Pixel location
in each mask
Image
\label{image-1}
Figure 4.7: Implementation of Prewitt operators using a 3 × 3 2–D mask
First, the gradient value at each pixel is conventionally given as
s5 = 1 . 1 1
1 1 1 1 1 1 1
. . .
2 . 2 . 2 (z7 + z8) + z9. − 2 . 2 (z1 + z2) + z3.. + 2 2 2 (z3 + z6) + z9 − 2 2 (z1 + z4) + z7
= 1
8 (|z7 + z8 + 2z9 − z1 − z2 − 2z3| + |z3 + z6 + 2z9 − z1 − z4 − 2z7|) ,
(4.6)
where s5 is the pixel value at center location of the processed image. We consider the cofficients 1/2 so as to ensure the pixel values being into the range of the grayscale pixel value [0, 255].
0.5 0.5
Figure 4.8: Design of Prewitt operators using SC
Now, we implement this operator by using SC. It can be shown in Fig.4.8 that the architecture of the Prewitt operator in Stochastic domain. In this configuration, Pini = zi/256 represents SN in unipolar form which encodes the pixel value at location i, and Pout5 perform the output value in the
same format of center pixel. We assume that Bini = 2Pini − 1, Bout5 = 2Pout5 − 1 being the bipolar
representation of pixel values at the input and the output, respectively. Therefore, according to this
design, the output is given as
1
Bout5 = 8 (|Bin7 + Bin8 + 2Bin9 − Bin1 − Bin2 − Bin3 | + |Bin3 + Bin6 + 2Bin9 − Bin1 − Bin4 − 2Bin7 |)
(4.7)

(a) (b) (c) (d)
Figure 4.9: Simulation results of edge detection based on Prewitt operator on (a) original image using: (b) conventional approach and stochastic computing of lengths (c) 2048 bits and (d) 4096 bits
By combining (4.6), (4.7) and the above assumptions, it can be easily deduced as
s5 1
Pout5 = 512 − 2 (4.8)
Hence, it is similar to the result obtained in case of Roberts operator, the gradient value at center pixel can be received by counting the number of bit beging logic 1 at the output bit sequence.
It can be shown in Fig.4.9 and Fig.4.6 that the Prewitt operator provides an better result of edge detection compared that of the Roberts operator in conventional approach. Meanwhile, it seems to be converse in stochastic processing. This fact can be explained by the inaccuracy in multi–stage SC circuit. Because it is evident to be seen that the Prewitt operator require more stages than the Robert operator, and when the signals pass more stages, their PDFs becomes different from the expected ones (the uniformly distributed form at the output is required). This issue will be studied as an important problem of SC in the future.