011)
Abstract
Spectral images as well as color images observed from object surfaces are much influenced by various illumination conditions such as shading and specular highlight. Several invariant representations were proposed for these conditions using the standard dichromatic reflection model of dielectric materials. However, these representations are inadequate for other materials like metal. This article proposes an invariant representation that is derived from the standard dichromatic reflection model for dielectric and the extended dichromatic reflection model for metal. We show that a normalized surfacespectral reflectance by the minimum reflectance is invariant to highlights, shading, surface geometry, and illumination intensity. Here the illumination spectrum and the spectral sensitivity functions of the imaging system are measured in a separate way. As an application of the proposed invariant representation, a segmentation algorithm based on the proposed representation is presented for effectively segmenting spectral images of natural scenes and bare circuit boards.
Introduction
Spectral images contain large amount of information, compared with color images, are useful for a variety of applications such as material identification, natural scene rendering, colorimetric analysis, and machine vision tasks [1]. Spectral images as well as color images observed from object surfaces are much influenced by various illumination conditions. The observed spectral images do not only depend on surfacespectral reflectance and illuminant spectrum, but also include various reflection effects such as shading and specularity, which mainly depend on illumination geometries and surface materials. Therefore, image representations invariant to shading, shadow, lighting, and specularity have been proposed for color images [2–6] and for spectral images [7, 8] so far in several ways.
Geusebroek et al. [3] investigated the differential photometric invariance. Van de Weijer et al. [4] introduced the photometric quasiinvariants which were a set of photometric invariant derivatives with better noise and stability characteristics. Combining the photometric quasiinvariants with derivativebased feature detectors led to features which could identify shadow corners and object corners [5]. Stokman and Gevers [7] proposed a method for edge classification from spectral images. Their method aimed at detecting edges and assigning one of the types of shadow, highlight, and material edge. However, most of those methods were constructed based on the standard dichromatic reflection model by Shafer [9]. This model assumes that an object surface is composed of inhomogeneous dielectric material, and reflected light from the surface is decomposed into two additive components of body (diffuse) reflection and interface (specular) reflection. This decomposition results in the classification of physics events, such as shadows and highlights. However, the modelbased method is valid for such limited materials as plastics and paints [10–12]. It should be noted that there are metallic objects in realworld scenes, which cannot be described by the standard dichromatic reflection model.
The present article proposes an invariant representation for spectral reflectance images. The invariant representation for a variety of objects in a real world is derived from the standard dichromatic reflection model for dielectric [9] and the extended dichromatic reflection model for metal [13, 14]. We show that the invariant formulas for spectral images of both artificial object and natural ones preserve surfacespectral reflectance information and are invariant to highlight, shadow, surface geometry, and illumination intensity. Here we measured the illumination spectrum by using a Spectroradiometer and a standard white reference and measured the spectral sensitivity functions of the imaging system by using a Spectroradiometer and a Monochromator.
As an application of the proposed invariant representation, the transformed spectral image with the invariant properties is used as an invariant operator for the image segmentation problem. Several algorithms were recently proposed for segmenting the spectral images into distinct surface areas [15–19]. However, those algorithms were not always robust for highlight and shading effects occurred for different surface materials, and required a careful adjustment of the lighting position. As a typical example, it was difficult to segment materials on a bare circuit board with various tiny elements [20–23]. In this article, we present a segmentation algorithm based on the proposed representation for effectively segmenting spectral images of natural scenes and bare circuit boards. The proposed invariant representation is also applied to other alternative segmentation algorithms. Experimental results using realworld objects including metals and dielectrics show that the proposed representation is invariant to highlight, shadow, and object surface geometry, and effective for image segmentation.
The reminder of this article is organized as follow: 'Dichromatic reflection models' section describes two types of dichromatic reflection models. 'Proposed invariant representation' section develops the proposed invariant representation in detail. 'Test of invariant representation' section tests the invariant properties for the captured spectral images of several sample objects. 'Application to image segmentation' section discusses image segmentation based on the transformed spectral images to the invariant representation. This article ends with 'Conclusions' section.
