Design and Implementation - Segmentation of Hand Bone for Bone Age Assessment - page 68

variable and the frequency of mode as responsive variable. The data is fitted to
normal distribution to ease visual comparison and explanation.
From the figure, it is observable that the overall number of pixels in origi-
nal image is shifted to the left as the iteration begins without exception. This
shifting is due to the diffusion indicating that the overall regions have been
smoothed and therefore the local variance is getting smaller. This smoothing
process, in fact is the main purpose of diffusion. The most important observa-
tion is that the number of the distribution of local variance; from Fig. 3.7 , it is
observable that the mode of the local variance is monotonically increasing as the
iterations begins until the fourth iterations (pink), and then it begins to drop at
fifth iterations. Beyond fifth iterations, the mode begins to fluctuate instead of
monotonically deceasing; this fluctuation is analogous to the concept of steady
state. The reason the decision is to halt the diffusion at third iteration instead of
fourth iteration in this scale selection is due to the reason that this small incre-
ment between the third and fourth is very likely attributable to the smoothing
of pertinent information such as edges in the image. These observations are not
randomly occurring only in certain hand bone radiographs, but occur with cer-
tainty in all hand bone radiographs in database without exception. We formulate
the solution in general terms in next paragraph and explain how this scheme ful-
fils the desired criteria.
Let the diffused image at T-iteration to be depicted as D(T) . We compute the
local variance of D(T) and depict it as V(T) . Then we compute the difference
between the mode of V(T) and V(T − 1) , if the difference is negative at iteration
T = m, then the scale selection is m − 1. This scale selection scheme conforms
to the mentioned criteria. First of all, it is simple and hence has excellent perfor-
mance in computational complexity and this fulfil the P2 criterion; then, it utilizes
prior knowledge that edge will be smoothed out if the iterations exceed a certain
number, and utilizes as well the local variance obtained in previous step of estab-
lishing the diffusion strength function; hence it fulfils the P4 criterion and in turn
fulfil the P2 criterion. Besides, the scale selection is autonomous and requires no
training procedure or user intervention which fulfils the P6 criterion. Lastly, it ful-
fils P10 criterion because it is adaptive to the image statistics instead of using any
rigid thresholding technique.
Gerig et al. [ 18 ] has made an analysis on the diffusion filter integration con-
stant, Δ t, and concludes that in two dimensional discrete implementations of eight
neighboring pixels, the constant range should be in between 0 and 1/7 to ensure
the stability. The more Δt is to zero, the better the integration approximates the
continuous case. Nevertheless, more iterations are required by the filter to diffuse
the image to a certain extend. This indicates that any number in between 0 and
1/7 will not affect significantly the resultant image as long as the iterations vary
accordingly. The value of the constant is set empirically as 1/7 in the implementa-
tion of the diffusion.
The flow chart of conventional anisotropic diffusion is presented in Fig. 3.8 .
A diffusion function has to be chosen, followed by determining a constant. This