This paper presents solutions to the local patch based Active Content Fingerprint (aCFP) with linear modulation, general linear feature map and convex constraints on the properties of the local feature descriptor. A direct approximation of the linear feature map such that the image distortion is as small as possible and the approximate linear feature map is as close as possible to the original map is proposed. Then an explicit regularization of the trade-off between the modulation distortion and the robustness of the local feature is introduced trough a novel problem formulation. A computer simulation using local image patches, extracted from publicly available data set is provided, demonstrating the advantages under: additive white Gaussian noise (AWGN), lossy JPEG compression and projective geometrical transform distortions.