Dichromatic reflection models
The standard dichromatic reflection model [9] suggests that light reflected from the surface of an inhomogeneous dielectric object is composed of two additive components, the interface reflection and the body reflection. The radiance of the reflected light Y (θ, λ) is a function of the wavelength λ, ranging over a visible wavelength 400 to 700 nm, and the geometric parameters θ, including the direction angles of the viewing angle and the phase angle. The standard dichromatic reflection model describes the reflected light in the form
where L_{I}(λ) and L_{B}(λ) are the spectral power distributions of the interface and the body reflection components, respectively. The weights c_{I}(θ) and c_{B}(θ) are geometric scale factors.
The reflection model is also described in terms of spectral reflectance. Let E(λ) be the spectralpower distribution of the incident light. The spectral reflectance function defined as S(θ, λ) = Y (θ, λ)/E(λ), independent of illumination, can be expressed as
where S_{I}(λ)and S_{B} (λ) are surfacespectral reflectances for the interface and the body components, respectively. The standard model incorporates the neutral interface reflection (NIR) assumption which states that the interface reflection component S_{I}(λ) is constant over the range of visible wavelength as S_{I}(λ) = S_{I}^{c}. This allows Equation 2 to be written as
where c'_{I} (θ) = c_{I}(θ)S_{I}^{c}. It is shown that this reflectance model is valid for a variety of natural and artificial dielectric objects including plastic and paint (e.g., see [10–12]).
Metal is a homogeneous material that indicates essentially different reflection properties from the inhomogeneous dielectric materials. It consists of only interface reflection with the Fresnel reflectance. Therefore, if the surface is shiny and stainless, the body reflection component in the reflected light is negligibly small. A sharp specular highlight is observed only at the viewing angle of the mirrored direction. Thus, the surface reflection depends on the incident angle of illumination. Tominaga [13] shows that the surfacespectral reflectance function of a metal can be approximated by a linear combination of two interface reflection components as
where the first term on the right hand side corresponds to the specular reflectance at the normal incident. The second term, which is constant over the visible wavelength range, corresponds to the grazing reflection at the horizontal incident. It is important to note that the observed spectral reflectance can be expressed in a linear combination of the reflectance function at the normal incidence and a constant reflectance. This type of surface reflection is called the extended dichromatic reflection model.
Proposed invariant representation
We briefly explain an invariant representation for spectral images [7, 8], which was limited to the standard reflection model for inhomogeneous dielectric objects, and then extend the invariant representation for all materials including inhomogeneous dielectric and homogeneous metal.
Let us suppose that a spectral image is captured at N points in the visible range 400 to 700 nm. Let i and j be two different wavelengths (spectral bands) in the range. From Equation 3 of the standard reflection model, subtraction of one band from another provides a reflectance representation that is independent of specular highlight as
Moreover let l and n be two other wavelengths. Then the following ratio of two subtractions between wavelengths can be spectral invariant, that is, invariant to highlight, shading, and surface geometry [7, 8].
Next, we newly consider the reflection model of metal objects. The extended reflection model in Equation 4 is the same fashion mathematically as the standard model in Equation 3 for dielectric, although the two reflection models are physically different. Therefore, we derive a unified invariant representation for all materials including inhomogeneous dielectric and homogeneous metal. In fact, the following invariant equation can be derived for metal from Equation 4,
where the geometric weighting coefficients c_{I1}(θ) and c'_{I2}(θ) in Equation 4 are eliminated. We note that the operation of Equation 7 results in an equivalent operation to Equation 6, that depends on only surfacespectral reflectance and is invariant to highlights, shading, and geometries.
The above spectral operations based on subtraction and division provides mathematically simple and robust spectral invariant representations. However, the above computation depends on wavelengths such as λ_{ l } and λ_{ n } , and therefore, for real spectral data it is unstable because the denominator includes subtraction and we can get negative values. There is no method that enables us to choose the appropriate bands. Thus, the invariant is unstable and depends on wavelengths. The above operation can be generalized to a stable form. Note that use of the minimum value of spectral reflectance preserves the original spectral characteristics. Then an invariant representation for spectral reflectance is defined as
For practical computation, let us sample spectral reflectance at N wavelengths in the visible range 400 to 700 nm. Then the discrete version of the invariant representation is written as
This representation is spectral invariant for all materials, including inhomogeneous dielectric and homogeneous metal. The observed spectral reflectance S(θ, λ) can be reduced into a normalized surfacespectral reflectance S'(θ, λ) = S'(λ) that is independent of the geometric parameter θ. This representation is useful as an invariant operator for a variety of spectral image analysis, including material classification and image segmentation. The properties are summarized in the following lemmas.
