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detector simply retrieves the LSB of pixelsâ€™ luminance values defined in the

predetermined pixel set. Since the luminance values are not significantly changed, the

marked workâ€™s fidelity remains relatively high. The capacity of such a scheme is large as

well since one bit of information can be stored for every pixel in the original work.

In another related spatial watermarking technique, a work is divided into n ordered,

disjoint subsets as shown in Figure 20. The embedding is done by ensuring that the parity

of the luminance LSBs for all the pixels in the nth set is equal to the nth bit of the message.

Katzenbeisserâ€™s and Petitcolasâ€™s (2000) work presents other spatial techniques based on

quantization and dithering.

In general, spatial techniques are only appropriate for fragile watermarks. Typical

operations such as compression or translation can completely destroy a spatial based

mark. If a watermarking systemâ€™s goal is to provide robust watermarks, it must generate

marks that are invariant to most common types of distortions.

40

Frequency domain techniques have been the subject of most current research. The

Discrete Fourier Transform (DFT) mentioned earlier in this paper provides the basis for

approaches of this nature. Unfortunately, DFT becomes problematic because its

calculation does not always result in real numbers. Usually, derived forms of the DFT

such as the Discrete Cosine Transform (DCT) are used in practical watermarking systems

(Katzenbeisser & Petitcolas, 2000). Marks embedded in the frequency domain tend to

survive common compression methods such as JPEG and MPEG. Since DFT magnitudes

are not affected by translation, watermarking techniques based on DFT algorithms also

tend to be immune to translation attacks. Since wavelets add the concept of spatial

location to frequency information, Discrete Wavelet Transform (DWT) systems add the

additional flexibility of embedding marks in favourable positions within a work.

Embedding a mark in the frequency domain is done by modifying the magnitude of

various frequency coefficients in the cover work (Nikolaidis & Pitas, 1999). Ã“ Ruanaidh

and Pan (1998) present an example of this based on the DCT. In other frequency domain-

based systems, such as the ring-shaped watermark approach (Nikolaidis & Pitas, 1999),

this is accomplished by creating defined relationships between certain coefficients after

calculating the frequency domain representation of an image.

More complicated systems address the more general problem of creating watermarks

invariant to all geometric attacks. The equation below is an affine transformation of the

41

point (xo, yo) to (x, y). It encapsulates the geometric operations of translation (via the [e

f] matrix), scaling (via the [a b c d] matrix) and rotation (via the [a b c d] matrix).

ï£®xï£¹ ï£®a ï£¹ï£®x ï£¹ ï£® ï£¹

b e

o

= +

ï£¯c ï£ºï£¯y ï£º ï£¯ ï£º

ï£¯yï£º d f

ï£° ï£» ï£° ï£»ï£° ï£» ï£° ï£»

o

One approach to solving this problem is to reverse any distortions performed on the

marked work before it is presented to the extraction mechanism. Various permutations of

the affine transformation parameters can be used to derive the undistorted work. Two

problems exist with this approach; complexity and the possibility of getting false

positives (Katzenbeisser & Petitcolas, 1999). Given the search space of six independent

variables, calculating every possible transform and running the result through an

extraction algorithm is computationally expensive. In addition, the larger the search

space, the more likely a transform will yield a valid but incorrect watermark. If the

marked cover work is known to have reference points, this fact can be used to

significantly reduce the size of the search space.

An alternative approach is to ensure that the watermarking system is invariant to all

possible affine transformations. As stated earlier, DFT schemes tend to be invariant to

translation, however, their watermarks can be destroyed by other geometric operations.

Watermarks embedded using the Mellin-Fourier Transform have been noted to be

invariant to affine transformations. This process involves a DFT transformation followed

by a log-polar mapping. The results of this operation are subsequently put through a

second DFT process to arrive in the invariant marking space. Ã“ Ruanaidh and Pun (1997)

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provide more detail about the Mellin-Fourier Transform in their works. This appears to

be an optimal solution however watermarks embedded in this space are not resistant to

lossy image compression (Ã“ Ruanaidh & Pun, 1997). In addition, the log-polar mapping

and its inverse performed during the Mellin-Fourier Transform degrade the fidelity of the

work.

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