## 2D Shape Classification Using Multifractional Brownian Motion

Citations: | 7 - 5 self |

### BibTeX

@MISC{Bicego_2dshape,

author = {Manuele Bicego and Ro Trudda},

title = {2D Shape Classification Using Multifractional Brownian Motion},

year = {}

}

### OpenURL

### Abstract

Abstract. In this paper a novel approach to contour-based 2D shape recognition is proposed. The main idea is to characterize the contour of an object using the multifractional Brownian motion (mBm), a mathematical method able to capture the local self similarity and long-range dependence of a signal. The mBm estimation results in a sequence of Hurst coefficients, which we used to derive a fixed size feature vector. Preliminary experimental evaluations using simple classifiers with these feature vectors produce encouraging results, also in comparison with the state of the art. 1

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Citation Context ... can be shown that, for fractal or self-similar processes, the Horst exponent H can be related to a fractal dimension Dh [10] through the equation Dh =2− H, whereDh is the fractal Hausdorff dimension =-=[14]-=-. Nevertheless, from a shape recognition point of view, we consider it too strong to assume contours showing a complete self similarity, and modeling them with a fBm process (or fractals) could be too... |

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Citation Context ...ng presented over the past years, each with different characteristics, like robustness to noise and occlusions, invariance to translation, rotation and scale, computational requirements, and accuracy =-=[2,3]-=-. In this paper a novel approach to contour-based 2D shape recognition is proposed. The main idea is to characterize the contour of each object using the multifractional Brownian motion (mBm) [4,5,6],... |

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Citation Context ... as K(α) = απ sin 2 Γ (α +1) π ,andB(·) stands for the ordinary Brownian motion process – introduced by Robert Brown in 1872 to describe the random movements of particles suspended in a liquid or gas =-=[18]-=-. The process is self-similar1 of parameter H and has stationary increments. Its covariance function reads as E (BH(t)BH(s)) = c2 2 ( |t| 2H + |s| 2H −|t − s| 2H) The fBm can be generalized by allowin... |

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Citation Context ...≤ 1 to ensure the continuity of the motion. Notice that since H(t) is the punctual Hölder exponent of the mBm at point t, the process is locally asymptotically self-similar with index H(t) (see, e.g. =-=[20]-=-) in the sense that, denoted by Z(t, au) :=M H(t+au)(t + au) − M H(t)(t) the increment process of the mBm at time t and lag au, itholds lim a→0 + a−H(t) Z(t, au) d = BH(t)(u), u ∈ R. (4) 1 We recall t... |

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Citation Context ...we explore the use of the mBm function for characterizing the shape contours with the aim of 2D shape classification. In the proposed approach, the contour is modeled with the curvature (similarly to =-=[15,16]-=-), which was fitted with a mBm function. The sequence of Horst coefficients computed was subsequently employed to derive a fixed length vector, which characterizes each shape. Some experimental evalua... |

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Citation Context ...we explore the use of the mBm function for characterizing the shape contours with the aim of 2D shape classification. In the proposed approach, the contour is modeled with the curvature (similarly to =-=[15,16]-=-), which was fitted with a mBm function. The sequence of Horst coefficients computed was subsequently employed to derive a fixed length vector, which characterizes each shape. Some experimental evalua... |

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Citation Context ...cy [2,3]. In this paper a novel approach to contour-based 2D shape recognition is proposed. The main idea is to characterize the contour of each object using the multifractional Brownian motion (mBm) =-=[4,5,6]-=-, a mathematical method employed in the finance community to characterize and model the financial series. The simplest way of introducing the multifractional Brownian motion is to consider it as an ex... |

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Citation Context ...rot and Van Ness [7], the fBm is characterized by a slowly decaying autocorrelation function depending on the parameter H ∈ (0, 1], named Hurst exponent. Following the definition that can be found in =-=[17]-=-, the process has the following moving average representation with BH(t) =C{πK(2H)} 1/2 ∫ R ft(s)dB(s) (1) 1 ft(s) = Γ ( H + 1 ) 2 { } 1 1 H− 2 H− 2 |t − s| 1]−∞,t](s) −|s| 1]−∞,0](s) where C is a pos... |

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Citation Context ...iable estimates and over a long period of time H is likely to change even further. So, more efficient estimators are needed in the case of the mBm. An answer to this problem is provided by Bianchi in =-=[21]-=-, who develops the work of [22] and defines a family of ”moving-window” estimators of H(t) based on the k-th absolute moment of a Gaussian random variable of mean zero and given variance VH (the varia... |

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Citation Context ... period of time H is likely to change even further. So, more efficient estimators are needed in the case of the mBm. An answer to this problem is provided by Bianchi in [21], who develops the work of =-=[22]-=- and defines a family of ”moving-window” estimators of H(t) based on the k-th absolute moment of a Gaussian random variable of mean zero and given variance VH (the variance of the unit lag increment o... |

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Citation Context ...ed using an 3 Approximate values are shown since in the paper no numerical values were given, justagraphicaltable–Tab.5(b).914 M. Bicego and A. Trudda efficient cyclic string edit distance algorithm =-=[25]-=-. The accuracy, computed by splitting the set in Training/Testing/Validation was 74.3% 4 . 4. Hidden Markov Models: recognition of 2D shapes is a quite unconventional application of HMMs, even though ... |

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Citation Context ...cy [2,3]. In this paper a novel approach to contour-based 2D shape recognition is proposed. The main idea is to characterize the contour of each object using the multifractional Brownian motion (mBm) =-=[4,5,6]-=-, a mathematical method employed in the finance community to characterize and model the financial series. The simplest way of introducing the multifractional Brownian motion is to consider it as an ex... |

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Citation Context ...d some characteristics of the financial series), some works applying the fBm function have also been presented in the pattern recognition context, like in medical imaging [8] or in speech recognition =-=[9]-=-. It is important to note that fBm is related to the fractal theory [10] – a theory largely employed in image analysis [11] or even object classification [12,13] – since it is modeling self-similarity... |

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Citation Context ...dical imaging [8] or in speech recognition [9]. It is important to note that fBm is related to the fractal theory [10] – a theory largely employed in image analysis [11] or even object classification =-=[12,13]-=- – since it is modeling self-similarity. Actually it can be shown that, for fractal or self-similar processes, the Horst exponent H can be related to a fractal dimension Dh [10] through the equation D... |

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Citation Context ...e fBm function have also been presented in the pattern recognition context, like in medical imaging [8] or in speech recognition [9]. It is important to note that fBm is related to the fractal theory =-=[10]-=- – a theory largely employed in image analysis [11] or even object classification [12,13] – since it is modeling self-similarity. Actually it can be shown that, for fractal or self-similar processes, ... |

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Citation Context ...dical imaging [8] or in speech recognition [9]. It is important to note that fBm is related to the fractal theory [10] – a theory largely employed in image analysis [11] or even object classification =-=[12,13]-=- – since it is modeling self-similarity. Actually it can be shown that, for fractal or self-similar processes, the Horst exponent H can be related to a fractal dimension Dh [10] through the equation D... |