Lemma 1: Assuming the standard dichromatic reflection model for inhomogeneous dielectric materials, S'(θ, λ) is independent of highlight, shading, surface geometry, and illumination intensity.
Proof: By substituting Equation 3 in Equation 9 and factoring out dependencies on highlight, shading, surface geometry, and illumination intensity, we have
Lemma 2: Assuming the extended dichromatic reflection model for homogeneous metal, S'(θ, λ) is independent of highlight, shading, surface geometry, and illumination intensity.
Proof: By substituting Equation 4 in Equation 9 and factoring out dependencies, we have
Test of invariant representation
Spectral imaging system
The proposed invariant representation was tested for spectral images of different metal and dielectric objects. Use of a liquid crystal tunable (LCT) filter provides precise filtration for spectral imaging [24–27]. We have developed a spectral imaging system, shown in Figure 1a, which consists of a monochromatic CCD camera with an air cooling system and 12bit dynamic range, a VariSpect LCT filter, an IRcut filter, and a personal computer. Accuracy decreases when spectral sensors have certainly broadband. Our system with LCT filter has the spectral properties of narrowband filtration of 10 nm and wavelength range 400 to 700 nm. This is not ideally narrowband, but excellent result can be obtained. The LCT filter tune to any wavelength in the visible range without vibration or noise.
The imaging system automatically captures and saves spectral images with arbitrary number of bands and shutter speeds. The actual measurement time required for capturing one spectral image with 31 bands is 4.75 s. Incandescent lamps are used for effective surface illumination. Figure 1b shows the total spectral sensitivity functions of the imaging system. We measured the spectral sensitivity functions of the imaging system by using a Spectroradiometer and a Monochromator. Spectral images calibration to compensate the undesirable effects due to uncontrolled acquisition conditions is done using a standard white reference and a black reference. The camera noise is captured using the black reference.
The dynamic range of 12bits looks adequate for imaging of most objects surfaces. However, when we observe object surfaces including strong specular highlights, such as smooth plastic and metal surfaces, we need highdynamic range (HDR) imaging. The HDR imaging allows spectral images to more accurately represent the range of intensity levels found in real scenes. Without HDR, areas that are too dark are clipped to black and areas that are too bright are clipped to white. This will strongly affect the invariant representation results. In this case, we extend the original dynamic range as follow: Suppose that three pictures of the same scene are automatically taken with different shutter speeds (different exposures). Then we combine the multiple images into a single image to extend the dynamic range of the capture image to 14bits equivalently.
If the illumination spectrum is known in advance, the surfacespectral reflectance can be estimated in a straightforward way from the camera outputs under narrowband filtration. Let λ_{1}, λ_{2,...,}λ_{31} be 31wavelengths of filtration. The surfacespectral reflectance can be estimated from the sensor outputs Y(θ, λ_{1}), Y(θ, λ_{2}),..., Y(θ, λ_{31}), as
where E(λ) is the illuminant spectralpower distribution, and R_{ i } (λ) is the ith sensor spectral sensitivity function in Figure 1b. The sensor output at each wavelength in Equation 12 is normalized with a factor on illuminant and sensing sensitivity. Thus, the spectral reflectance is recovered by eliminating the lighting and sensing effects from the sensor outputs. In a general computer vision problem we have to estimate the illumination spectrum. This problem is discussed in 'Illumination estimation and invariant representation' section.
Test result
We have used a test scene including a metal object of copper and two dielectric objects of ceramic (cup) and plastic (frog). Figure 2a shows the color image of the original surfacespectral reflectances. This color image was obtained using the CIEcolor matching functions, which are the spectral sensitivity curves of the CIE XYZ tristimulus values X, Y, and Z, of the observed reflectance S(θ, λ). Then the XYZspace was converted to a RGBspace. We note the scene has the illumination effects of shadows and highlights observed over different parts on the object surfaces. From this scene, we obtained a set of spectral reflectance images with the size of 439 × 297 × 31. Figure 2b shows the color image of the proposed invariant representation of the normalized spectral reflectance. Shadows, highlights, and surface geometry are much reduced in the invariant representation of both the dielectric objects and the metal object.
