A Stochastic Approach to Content Adaptive Digital Image Watermarking by cali0998

VIEWS: 62 PAGES: 19

									      A Stochastic Approach to
          Content Adaptive
     Digital Image Watermarking

S. Voloshynovskiy*#, A. Herrigel#, N. Baumgaertner#, T. Pun*


           *Computer Science Department (CUI)
           University of Geneva, Switzerland
           Contact: http://cuiwww.unige.ch/~vision


           #
               Digital Copyright Technologies (DCT)
           Stauffacher-Strasse 149
           CH-8004 Zurich, Switzerland
                                                        1




Content

1. Introduction
1.1. Why deal with content adaptive watermarking
1.2. Empirical perceptual models used in watermarking
     applications

2. Image watermarking – MAP formulation
2.1. Stochastic models of watermark and cover image
2.2. Generalized solution of MAP – image denoising
2.3. Image denoising and lossy compression

3. Adaptive image watermarking
3.1. Noise visibility function (NVF)
3.2. NVF based on non-stationary Gaussian model
3.3. NVF based on stationary GG model
3.4. Content adaptive watermarking

4. Results of computer simulation

5. Conclusions
                                                                           2



  1. Introduction

     1.1. Why deal with content adaptive watermarking


     3 linked criteria for watermarking




                                Visibility V




                Capacity C                     Robustness R




                     Why stochastic approach


                             Perceptual
                               mask




                     Stochastic Image modeling


 Image                                                         Detector
                    As a model of channel noise               robustness
Capacity
                                                                                 3


       1.2. Empirical perceptual models used in watermarking
            applications

                 Transfer modulation function of HVS
                 J. Ruanaidh, T. Pun (Signal Processing, No3, 1998) - CUI

                 Luminance sensitivity
                 M. Kutter (Proc. SPIE, 1998) - EPFL

                 Luminance and texture masking - L&T
                 M. Kankanhalli, R. Ramakrishman (ACM Multimedia, 1998)
                 F. Bartolini, M. Barni, V. Cappelini, A. Piva (ICIP, 1998)
                 use results of perceptual image compression (N. Jayant, J.
                 Johnton, R. Safranek, Proc. IEEE, 1993)
          CUI                       EPFL                         L&T
Basic Idea:                 Basic Idea:                 Basic Idea:
Embed in middle band        Embed according to          Embed according to
as trade-off between        Luminance Function          Lumainance      and
visibility & distortions                                Texture masking (as
(lossy compression)                                     edge detection)




                             Basic drawbacks:
It does not take into       Watermark strengh is        -A set of empirically
account spatial             different only for darker   adjusted parameters
anisotropy of image         and lighter regions.        is required
spectra and as the          Pattern masking             -No closed form
result is not adaptive to   properties are not taken    expression for optimal
image content               into account                detector design and
                                                        capacity estimation
                                                                          4



            Complete content adaptive model should include:

                Luminance masking

                Contrast sensitivity

                Texture masking including spatial anisotropy (and MTF)

                Color masking

    We concentrate on texture masking function in coordinate
    domain with further extension to wavelet transform and
    steerable pyramids to incorporate spatial anisotropy


                The main advantages of the proposed approach:

    - closed form solution

    - automatically adjusted to wide set of images

    - simple computation in comparison with the theoretical models
    of HVS

    - a few model parameters needed to be estimated

    - criterion is stochastically based

    - takes into account statistics of cover image and distribution of
    watermark

    - makes possible to design optimal watermark detector and
    resolve 3 contradicting requirements: Visibility-Robustness-
    Capacity

- opens new opportunities to perceptual image quality criterion development
                                                                          5



          2. Image watermarking – MAP formulation

                       Two problems:
                                      - watermark estimation/detection
                                      - watermark removal

Generalized model
                    Notations:

                        y: stego image;
                        x: original (cover) image;
                        n: watermark (considered as noise-like signal)

                                      y = x+n
                      Maximum A Posteriori (MAP) estimation

                    Watermark estimation

                      n = argmax{ln p X ( y ¦ n ) + ln pn (n )}
                      ˆ                       ~            ~
                            ~
                            n ∈ℜ N




