In order to further improve the effectiveness of image processing, it is necessary that an efficient invariant representation is stable to deformation applied to images. This motivates the study of image representations defining an Euclidean metric stable to these deformation. This paper mainly focuses on two aspects. On the one hand, in this paper,two properties of expected scattering and averaged scattering, i.e., Lipschitz continuity and translation invariance, are proved in detail. These properties support that excepted scattering and averaged scattering are invariant, stable and informative representations. On the other hand, the issue of texture classification based on expected scattering and averaged scattering has been analyzed respectively in this study. Energy features, which are based on expected scattering and averaged scattering, are calculated and used for classification.Experimental results show that starting with the seventh feature, the two approaches can achieve good performance in texture image classification.