Abstract—Use of digital multimedia data on internet is common due to huge number of applications, websites and especially social media websites. Watermarking is a technique that provides copyright protection of digital assets. Singular Value Decomposition (SVD), Discrete Wavelet Transform (DWT), Thin SVD, Compact SVD and Hybrid DWT-SVD are used for watermarking as they are robust and imperceptible techniques. These techniques can’t approximate the complexity and capacity of watermarking because left singular matrix, diagonal matrix and right singular matrix returned by SVD, Thin SVD and Compact SVD have a defined order that depends on size of the input image. In this article, watermarking technique for images is proposed to improve robustness and imperceptibility, to approximate the complexity and capacity of the watermarking. Hybrid DWT and Truncated SVD (TSVD) are used for watermarking and the watermark is embedded on singular values of diagonal matrix of TSVD. TSVD requires a parameter for approximation and return approximated diagonal matrix according to that parameter. Comparisons among DWT-SVD, DWT-Thin SVD, DWT-Compact SVD and DWT-TSVD show that Hybrid DWT-TSVD for watermarking outperform imperceptibility, robustness, capacity and complexity. Keywords—Singular Value Decomposition; Thin SVD; Compact SVD; Truncated SVD, Discrete Wavelet Transform; Normalized Coefficients (NC); Peak Signal to Noise Ratio (PSNR) I. INTRODUCTION IGITAL data on Internet is common now days. Everyone has its own digital data (images, videos and audios). Due to social media sites and other sites, everyone is sharing digital data. It is easy to copy shared digital data on internet that can be used for profit or other purpose without getting authentication and copyrights. Watermarking is used to prove ownership of digital assets. There are many secure methods for watermarking. Digital watermark can be embedded in two domains: transform domain and spatial domain. In spatial domain, watermark is inserted by changing the pixel values of host image. These techniques are easy to implement but not robust for different kind of signal processing attacks [1] [2]. In transform domain, transform coefficients are changed to embed watermark. DWT, Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT) and SVD are various transform methods. These techniques are complex but robust against different kind of signal processing attacks [1-19]. Discrete Wavelet Transform (DWT) is a strong multi resolution technique for decomposition and it is consistent with the human eye’s visual perception process [6] [14]. DWT has temporal resolutions properties than other transforms that can capture information of both location and time of a signal [7]. DWT divides an image into four bands: one as low pass band (LL) and others as high pass bands (diagonal HH, horizontal HL and vertical LH) as shown in Fig. 1. Three-level DWT is shown in Fig. 1. LL is a low frequency band where any change will be visible. Rather than inserting watermark in this band, it is decomposed into sub bands. HL and LH are intermediate frequency bands and contain edges information. Watermark is inserted in these bands. HH is diagonal component. SVD of a real matrix ‘A’ of order mxn is represented as =ܣ  . (1) Where U is mxm left singular vector matrix, S is mxn diagonal matrix with singular values and V is nxn right singular vector matrix. Here both the matrices U and V are orthogonal matrices [1] [3]. Thin SVD of a real matrix ‘A’ of order mxn and n << m is represented as =ܣ ௡  ௡   . (2) Only the n column vectors of U corresponding to the row vectors of   [13] [9].  ௡ is mxn,  ௡ is nxn diagonal and V is nxn. Compact SVD of a real matrix ‘A’ of order mxn, rank r and r << n is represented as =ܣ ௥  ௥  ௥  . (3)  ௥ is mxr,  ௥ is rxr diagonal and  ௥  is rxn . Random Approximation of Capacity of watermarking using Hybrid DWT and Truncated SVD Author: Hafiz Muhammad Mudassir (hmmudassir82@gmail.com), Co-Authors: Basheer Ahmad (drbasheer@iqraisb.edu.pk), Usman Hashmi (usman16288@yahoo.com) and Aihab Khan (aihabkhan@yahoo.com) IQRA University Islamabad Pakistan D International Journal of Computer Science and Information Security (IJCSIS), Vol. 14, No. 11, November 2016 476 https://sites.google.com/site/ijcsis/ ISSN 1947-5500