687Journal of Real-Time Image Processing (2019) 16:685–695
1 3
of each block may be modified in the data embedding pro-
cedure, for each
,
𝜎
can be recovered as
p
𝜎
itself.
Furthermore, the smallest pixel value
𝜎
in each block
can also be modified (either decreased by 1 or unchanged) to
embed data. The similar data embedding and extraction pro-
cedures are omitted here.
2.2 Improved PVO‑based RDH [35]
In the original PVO-based RDH method, the blocks with
are not utilized to carry data. However, these blocks
are usually smooth and suitable for reversible embedding.
Based on this consideration, to take the advantage of the
blocks with
, an improved PVO-based method is
proposed by Peng etal. [35].
For the data embedding of this method, first, for a given
block with sorted values
𝜎
𝜎
, instead of comput-
ing the prediction-error
in (1) as the original PVO-based
method does, it is redefined as follows considering the order
of
and
Clearly, one can verify that the redefined prediction
error satisfies
≥
if
and
if
. That is to say, the prediction error defined
in this way is ranged from
to
. For example, for the
Lena image, the histogram of the redefined prediction error
is shown in Fig.2. This histogram is a Laplacian-like
distribution centered at 0 with two sides decay. Then, the
bins 0 and
are expanded for data embedding. More spe-
cifically,
is modified to derive the marked prediction-
error
in the following way:
(6)
max
=
p
𝜎(n)
− p
𝜎(n−1)
, if 𝜎(n) >𝜎(n − 1)
p
𝜎
− p
𝜎
, if 𝜎(n) <𝜎(n − 1)
(7)
max
=
⎪
max
max
d
max
− b, if d
max
=−1
d
max
+ 1, if d
max
≥ 1
−
≤ −
where
is a to-be-embedded data bit. Accordingly,
the largest pixel value
𝜎
is modified as:
to derive the marked pixel value.
For this improved method, a key issue is that, unlike other
expansion-shifting based RDH methods, the expansion bins
cannot be arbitrarily selected. To guarantee the reversibility,
the sign of each prediction error (i.e., “
” or “
”) should
not be changed after data embedding.
Similar to [34], in each block, only the largest pixel value
𝜎
is either increased by 1 or unchanged, while other
pixel values remain unchanged. The PVO of each block is
unchanged as well and, thus, the recovery and extraction
process can be conducted accordingly. Specifically, for the
decoder, the marked prediction-error
is first computed
for a marked block with sorted values
p
𝜎
, ...,
p
𝜎
as
follows:
Then, recover the original pixel value
𝜎
as:
In addition, for each
,
𝜎
is recovered as
𝜎
itself. And, the embedded data bit is 0 if
∈{0, −1
or
1 if
∈{
−
.
Besides, in this method, the smallest pixel value
𝜎
of
each block is also modified for data embedding, by consider-
ing the prediction error defined as:
One can verify that
≥
if
, and
if
. The histogram of
is also a Laplacian-like
distribution centered at 0 with two sides decay. For brevity,
the similar data embedding and extraction procedures by
modifying the smallest pixel value are omitted here and the
details can be found in [35].
2.3 Pairwise PEE [18]
To better utilize the image redundant and improve the
conventional PEE-based reversible embedding, the so-
called pairwise PEE is proposed in [18] by modifying the
2D prediction-error histogram for data embedding. In this
method, the prediction errors are jointed into pairs to gener-
ate a 2D prediction-error histogram and, based on a specifi-
cally designed 2D mapping, it aims to reduce the embedding
(8)
𝜎
= p
𝜎
+
d
(9)
max
=
�p
𝜎(n)
− �p
𝜎(n−1)
, if 𝜎(n) >𝜎(n − 1)
�p
𝜎
− �p
𝜎
, if 𝜎(n) <𝜎(n − 1)
(10)
𝜎(n)
=
p
𝜎(n)
, if
d
max
∈{0, −1}
p
𝜎
− 1, otherwise
(11)
min
=
p
𝜎(2)
− p
𝜎(1)
, if 𝜎(2) >𝜎(1)
p
𝜎
− p
𝜎
, if 𝜎(2) <𝜎(1)
Fig. 2 Histogram of
defined in (6), for the standard
sized gray-scale image Lena with block size of