For detailed inspection, Figure 2c depicts a 3D view of the component image of spectral reflectance at 550 nm for a small rectangular area including metal and dielectrics. The observed reflectance S(θ, λ) is strongly influenced by shading and illumination effects. Figure 2d shows the 3D view of the invariant representation S'(θ, λ) = S'(λ) for the same part at 550 nm. Shadows and highlights disappear from the test part, so that the transformed spectral image depends on only the inherent spectral reflectance to each object surface. Therefore, the image is clearly segmented into two different material regions. Thus, the proposed invariant representation is valid for both dielectric and metal objects, and is much more robust under a variety of illumination effects than the observed reflectance data.
To confirm the effectiveness of the proposed invariant representation, we tested another spectral image including a metal object and two dielectric objects of plastic. Figure 3a shows the color image of the original surfacespectral reflectances. The image has the illumination effects of shadows and highlights. Figure 3b shows the color image of the proposed invariant representation. We note that shadows, highlights, and surface geometry are much reduced in the invariant representation of both the dielectric objects and the metal object.
Application to image segmentation
Algorithms for spectral image segmentation
As an application of the proposed invariant representation, the transformed spectral image with the invariant properties is used as an invariant operator for the image segmentation problem. Several algorithms have been proposed for segmenting the spectral images into distinct surface areas [15–19]. Paclík et al. [15] proposed a segmentation method of spectral images acquired by imaging crosssections with scanning electron microscopy using energydispersive Xray microanalysis (SEM/EDX). An effective method was introduced by MartínezUsó et al. [16] for spectral image segmentation in fruit inspection applications. A hierarchical Markovian model for hyperspectral image segmentation was proposed by MohammadDjafari et al. [17]. However, those algorithms were not always robust for the highlight and shading effects occurred for different surface materials, and required a careful adjustment of the lighting position.
In our previous study, we introduced a segmentation algorithm 'PCA+Ncut method' for material classification using dimensionreduced spectral information [18]. This method was used to execute the normalized cut (Ncut) [28] in a lowdimensional space obtained with the principal components analysis (PCA). In Ref. [19], we proposed another algorithm which was fundamentally based on the normalized cut approximated by the Nyström method [29] to incorporate both spectral information and spatial information. Those previous methods were useful for material classification of printed circuit boards. However, the material classification was not invariant to illumination effects, but was performed under the ideal lighting condition of eliminating specular reflection and shadow.
In this article, we introduce an image segmentation algorithm based on the proposed invariant representation, which results in reliable segmentation results for natural scenes and circuit boards under arbitrary lighting conditions. The image segmentation process is composed of the following four steps:

(1)
The surfacespectral reflectances are estimated from the camera outputs, and then transformed into the invariant representation to highlight, shadow, and surface geometry.

(2)
The similarity matrix between pixels is constructed in the normalized cut scheme.

(3)
The computation burden is reduced by the Nyström approximation.

(4)
The Kmeans algorithm is applied to cluster the leading normalized eigenvectors to get the final segmentation.
Illumination estimation and invariant representation
The estimation of scene illumination from image data is one of interesting topics in image analysis and computer vision [1]. Although many algorithms were proposed for scene illuminant estimation, most algorithms assumed spatially uniform illumination from a single light source [30, 31]. Concerning spectral imaging, we presented an illuminant estimation method from the outputs of a multichannel vision system [26, 30], where the dichromatic reflection model was assumed to object surfaces. Therefore the illuminant spectrum could be estimated from the interface reflection component. Recently, the authors proposed a method for estimating the illuminant spectralpower distributions from omnidirectional observations by a multichannel omnidirectional imaging system [32, 33].
An easy way to estimate the illuminant spectrum is to place a standard white reference in a scene and capture the corresponding sensor outputs from the spectral image of the scene. The sensor outputs are then described as (i = 1, 2,..., 31). When the spectral sensitivities R_{ i } (λ) are narrow, the illuminant is estimated directly from the sensor outputs. This method is utilized in inspection systems for industrial application.
