Cryptography Reference
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traces of both shadow images. One can recognize a logo as well as shadow
images at the same time.
Myodo et al. extended this approach to be able to handle a photograph as
a secret image [25]. It can reconstruct a secret photograph image with very
little or no trace of shadow images by properly adjusting tones of images. This
aspect of tone adjustment will be discussed in Section 4.4.7.
4.4.6 Simultaneous Iterative Search
This type of method was proposed by Wu et al. in 2004 [42]. It can handle
three different photographs as two shadow images and a secret image without
pixel expansion, m = 1. This method consists of two major steps, tone ad-
justment and simultaneous iterative search. It first adjusts the tones of three
input images to satisfy a condition on relative differences of shadow and se-
cret images. Then it simultaneously searches three halftoned images. Since
[42] contains very little explanation about the two steps, it is hard to know
the exact algorithm. But it would take a certain amount of time to obtain a
result if it process is images in a brute-force manner.
4.4.7 Tone Adjustment
This type of approach attempts to improve image quality with image process-
ing technique. It tries to enhance contrasts, namely, dynamic ranges, of the
resulting images as much as possible. The conventional visual cryptography
studies consider relative differences, S and R , which represent a limitation
of possible pixel values. However, the pixel values of shadow and secret images
are actually limited by lower and upper limits, dynamic ranges, and there is
a certain interaction among them.
Nakajima and Yamaguchi precisely examined the interactions of pixel val-
ues in (2; 2) EVCS [28]. There exist constraints among the pixel values of
three corresponding pixels in shadow and secret images. Let us call the three
corresponding pixels a triplet. The constraints among values of a triplet are
represented as below:
o R 2 [max (o 1 ;o 2 ); min (o 1 + o 2 ; 1)] ;
(4.1)
where o R denotes pixel opacity of reconstructed secret image and o 1 and o 2
denote pixel opacities of resulting shadow images. 3 This expression indicates
that any reconstructed pixel must be equal to or more opaque than the most
opaque corresponding shadow pixel, max (o 1 ;o 2 ). It also indicates that the
reconstructed pixel must be equal to or less opaque than the sum of opacities
of corresponding shadow pixels, o 1 + o 2 .
3 In [28], Nakajima and Yamaguchi discussed pixel transparency instead of opacity. How-
ever, as we already indicated, people usually use 0 for a transparent (white) pixel and 1 for
an opaque (black) pixel in visual cryptography studies. Thus, here we consider pixel opacity
rather than pixel transparency.
 
 
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