Basic Message Coding 《 Digital Watermarking: Principles & Practice 》 Chapter 3 Multimedia Security

2 Multi-Message Watermarking Mapping messages into watermarking vectors –Analogous to mapping between messages and transmitted signals Source coding –Mapping messages into sequences of symbols Modulation –Mapping sequences of symbols into physical signals

3 Direct Message Coding Assign a unique, predefined message mark to represent each message Let the set of messages to be denoted M and the number of message in the set |M|. –We design a set of |M| message marks, W, each of which is associated with a message. –Encoding To watermark a work with the message m, the embedder simply embeds message mark W[m] –Decoding Detection values are computed for each of the |M] message marks. the most likely message is the one corresponding to the message mark with the highest detection value.

4 Code Separation How to design the message marks? –The message marks should be chosen to have good behavior with respect to predictable false-alarm rate, fidelity, robustness… –One message mark shall not be confused with another Marks should be far apart from one another in marking space If the corruption is too serious, the watermark might be erroneously decoded as a different message Message mark Channel Corrupted message mark Decoding ?

5 Message Coding as an Optimization Problem Designing an optimal set of |M| N-dimensional message marks is equivalent to the problem of placing |M| points on the surface of an N- dimensional sphere such that the distance between the two closet points is maximized

6 Randomly Generated Codes (1/2) If the number of messages is large compared to the dimensionality of marking space, randomly generated codes usually result in good code separation Distribution of average angles between three message vectors in randomly generated, three-dimensional codes

7 Randomly Generated Codes (2/2) If the dimension of marking space is large, randomly generated message vectors are likely to be orthogonal to one another Distribution of average angles between three message vectors in randomly generated, 256-dimensional codes

8 Problems of Direct Message Coding Direct message coding does not scale well –The detector must compute the detection value for each of |M| reference marks –e.g. If 16 bits of information (65,536 possibilities) is embedded, the detector must compute detection value for each of 65,536 reference marks. The problem can be dramatically reduced by first by representing each message with a sequence of symbols, drawn from an alphabet

9 Multi-Symbol Message Coding If each message is represented with a sequence of L separate symbols, drawn from an alphabet of size |A|, we can represent up to |A| L different messages. –E.g. |A|=4, message length |L|=8 Only 32 comparisons is required

10 Time and Space Division Multiplexing The Work is divided into disjoint regions, either in space or time, and embed a reference mark for one symbol into each part. –The message mark is constructed by concatenating several reference marks –e.g. To embed 4 symbols into an image of dimensions w pixels x h pixels, we would use reference marks of dimensions w/2 x h/2

11 Frequency Division Multiplexing The Work is divided into disjoint bands in the frequency domain, and embed a reference mark for one symbol in each –The message mark is constructed by adding together several reference marks of different frequencies

12 Code Division Multiplexing Several uncorrelated reference marks embedded into the same work have no effect on one another in a linear correlation system –Code division multiplexing in spread spectrum communication

13 Code Division Multiplexing (cont.) If we represent a message with sequences of L symbols drawn from an alphabet of size |A|, we define a set of Lx|A| reference marks, W AL All reference marks added to W m should be as orthogonal as possible –W AL [i,a] ‧ W AL [j,b] =0,i≠j –W AL [i,a] and W AL [j,b] must be maximally distinguishable –We may add W AL [1,3] and W AL [2,1] but we would never embed both W AL [1,3] and W AL [1,1] In a given location Example: –|A|=4, L=5 –Represent the message sequence 3, 1, 4, 2, 3 –W m =W AL [1,3]+W AL [2,1]+ W AL [3,4]+W AL [4,2]+W AL [5,3] W AL index Symbol number WmWm

14 Equivalence of Code Division Multiplexing to Other Approaches Viewing time division multiplexing as special cases of code division multiplexing Time division multiplexing Code Division Multiplexing

15 Example: Simple Eight-bit Watermarks (1/2) Mapping a message into bit stream A single reference pattern w ri, i=1 to 8, for each bit location i is used. w ri is generated pseudo-randomly according to a given seed and normalized to have zero mean. For embedded watermark pattern, –w mi =w ri if m[i]=1, w mi =-w ri if m[i]=0 –w tmp =∑w mi –w m =w tmp /s wtmp

16 Example: Simple Eight-bit Watermarks (2/2) Embedding –c w =c 0 +aw m Detection –The detector correlates the received image c against each of the eight reference pattern, and uses the sign of each correlation to determine the most likely value for the corresponding bit.

17 Problem with Simple Multi-Symbol Messages Example: (L=3, dimensionality of marking space=N) –W 312 =W AL [1,3]+W AL [2,1]+W AL [3,2] –W 314 =W AL [1,3]+W AL [2,1]+W AL [3,4] –W 312 ． W 314 = …=N The smallest possible inner product between two message marks that differ in h symbols is N(L-2h) –L-h: inner product for # of same symbols –h: inner product for # of differences As L increases, the inner product above increases, the angle between two vectors decreases, the message marks of the closet pair become progressively similar