This paper explores new aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and v, where u holds the geometrical information and v holds the textural information. Our aim is to provide the necessary variational tools and suggest the suitable functional spaces to extract specific types of textures. Our modeling uses the total-variation semi-norm for extracting the structural part and a new tunable norm, presented here for the first time, based on Gabor functions, for the textural part. A way to select the splitting parameter based on the orthogonality of structure and texture is also suggested.