Image Secret Sharing - Visual Cryptography and Secret Image Sharing - page 384

However, the Lagrange polynomial based image sharing has a potential
practical problem. It cannot yield too many shares. This is because the La-
grange polynomial curve with a high degree has severe oscillations once the
degree of the polynomial is greater than nine [16]. The consequence of this
oscillations phenomenon is that it is impossible to constrain the pixel values to
lie between 0 and 255. As a result, the resultant shares are not proper images
anymore as shown in Figure 14.4.

(a) Original image (b) Shares 1 (c) Shares 2

FIGURE 14.4
(See color insert.) The image sharing by using a high degree polynomial in-
terpolation in (a)-(c).

In Figure 14.4, the resultant image Figure 14.4(a) is the original secret im-
age, Figure 14.4(b) is the innocuous image and Figure 14.4(c) is the generated
share while the rest of the shares are all-white images. We use a polynomial
of degree 8 for generating Figure 14.4(c). Figure 14.4(c) has obvious color
overflow problems because some pixel values are more than 255 and some are
less than zero, hence the image quality is severely degraded. Note that this is
not a problem in Shamir's original secret sharing scheme because they con-
sider the secret to be shared as a binary integer, and thus a share can take on
any value. In our case, this binary integer has some constraints because they
denote image pixel values. In order to overcome this serious limitation, it is
obvious that some form of a piece-wise polynomial interpolation is required
in order to bound the degree of the polynomial and thus constrain the oscilla-
tions. We have developed a new color image sharing scheme based on moving
lines, which is a rational implicit curve.

14.3.3 Color Image Sharing Based on Moving Lines
It is well known that the equation of a line in the homogenous form in pro-
jective geometry [22] is:

aX + bY + cW = 0 (14.7)