A Spectral Invariant Representation of Spectral Reflectance Abdelhameed IBRAHIM 1;2 , Shoji TOMINAGA 1 , and Takahiko HORIUCHI 1 1 Graduate School of Advanced Integration Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan 2 Computers and Systems Engineering Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt (Received October 21, 2010; Accepted January 5, 2011) Spectral image acquisition as well as color image is affected by several illumination factors such as shading, gloss, and specular highlight. Spectral invariant representations for these factors were proposed for the standard dichromatic reflection model of inhomogeneous dielectric materials. However, these representations are inadequate for other characteristic materials like metal. This paper proposes a more general spectral invariant representation for obtaining reliable spectral reflectance images. Our invariant representation is derived from the standard dichromatic reflection model for dielectric materials and the extended dichromatic reflection model for metals. We proof that the invariant formulas for spectral images of natural objects preserve spectral information and are invariant to highlights, shading, surface geometry, and illumination intensity. It is proved that the conventional spectral invariant technique can be applied to metals in addition to dielectric objects. Experimental results show that the proposed spectral invariant representation is effective for image segmentation. # 2011 The Japan Society of Applied Physics Keywords: spectral invariant, dichromatic reflection models, spectral imaging system, spectral image segmentation 1. Introduction Spectral reflectance observed from object surfaces pro- vides crucial information in computer vision and image analysis which include the essential problems of feature detection, image segmentation, and object recognition. 1) However, in real-world applications there are various illumination factors that can affect spectral images. During spectral image acquisition there are factors that affect the captured spectral information. Some of these factors are illuminant intensity, non-uniformity, specular highlight, shading, and surface geometry. Image representations invariant to these factors have been proposed for spectral images 2–5) and for color images 6–10) in several ways. Stokman and Gevers 2) proposed a method for edge classi- fication from spectral images. Their method aimed at detecting edges and assigning one of the types of shadow, highlight, and material edge. Montoliu et al. 3) proposed a spectral invariant representation for dielectric materials. This method used for edge detection of spectral images. However, most of those methods were constructed based on the dichromatic reflection model by Shafer. 11) This model assumes that an object surface is composed of inhomoge- neous dielectric material, and the 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 model-based method is valid for such limited materials as plastics and paints. 12–14) It should be noted that there are metallic objects in real-world scenes which cannot be described by the standard dichromatic reflection model. The present paper proposes a more general spectral invariant representation for obtaining reliable 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 and the extended dichromatic reflection model for metal. We show that the invariant formulas for spectral images of both artificial object and natural ones preserve surface-spectral reflectance information and are invariant to highlight, shadow, surface geometry, and illumination intensity. The overall performance of the proposed spectral invar- iant representation is examined in experiments using real- world objects including metals and dielectrics in detail. Experiments are done with real spectral images to examine the performance of the proposed method. The results show that the proposed representation is invariant to highlight, shadow, and object surface geometry, and effective for image segmentation. 2. Dichromatic Reflection Models The standard dichromatic reflection model 11) 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 ð;  Þ is a function of the wavelength  , ranging over a visible wavelength [400, 700 nm], and the geometric parameters , including the direction angles of the viewing angle and the phase angle. The standard dichro- matic reflection model describes the reflected light in the form Y ð;  Þ¼ C I ðÞL I ð Þþ C B ðÞL B ð Þ; ð1Þ where L I ð Þ and L B ð Þ are the spectral power distributions of the interface and body reflection components, respectively. The weights C I ðÞ and C B ðÞ are the geometric scale factors. The reflection model is also described in terms of spectral reflectance. Let Eð Þ be the spectral-power distribution of a uniform illumination. The spectral reflectance function defined as Sð;  Þ¼ Y ð;  Þ=Eð Þ, independent of illumi- nant, can be expressed as  E-mail address: ibrahim@graduate.chiba-u.jp OPTICAL REVIEW Vol. 18, No. 2 (2011) 231–236 231