Here suppose that the spatially uniform illuminant spectrum E(λ) is estimated from multiple light sources by such a method as described in the above. Then the surfacespectral reflectance S(θ, λ) of an object is obtained in a straightforward way using Equation 12. Moreover, the normalized spectral reflectance S'(λ) for invariant representation is obtained from the transformation of Equation 9. The following segmentation method is adapted to the invariant representation for classifying each pixel to a specific object.
Similarity matrix construction in the normalized cut scheme
In the normalized cut scheme, a similarity matrix W is constructed by all combination of two pixels based on the input Ndimensional feature vector, which corresponds to the invariant representation. The similarity w(x, y) ∈ W between two pixels x and y is defined as product of two similarity measures of the normalized spectral reflectance and the spatial location. Using the Gaussian kernel function to represent the similarity measure, we describe
where F(x) ∈ [0, 1] ^{N} is the Ndimensional feature vector at pixel x, which corresponds to the invariant representation S'(λ). The function Z(x) represents the spatial location that is effectively used to connect different regions to a similar segment, and ranges from 0 to the tested image size. The vector norm operator ·_{2} is the Euclidean distance. The standard deviation σ represents the sensitivity of the Gaussian distribution. The spectral sensitivity σ_{S} depends on the materials appearance while the spatial sensitivity σ_{P} depends on the size of tested image. The value of the σ parameter is very important. The parameter σ determines the distance of reachable neighbor pixels in one single step. A small σ captures better the similarities of the pixels. However if σ is too small, the pixels become disconnected. Thus, we need to optimize the spectral and the spatial sensitivity parameters σ_{S} and σ_{P}.
Nyström approximation
Exact normalized cut classification requires calculations of eigenvalues and eigenvectors of the huge similarity matrix W. The Nyström method is a technique for finding a numerical approximation to the eigen decomposition, and it was widely applied to areas involving large dense matrices. Here the Nyström method estimates the eigenvalues and eigenvectors from smaller matrices. Let L be the number of pixels. We empirically select number of sample pixels m, less than 1% of all data and more than number of segments, enough for choosing all objects in the tested image. Let A ∈ ℝ^{m × m}represents the similarities amongst the sample pixels and B ∈ ℝ^{(Lm) × m}represents the similarities between the m sample pixels and the (Lm) remaining pixels. Two matrices A_{S} and B_{S} are constructed using F(x)  F(y)_{2} from the invariant representation S'and two other matrices A_{P} and B_{P} are constructed using Z(x)  Z(y)_{2} from the spatial location. Then, from the minimum average of the spectral matrices A_{S} and B_{S} and the spatial matrices A_{P} and B_{P}, we estimate the spectral and the spatial sensitivity parameters as σ_{S} = min{avg(A_{S}), avg(B_{S})} and σ_{P} = min{avg(A_{P}), avg(B_{P})}.
By using Equation 13, the two matrices of A and B are constructed from the estimated sensitivity parameters, and the spectral and the spatial matrices as and . Then the approximated eigenvectors matrix V is defined as
where Q = A + A^{1/2}BB^{T}A^{1/2} and it is diagonalized as Q = U_{Q}Λ_{Q}U^{T}_{Q}. We select the largest k eigenvectors from matrix V according to the required number of k materials.
Kmeans to cluster the leading Eigenvectors
We use the k eigenvectors stacked in columns simultaneously to get the final clusters for k objects. From the matrix V ∈ R^{L × k}form matrix by renormalizing each of V's rows to have unit length [34] as
where L is the number of pixels. By treating each row of the leading normalized eigenvectors as a point in R^{k} , we cluster them into k different objects via Kmeans that attempts to minimize the distortion. Finally, we assign the pixel S'_{I} of the normalized spectral reflectance to cluster j if and only if row i of the matrix was assigned to cluster j. Thus, the input normalized spectral reflectance data S'(λ) of spectral images is assigned to different segments to get the final segmentation.