                    Watermark removal (or ≡ image estimation/denoising)
                       x = arg max{ln pn ( y ¦ ~ ) + ln p X (~ )}
                       ˆ                       x             x
                             ~ ∈ℜ N
                             x
                                                                                       6




y                              ˆ
                               x              ˆ
                                              n     Robust              Message
                                      -
           Estimator                      ⊕       detector and
                                                  demodulator




                                                      Key

    2.1. Stochastic models of watermark and cover image

2.1.1. Watermark models: r.v. ind. ident. distributed samples
                                - p n (n ) ∝ i.i.d . N (0, Iσ n ) for spread spectrum wm
                                                              2



                                - pn (n ) ∝ i.i.d . GGD       for binary wm + Masking
2.1.2. Cover image models: r.v. ind. ident. distributed samples

                                    AR formulation

                                   x = Ax + ε = x + ε

                            ε = x − x = (I − A)x ≡ Cx

    A - neighborhod set; x - predicted value; ε - residual (i.i.d.)
                A – LP operator; C – HP operator



         (Homogeneous)                                  (Inhomogeneous)


       Globally i.i.d. GGD                           Locally i.i.d. Gaussian


           σ x2
                   0       L  0                          σ x1
                                                             2
                                                                    0     L  0 
                                                                              
             0     σ   2
                           0  M                            0     σ x22   0  M 
      Rx =            x
                                                     Rx = 
            M     0       O 0                            M       0     O 0 
                                                                           2 
            0
                  0       0 σ x2 
                                                          0
                                                                   0     0 σ xN 
                                                                                 
                                                                                             9



        2.2. Generalized solution of MAP – image denoising

              Consider problem of image estimation in AWGN.

                                                       (
                                    p n (n ) ∝ i.i.d . N 0, Iσ n
                                                               2
                                                                )
                 p X (x ) ∝ i.i.d . GG (x , R x ) or p X (x ) ∝ i.i.d . N (x , R x )




                                      Known solutions



          Globally i.i.d. GGD                              Locally i.i.d. Gaussian

        (non-convex function)                                 (convex function)

General close form solution does not exist!!!                 Close form solution


           Iterative solutions                                Adaptive Wiener (Lee filter)

                         ICM
                                                      Reweighted
              Stoch.& determ. anneling                Least
              Graduated nonconvexity                  Squares
                                                      (RLS)
                       ARTUR

           Particular close form
           solution in wavelet domain

                 Shrinkage methods
                                                                                       10



                            Generalized solution

                                1               
                   x = arg min  2 y − ~ + ρ (r )
                                         2
                   ˆ                   x
                         ~∈ℜ N
                         x      2σ n            

                                    RLS solution

                                 1                   2                2
               x k +1 = arg min  2 y − ~ k
               ˆ                        x                 + w k +1 r k 
                          ~∈ℜ N
                          x      2σ n                                  

              Iterative solution: x k → w k +1 → x k +1
                                  ˆ              ˆ

Notations for k-iteration:
           ρ (r ) = [η (γ ) ⋅ r ]
                                γ
                                               - GGD energy function
              x−x
           r=                                  - residual
              σx
                 ′
           ρ (r ) = γ [η (γ )]
                              γ         r
                                        2 −γ   - derivative from GGD energy function
                                    r
                       ′
                 ρ (r )
               1
           w=                 - weighting function
               r
           with γ estimated using moment matching method

                      w is constant for k-iteration:

    RLS solution in the form of the adaptive Wiener filter

                             wσ n2
                                            σx2
                    x=
                    ˆ                 x+            y
                           wσ n + σ x
                              2     2
                                         wσ n + σ x
                                            2     2



              RLS solution in the form of Lee filter

                                           σx2
                         x=x+
                         ˆ                         (y − x )
                                        wσ n + σ x
                                           2     2
                                                                        11



Comparative analysis of different distributions in RLS estimate

     Estimated, Gaussian and           Corresponding energy functions
     Laplacian pdfs




  Derivatives from energy functions   Weighting functions
                                                   12



2.3. Image denoising and lossy compression

                    y = x+n
                  Assumptions:

                                 (
             p n (n ) ∝ i.i.d . N 0, Iσ n
                                        2
                                            )
             p X (x ) ∝ i.i.d . GG (x , R x )




      RLS solution for fixed iteration:

                        σx2
         x=x+
         ˆ                      (y − x )
                     wσ n + σ x
                        2     2




