Transition Bit String (TBS)

For any applications in digital image processing, we offer a very useful feature index extracted from two-dimensional signals, making it the desired descriptions of texture patterns. This defined function TBS, can be taken as a projection mapping from a 2-d input signal to a 1-d output signal. For any different kinds of combinations of feature series, the characteristic function TBS could be defined as (16-1) or (16-2)

Input (Texture patterns)

Query Function

no

Output (Classified Textures) Pre-Classification

Mechanism

Decision Engine yes

no Judge if the

number of clusters was given.

yes

Thresholding

Cluster Decision yes

no Feature Extraction

Mechanism

depending on the type of input signals. Equation (16) can also be expressed in terms where TBS indicates the sum or union of differently-combined feature curves after scanning the original 2-d input signals in some orientation k over its axis x , while ^k n is the total number of feature curves. Here we use the superscript k to avoid f

confusion with the overall domain of axis x. The domain of TBS should be the unions of x with the numerical value k from 1 to^k n . If the summation in (16-1) is replaced _f by the “OR” operation, TBS stands for the union of feature series in some sense and (16-2) will replace (16-1). Whether to select (16-1) or (16-2) is usually determined by the sensibility in orientations of texture patterns and output signals. Apparently, (16-2) would be more applicable to the applications of CNN’s since we simulate the CNN outputs in a more logical form, which also keeps more information from the extracted feature curve. The derivative equation (17) kicks off the normalization of histogram by dividing the value n, the total number pixels in the images of CNN outputs, to keep from the disturbance of various image sizes. Also, the maximum value x in the ^k

orientation k indicates the total number of continuous strings listed in the above-mentioned equations and here a string is referred to as a digital image with its corresponding pixels.

In addition, equation (18) describes the kernel part of this characteristic function where )n_k(x^k represents the length of the current continuous string for the steady state signals (y_∞). Andx^k(y_∞)⁽ⁱ⁾ means the transformed signals in orientation k fromy_∞ in location i. In our application, we use the output of CNN’s for this kind of input signals. The numerical value (n_k(x^k)) hence indicates the length of the x^k th continuous string for the kth feature curve in the orientation k. This value will be accumulated only when the difference ratio between the current string (x_i^k) and the previous one (x^k_i−1) is lower than some predefined threshold (ε ). This threshold depends on the types of image and how we tolerate noises of the processed images.

This explains the difference ratio is higher for more noisy textures. In our case, ε is set to be 1 for the CNN binary output. Our featured function regarding x^k projects the original 2-d signal (CNN outputs here) onto a 1-d curve so as to re-classify the binary strings according to our requirement. Finally, the defined characteristic function TBS can be determined after analyzing the combination of feature curves in all considered orientations. As we mentioned, the pre-defined threshold is frequently set to remove more noises while deciding the current continuous string. This featured function can be more flexible by selectively determining or adjusting this threshold.

Later in our experimental results, the feature curves in two dominant orientations (horizontal and vertical) are well-functioning enough to classify up to sixteen texture patterns in our database. This method is still applicable to the feature curves in more orientations if the texture patterns are distributed in some particular orientation. We named this feature curve TBS (Transition Bit String) since its horizontal axis

represents the times of transitions or the number of total continuous strings and the vertical axis shows the length of some specific string. To illustrate the use of TBS, we have some regular texture patterns in the ideal binary level at different frequencies and orientations as shown in Fig. 4.3_1 (a), where the proportion of whiter area in (a.1) is 2/3 distributed horizontally and each texture image is assumed in LxL size for simpler explanations. The proportion of whiter area in (a.2) is one half of the same area in (a.1), and (a.3) is again one half of (a.2). As to Fig. 4.3_1 (a.4) ~ (a.6), we have the same proportion between two different bright blocks while they are distributed vertically. The corresponding TBS in the horizontal orientation could be shown in Fig. 4.3_1 (b). We define the x-axis of Fig. 4.3_1 (b) as the number of continuous strings and y-axis as the length of each continuous string (denoted by counts). In Fig. 4.3_1, the numerical values (like 2L/3, L/3, etc…) represent the corresponding coordinate values of x-axis or y-axis on its feature curve. From the observations we make here, horizontal TBS plots indicate the same distributions for the same textured patterns regardless of frequencies and orientations; they also at the same time reveal different lengths and widths for those texture patterns. On the other hand, vertical TBS plots demonstrate the opposite conditions against the horizontal ones. In our case, our TBS combines the feature curves in two distinct orientations (horizontal and vertical) by (16-2). Hence, TBS in truth shows the capability in discriminating texture patterns at different scales or frequencies, and we could use TBS to classify and represent texture patterns of natural images as long as the images of texture patterns are spaced clearly by nature or by appropriate transformation. As a result, we may successfully incorporate the structure of CNN’s in the sense of the ability in generating distinct binary outputs for digital images to obtain the unique projected feature maps with respect to various textures. The reason of introducing CNN’s in this chapter will be given in the following section.

Fig. 4.3_1 Illustrations for TBS plots (a) The ideal texture patterns in various frequencies and orientations (b) The corresponding TBS plots orderly arranged from

left to right, and top to bottom (corresponding to (a.1) ~ (a.6)).

2L/3

No. of cont. strings

L/6

No. of cont. strings counts

No. of cont. strings

counts

No. of cont. strings L²/3