Experimental results
We have examined the performance of the segmentation algorithm based on the proposed invariant representation for spectral images. First we used the same spectral reflectance image as in Figure 2. Figure 4a is the original image of a copper object and two dielectric objects. Figure 4b depicts the ground truth of material classification by segmenting manually the image. Figure 4c shows the segmentation result using the original spectral reflectance data. The result contains many shadows and highlights effects. Figure 4d shows the segmentation result using the proposed invariant representation. The image is clearly segmented into different material regions and background. Thus, the segmentation based on the proposed invariant representation is independent of the illumination effects of highlight and shadow, and the geometry of object shape.
In order to confirm the effectiveness of the invariant representation, two alternative methods were also examined. Although we tried several different parameter settings in each method, only the best results are shown for the comparison. The Kmeans classification method [35] was used to segment the spectral image in Figure 4a by clustering the original spectral reflectance data in the spectral space using random seeds, wherein the number of clusters was set to 4. The result is shown in Figure 5a. The PCA + Ncut method [18] was used to execute the normalized cut in a lowdimensional space obtained with the principal components analysis. The segmentation result of clustering the original spectral reflectance data is shown in Figure 5b. We note that the both segmentations are strongly affected by different illumination events on the objects surfaces. On the other hand, Figure 5c, d shows the segmentation results by the Kmeans and the PCA + Ncut methods, respectively, based on the proposed invariant representation. The image is well segmented into different material regions and background, compared with the results in Figure 5a, b. However, the effect of interreflection appears in the segmented images.
Next, we have applied the proposed method to material classification of a printed circuit board with various tiny elements. A bare circuit board surface layer is composed of various elements, which are a mixture of different materials including dielectrics (i.e., photoresist, silkscreen print, and substrate) and metals (i.e., copper), as shown in Figure 6. We used a part of a real circuit board. Figure 6a shows the color image of the original spectral reflectance data. The board is illuminated by one light source from the left direction. The size of the captured spectral image by our imaging system is 433 × 363 × 31.
Figure 6b depicts the ground truth by manually segmenting the image. The ground truth image consists of four segments such as whitesilkscreen print, yellowmetal, greenresistcoated metal, and blackbackground (substrate). Figure 6c shows the material classification result based on the original spectral reflectance data with the illumination effects. We find that miss classification occurs around metal and resistcoated metal lines at the left side of the segmented image. This can be caused by specular reflection of these materials surfaces. Figure 6d shows the classification result on the proposed invariant representation without the illumination effects. The original scene is well segmented into four material regions.
The accuracy of the segmentation results between the segmented image and the ground truth image is numerically demonstrated by two different measures. First, we used the similarity measure with the window size 16 × 16 for labeled images shown in [36]. Second, a pixel by pixel comparison is used to calculate the segmentation quality numerically for the whole regions using
Tables 1 and 2 list the numerical accuracy of segmentation results for the natural image and the circuit board image. The accuracy using different window size for the similarity measure and the quality rate show that the segmentations based on the proposed invariant representation achieve high accuracy for both natural scene and circuit board scene, compared with the segmentation results using the original spectral reflectance data.
Conclusions
The present article has proposed an invariant representation for spectral reflectance images which can be applied to dielectric and metal objects. The invariant representation was derived from the standard dichromatic reflection model for dielectric and the extended dichromatic reflection model for metal. We have shown that the normalized surfacespectral reflectance by the minimum reflectance is invariant to highlights, shading, surface geometry, and illumination intensity. As an application of the proposed invariant representation, a segmentation algorithm based on the proposed representation was presented for effectively segmenting spectral images of natural scenes and bare circuit boards. Experimental results using realworld objects including metals and dielectrics showed that the proposed representation is invariant to highlight, shadow, and object surface geometry, and effective for image segmentation. We also explore as a future work the application of the proposed invariant representation in other fields and the effect of interreflection.
Abbreviations
 HDR:

highdynamic range
 LCT:

liquid crystal tunable
 Ncut:

normalized cut
 PCA:

principal components analysis.
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Ibrahim, A., Tominaga, S. & Horiuchi, T. Invariant representation for spectral reflectance images and its application. J Image Video Proc. 2011, 2 (2011). https://doi.org/10.1186/1687528120112
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Keywords
 Invariant representation
 dichromatic reflection models
 spectral imaging system
 spectral image segmentation