In this part, four features are introduced. Among them, RMS \( s \) is first proposed by us. It can describe the variation of 1D signature in vertical direction well.

### Root mean square roughness w

Root mean square roughness describes the irregular degree of 1D signature. The root mean square roughness is defined as: \( w \) is root mean square roughness defined as \( w = \sqrt { \langle h^{2}\rangle - \langle h \rangle ^{2} } \). Among the equation, 〈 〉 expresses the statistical average, \( w \) expresses the fluctuation degree of \( h \) in vertical direction. The shape is more regular with the value of more small. That is to say that the margin is more close to a circle or ellipse. The mass will more probably be benign than malign. So root mean square roughness may be used as a feature to classify the benign or malign breast mass.

### The \( \mu_{R} /\sigma_{R} \) ratio

The \( \mu_{R} /\sigma_{R} \) ratio (where \( \mu_{R} \) means mean radial distance of tumor boundary, and \( \sigma_{R} \) means standard deviation), describes the circularity of the breast mass contour. Malignant mass should have smaller values of circularity than benign mass. Haralick [24] proved that the \( \mu_{R} /\sigma_{R} \) ratio is a good feature in classifying malignant mass and benign mass. Polhman [15] applied this feature in his 1D signature and acquired the good result.

### Fractal dimension

According to the fractal geometry of Mandelbort, the fractal dimension can describe the property of self-similarity in some way. Many fractal models are proposed to analyze fractal phenomenon of nature. The popular fractal model is differential box-counting method. Studies prove that the differential box-counting method is appropriate to self-similarity fractal model. In medical image, the fractal Brownian motion (fBm) model has been shown to be suitable for the analysis of medical image because the intensity surface of a medical image can be viewed as the end result of random walk. The fBm model belongs to the class of statistically self-affine fractal concept and regards naturally occurring rough surfaces as the end result of random walks. Since the roughness of the intensity surface of a medical image can also be viewed as the end result of a random walk, the fBm model suits for the analysis of medical images. To the affine fractal random rough model, autocorrelation function and height-height correlation function can be expressed as [23]:

$$ R(\rho ) = w^{2} exp[( - \rho /\xi )^{2\alpha } ] $$

(1)

$$ H(\rho ) = \langle [h(n) - h(n^{\prime})]^{2} \rangle = 2w^{2} \{ 1 - exp[( - \rho /\xi )^{2\alpha } ]\} $$

(2)

where \( \alpha \) is the fractal exponent, the relative between \( \alpha \) and fractal dimension \( D \) is \( \alpha = d - D \), \( d \) is the space dimension, and \( \alpha \) is constraint by \( 0 \le \alpha \le 1 \). \( w \) is root mean square roughness expressing the fluctuation degree of \( h \) in vertical direction, and \( \xi \) is correlation length expressing the fluctuation degree of \( \rho \) in horizontal direction. The autocorrelation function \( R \) of \( h(i) \) is can be defined as:

$$ R(i + \rho ) = \langle h(i)h(i + \rho ) \rangle /w^{2} $$

(3)

Here, \( \rho = |i_{2} - i_{1} | \) is the interval between two points on signature. The autocorrelation function \( R \) has some characteristics such as: (1) If the signal is the smooth and steady random process, \( R(i + \rho ) \) is irrelevant to \( n \) and relevant to only \( o \) i.e. \( R(i + \rho ) = R(\rho ) \). With the increment of correlation interval \( \rho \), \( R(\rho ) \) decreases little by little and tends to be zero. The rate of decrease is decided by the distance between two points irrelevant to each other. The correlation length is defined by the value of correlation interval at the point that the autocorrelation function \( R(\rho ) \) decreases to \( e^{ - 1} \) of the maximum. The correlation length \( \xi \) expresses the speed that \( R(\rho ) \) decreasing with \( \rho \).If the interval between two points is less than \( \xi \), the two points are correlated. Otherwise, the two points are independent. The fluctuation in the horizontal direction is expressed with \( \xi \) and the fluctuation in the vertical direction is expressed with \( w \).

In the condition of \( \rho < < \xi \), self-affine fractal surface \( h(n) \) satisfies self-affine transform below:

$$ h(x_{0} ,y_{0} ) = \varepsilon^{2\alpha } h(\varepsilon x_{0} ,\varepsilon y_{0} ) $$

(4)

If the scale is small as \( 1/\varepsilon \), the average variation of height difference is \( \varepsilon^{2\alpha } \). This variation is corresponding to the power law variation of height-height correlation function during the short distance. The relationship is

$$ h(\rho ) \propto \rho^{2\alpha } ,\rho < < 1 $$

(5)

The power law variation of height-height correlation function can describe statistically self-similarity characteristic and local fluctuation. If \( \alpha \) is smaller, the local fluctuation is more violent and fractal dimension is larger. From the Eq. (5), we can conclude that in log–log coordinate system \( h(\rho ) \) is proportional to \( \rho \) when \( \rho < < 1 \). \( 2\alpha \) can be estimated from the slope of the line approximated by linear least squares fitting on \( log(H(\rho )) \) versus \( log(\rho ) \) when we choose a range of the lower scale \( \rho \). Figure 5 shows the curve of \( log(H(\rho )) \) versus \( log(\rho ) \) and the linear fitting for benign and malign mass. In this paper, we look the 1D signature of contour as height distribution of the affine fractal random surface. The fractal dimension indicates the self-similarity feature and it also expresses the local non-smooth fluctuation of the signature. The fractal dimension D is larger and larger; the local fluctuation of the signature is more and more drastic. Here we use the fractal exponent \( \alpha \) of 1D signature of contour as the third feature to distinguish the benign mass from the malign one.

### RMS slope \( s \)

Each point on the contour has different slope. The variation of slope describes the shape of contour. If the contour is smooth, the variation of slope is slow and regular; otherwise, variation of slope is drastic. When we transform 2D contour into 1D signature, the value in the Y-axis expresses the circularity. The absolute value of the slope shows the variation speed of contour. So we take the slope distribution of each point on the contour as one of the features to discriminate malign mass from benign mass. Slope is acquired by linear interval. Root mean square slope is defined as:

$$ s = \sqrt { \left\langle \left (\frac{dh(\rho )}{d(\rho )}\right)^{2} \right\rangle } $$

(6)

We can see from the Fig. 4 and Eq. (6) that the slope of benign mass has small value and the fluctuation is gentle. While the slope of malignant mass has big value and the fluctuation is violent. The variation range of the RMS \( s \) for malignant mass is wider than benign mass.