    Shrinkage solution (known in wavelet domain)

   x = x + max (0, y − x − T )sign( y − x )
   ˆ

        σn
         2
   T=        2 - for Laplacian image prior
        σx
                                                                      13




             Denoising – shrinkage
                   function

          ˆ
          x HF                                         ˆ
                                                       x HF




                         x−x                                        x−x
                    r=                                         r=
                         σx                                         σx




Hard-thresholding                          Soft-thresholding


                               ˆ
                               x HF



                                           γ =1
                                             γ = 0.7


                                           x−x
                                      r=
                                           σx




                         GGD prior
                                                               14



                    Denoising – Lossy compression


    As watermark estimation and corresponding attacks




 Denoising – shrinkage               Quantization




                             x−x
                        r=                               x−x
                             σx                     r=
                                                         σx




Laplacian image prior
                                              15




    3. Adaptive image watermarking

    3.1. Noise visibility function (NVF)


     Consider term from RLS solution
    (plays the role of texture masking)


                             σx 2
                    b :=
                           wσ n + σ x
                              2     2




 In our approach we define NVF function as

                                  wσ n2
          NVF = 1 − b =
                                wσ n + σ x
                                   2     2




      (
 n ∝ N 0, Iσn
            2
                )                       NVF
                      Perceptual
                        model
    σn = 1



Behavior:
  - flat regions:               NVF → 1
  - edges and textured regions: NVF → 0
                                                                       17




    (Homogeneous)                          (Inhomogeneous)


                            Advantages

Only two parameters of the
model should be estimated                  Convex energy function


                           Disadvantages
Concave energy function              - Local variance is estimated as ML
(NVF is ‚noisy‘,                     under locally i.i.d assumption that is
 but with good resolution)           not true for large windows, but small
                                          windows – bias in estimate
                                             - NVF is oversmooth




                          Original image




                              NVF
                                                                                  20



                        4. Results of computer simulation

Classical PSNR

                                                           2
                                                    255
                                 PSNR = 10 log 10
                                                    x− y
                                                           2




Weighted PSNR
                                                               2
                                                     255
                               wPSNR = 10 log 10           2
                                                    x− y   NVF




                        Barbara image: Gaussian distributed watermark
      Method of embedding                            Watermark strength
Non-adaptive                                5.0                    10.0   15.0
Adaptive non-stationary Gaussian NVF        6.5                    15.0   23.0
Adaptive stationary GG NVF                  8.0                    20.0   31.0
Adaptive Kankanhalli                        9.8                    19.6   30.4
           PSNR, dB                         34.1                   28.3   24.7



        Examples of embedding of binary watermark for the fixed PSNR=24.7dB
                            (in adaptive case Sf=3 for all embedding)


   wPSNR,dB:     26.45                          27.87                     29.28




     Strength:   15.0                     Se:       23.0                    31
                                                                                      21




            Examples of embedding of binary watermark for the fixed PSNR
                           (in adaptive case Sf=3 for all embedding)

            Non-adaptive                 adaptive nG                   adaptive sGG
PSNR, dB:




34.10




                  5.0                          6.5                          8.0




28.30




                  10.0                         15.0                         20.0




24.7




                  15.0                          23.0                         31.0
                                                                        22




        Examples of Gaussian watermark embedding for the fixed PSNR
                                (in adaptive case Sf=3)

             Kankanhalli’s method               Adaptive sGG




PSNR: 28.3 dB Strength: 19.6                              20.0




PSNR: 28.67 dB Strength: 21.7                                    21.0
                                                                         23



                             4. Conclusions

- stochastic approach to content adaptive watermarking is developed

- developed models require small amount of parameters to be estimated

- results show superior performance in comparison with the other known
  empirical adaptive approaches with respect to both visual quality and
  watermark strength

- the similarity between denoising and compression as the watermark
  removal attacks is shown

- models make possible to develop the optimal detector for the watermark
  and to estimate capacity of the images


Future work:

- new ML and MAP solutions for local image variance estimation

- design of the optimized robust watermark detector for the stationary non-
  Gaussian and non-stationary Gaussian channels (considered image
  models) and corresponded distribution of errors in watermark estimation

- unified approach to the compression/denoising attack analysis

								
To top