The present invention relates to an apparatus for multi-material decomposition of an object. Providing (310) at least one image of the object is described. The at least one image is derived from at least one spectral X-ray image of the object, and the at least one image includes a photoelectric total attenuation coefficient image and a Compton scattered total attenuation coefficient image. A plurality of photoelectric decay coefficients are provided for a plurality of materials (320), and each photoelectric decay coefficient is associated with a corresponding material. A plurality of Compton scattering attenuation coefficients are provided for the plurality of substances (330), and each Compton scattering attenuation coefficient is associated with a corresponding substance. A total volume constraint at an image position in the at least one image is set as a function of the sum of individual volumes of the plurality of materials at the image position (340). A volume fraction of the plurality of materials at the image location is determined according to a global function (350), the global function: photoelectric total attenuation coefficient at the image location taken from the photoelectric total attenuation coefficient image, Compton Compton scattering total attenuation coefficient at the image location taken from the scattered total attenuation coefficient image； the plurality of photoelectric attenuation coefficients for the plurality of substances； the plurality of Compton scattering attenuation coefficients for the plurality of substances； and the total volume Includes constraints. Data representing the volume fraction of the plurality of substances is output (360).本発明は、オブジェクトの複数物質分解のための装置に関する。オブジェクトの少なくとも一つの画像を提供する（３１０）ことが記載される。前記少なくとも一つの画像は、オブジェクトの少なくとも一つの分光X線画像から導出され、前記少なくとも一つの画像は、光電全減衰係数画像およびコンプトン散乱全減衰係数画像を含む。複数の物質について複数の光電減衰係数が提供され（３２０）、各光電減衰係数は対応する物質に関連付けられている。前記複数の物質について複数のコンプトン散乱減衰係数が提供され（３３０）、各コンプトン散乱減衰係数は対応する物質に関連付けられている。前記少なくとも一つの画像におけるある画像位置における全体積制約
