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(JASTEG2000 website) and experimentally
evaluated. In the worst cases, it gets up to 35%-45%
embedding rate with approximately 2 dB of
distortion, and up to 30%-40% embedding rate with
less than 4 dB of distortion, when applied on
benchmarks coded at 0.5 bpp and 1.0 bpp,
respectively. These results outperform those
achieved by other existing methods. In particular, we
will show that our method produces much less post-
embedding growth, getting a considerably lower
distortion, than JPEG2000-BPCS.
2 JPEG2000 STANDARD
JPEG2000 is a powerful coding technique for still
images, standardized as ISO 15444 in 2001. It
achieves impressive compression ratios even holding
a good image quality, overcoming the classical
JPEG. The key of this performance are the wavelets,
which constitute, in JPEG2000, the counterpart of
the Discrete Cosine Transform (DCT) in JPEG.
Wavelets (S. G. Mallat, 1989) use complex base
functions, with some coarse features akin to sine
waves. They also contain some detailed features like
pulse codes, thereby creating a set of fuzzy pixels
with variable-sized features, as opposed to DCT's
one-size-fits-all sine waves. Other interesting
(sometimes innovative) features of JPEG2000 are:
- it offers both lossless and lossy coding;
- it allows of modifying and coding any region of
the image, directly working on the compressed data
stream;
- it introduces the concept of Region of Interest
(ROI) of an image;
- it allows a flexible bitstream ordering;
- it has an improved error resilience of the
codestream;
- it provides a localized random access into an
image;
- it grants an efficient and accurate rate control.
More in detail, the processing steps of JPEG2000
are:
(a) DC-Shifting
: This step is applied on the
components having only positive values. It shifts the
range of these components from [0, 2
n
-1] to [-2
n-1
+1,
2
n-1
], n being the number of bits used for that
component.
(b) Multi-Component Transform
: This step is
needed in case of color images. It un-correlates the
color components either into YUV space (in case of
lossless coding) or into YCbCr space (in case of
lossy coding).
(c) Discrete Wavelet Transform (DWT)
: DWT is
the cornerstone of JPEG2000. The best way to
represent a signal using wavelets is to scan the entire
image for the “mother wave” that best represents
that particular image. However, this “mother wave”
would have to be attached to the image data, thereby
increasing the size of the compressed file. Instead,
JPEG2000 adopts an universal mother wave ahead
of time, eliminating the need to send it along with
the file. These ones are LeGall 5/3 (for lossless
coding) and Daubechies 9/7 (for “lossy” coding).
(d) Quantization
: Scalar and uniform
quantization is applied on DWT coefficients. The
standard does not specify thresholds values, since
these ones can be decided by the user, basing on the
particular case. Anyway, the standard proposes a
method for determining them.
(e) ROI Scaling
: This is an optional functionality
in which, the wavelet coefficients related to regions
that the user has indicated as “relevant” are scaled
up. By this way, these Regions Of Interest (ROI)
gain quality during the next coding steps.
(f) EBCOT (Tier 1)
: In JPEG 2000, coding is
performed in two steps (tiers). In tier 1, the
quantized coefficients of each subband are
partitioned into codeblocks. These ones are
independently coded using an Embedded Bit-planes
Coding with Optimized Truncation (EBCOT).
(g) EBCOT (Tier2)
: Tier 2 optimally truncates
the bit-stream of each codeblock minimizing the
distortion due to the bit-rate constraint. Firstly,
candidate truncation points are selected in the
convex hull of the rate-distortion curve. Afterward,
when some codeblocks are collected, and a statistic
is available, the truncation point is selected among
these candidates in order to minimize the distortion.
In JPEG2000, this step produces information loss,
as well as quantization.
3 A STEGANOGRAPHIC
METHOD FOR JPEG2000
When steganography is applied to classical codecs
(e.g., JPEG), the embedding is generally cascaded to
the quantization, since this one is the main (and
often, the unique) lossy step. Nevertheless, this
approach is unsuitable for JPEG2000. In fact, in this
standard, the quantized coefficients can successively
be truncated in EBCOT Tier 2 in order to match a
particular bit rate, that the user could have required.
Therefore, steganography when applied to
JPEG2000 cover images needs a “down-coding/up-
decoding” scheme, as that one shown in Fig. 1. In it,
SECRYPT 2006 - INTERNATIONAL CONFERENCE ON SECURITY AND CRYPTOGRAPHY
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