Feature extraction using wavelet and fractal

Abstract

In this paper, we are investigating the utility of several emerging techniques to extract features. A novel method of feature extraction is proposed, which includes utilizing the central projection transformation (CPT) to describe the shape, the wavelet transformation to aid in the boundary identification, and the fractal features to enhance image discrimination. It reduces the dimensionality of a two-dimensional pattern by way of a central projection approach, and thereafter, performs Daubechies' wavelet transform on the derived one-dimensional pattern to generate a set of wavelet transform sub-patterns, namely, curves that are non-self-intersecting. The divider dimensions are computed from these curves with a modified box-counting approach. These divider dimensions constitute a new feature vector for the original two-dimensional pattern, defined over the curve's fractal dimensions. We have conducted several experiments in which a set of printed Chinese characters, English letters of varying fonts and other images were classified. Based on the Euclidean distance between the different feature vectors, the experiments have satisfying results.

Introduction

Feature extraction is the heart of a pattern recognition system. In pattern recognition, features are utilized to identify one class of pattern from another. The goal of feature extraction is to find a transformation from an n-dimensional observation space X to a smaller m-dimensional feature space X_i^T=(x_i1,x_i2,…,x_in), i=1,…,m that retains most of the information needed for pattern classification. The coordinate axes that define the feature space X_i^T are called features. There are two main reasons for performing feature extraction. First, the computational complexity for pattern classification is reduced by dealing with the data in a lower dimensional space. Secondly, for a given number of training samples, one can generally obtain more accurate estimates of the class-conditional density functions and thus formulate a more reliable decision rule. Whether or not this decision rule actually performs better than the one applied in the observation space depends on how much information was lost in the feature transformation. In some cases, it is possible to derive features that sacrifice none of the information needed for classification.

In this paper, one may think that the best description of a pattern is itself. So, why do we need the feature extraction step? In other words, why not use the pattern themselves as features? Classification may be seen as a function $\text{C}$ from the feature space X_i^T to the pattern space $\text{X(}\text{C}\text{:X→X}{}_{\mathrm{i}}{}^{\mathrm{<mtext>T</mtext>}}\text{)}$, where $\text{C}$ is the set of all possible classes. If X has a big dimension, and this will be the case if bitmaps are used as features, the application of $\text{C}$ may require a lot of CPU time. By extracting features from the pattern we decrease the dimensionality of X_i^T and therefore speed up the classification. The second important reason is that normalization is faster and more accurate in feature space than in pattern space.

The “source” features can be extracted from the bitmaps. This is not the only possible source. Other commonly used methods can be described below (Jain et al., 1996; Sazaklis, 1997):

1. Extraction from the boundary curves of the pattern. A boundary following algorithm can be run on the bitmap in order to obtain the coordinates [x(t),y(t)] of the contour of the pattern. This curve is closed and therefore the functions x(t) and y(t) are periodic. Feature extraction from the boundaries can, however, easily be normalized to translation, scaling and rotation. In addition, they have low sensitivity to noise. Of course, methods based on these features cannot be used if noisy gaps “open” the characters, or if the characters are not connected.

2. Extraction from the HV-projections. From the bitmap b(x,y), 0⩽x⩽W−1, 0⩽y⩽H−1, the following two functions are computed:

$$\text{h(y)=∑}{}_{\mathrm{i=0}}{}^{\mathrm{W-1}}\text{b(i,y)}\text{and}\text{v(x)=∑}{}_{\mathrm{j=0}}{}^{\mathrm{H-1}}\text{b(x,j),}$$

h and v are, respectively, called the horizontal and vertical projections of b. Features computed from h and v cannot be normalized to rotation nor can the shapes of the characters be reconstructed from h(y) and v(x).

3. Pattern profiles. A symbol can be described in terms of its four profiles: left L(y), right R(y), top T(x), and bottom B(x). The profiles give a measure of the variations of the shape on each side of the character. For the bitmap b(x,y), 0⩽x⩽W−1, 0⩽y⩽H−1, and the left profile L(y) may be defined by

$$\text{L(y):}\text{x}\text{if}\text{b(x,y)}\text{is}\text{the}\text{first}\text{set}\text{of}\text{pixels}\text{on}\text{the}\text{row}\text{y,}\text{W}\text{if}\text{there}\text{is}\text{no}\text{set}\text{of}\text{pixel}\text{on}\text{the}\text{row}\text{y.}$$

Fig. 1 shows the left and right profiles of the character `R'. The profile functions are very sensitive to noise, especially if it is present all over the bitmap. The contours cannot keep the information about the inside patterns.

4. Direct extraction from the bitmap. No special remark needs to be done about this source, all depend on the features that are extracted.

5. Extraction from gray-level rasters. The four previous schemes are used in bitmaps. Recently some methods have been described that use gray-level rasters instead. Variations in the gray-levels are used either directly as features or for generating functions from which features are computed.

6. Structural features. Structural features aim at capturing the essential shape information of the characters, generally from their skeletons, and sometimes from their contours. The features include: loops, junctions, crossing and end points, concavity and convexity, arcs and strokes.

Pattern recognition requires the extraction of features from regions of the image, and the processing of these features with a pattern classification algorithm. Many of the features used in our applications tend to be local in nature, which means their calculation requires a connected region of the image over which an average or other statistics is extracted. In this paper, we present a novel approach to extracting features in pattern recognition that utilizes a central projection transformation which combines the wavelet and fractal theories. In particular, this approach reduces the dimensionality of a two-dimensional pattern by way of a central projection method, and thereafter, performs Daubechies' wavelet transform on the derived one-dimensional pattern to generate a set of wavelet transformation sub-patterns, namely, curves that are non-self-intersecting. The divider dimensions are readily computed from the resulting non-self-intersecting curve. These divider dimensions constitute a new feature vector for the original two-dimensional pattern, defined using the curves' fractal dimensions. Once these feature vectors have been captured, we can compare them to the training set by calculating the Euclidean distance between the different feature vectors. The smallest distance was considered the match. Fig. 2 illustrates the overall approach.