Objective

2

The theoretical research and application of fractal geometry is continuously making progress. Traditional methods are formed in the spatial domain through iterative operations on generators. To extend the fractal generation method

we introduce the spectrum analysis method into fractal geometry. In signal processing tasks

such as filtering

data compression

model editing

and object retrieval

the spectrum analysis method has been widely and successfully applied

especially in the field of computer-generated fractals. However

spectrum analysis based on orthogonal transform is rarely studied in fractal generation and analysis. One reason is the lack of a suitable orthogonal function system. We should choose to express the fractal accurately with a few orthogonal basis functions. Polygonal fractal curves should be chosen as orthogonal expressions of the same type of orthogonal function systems. From the point of view of the spline function

the polyline is a linear spline function. Therefore

we make a fractal curve on the basis of a class of one-time orthogonal spline function system (Franklin function) to obtain the Franklin spectrum. Compared with other orthogonal function systems

the Franklin spectrum has inherent advantages in the expression and generation of fractal curves.

Method

2

We first show the orthogonal decomposition and reconstruction algorithm of fractal curves under the Franklin function system. Then

with the classic von Koch snowflake curve taken as an example

the Fourier series

the V system

and the Franklin orthogonal system are compared to express the fractal curve. Lastly

the Franklin spectrum is modified

and then orthogonal reconstruction is performed to generate rich and diverse fractal curves. The advantages and characteristics of this method are compared and demonstrated. The traditional generator iteration is a process-based fractal expression method

but we express the fractal from the perspective of spectrum

which is the essential difference. One of the fundamental core issues of spectrum analysis is the choice of orthogonal function systems. Considering that the fractal curve is a continuous and unsmooth polyline type

the traditional orthogonal function system has the following two problems. On the one hand

the usual trigonometric functions (Fourier transform) and wavelet transform are only suitable for smooth objects. On the other hand

the orthogonal piecewise polynomial function system represented by the V system is suitable for continuous and discontinuous objects. Neither method is suitable for fractal spectrum analysis. Therefore

this study introduces a type of continuous orthogonal function system

i.e.

the Franklin function system. Through the orthogonal decomposition of fractal curves

the corresponding Franklin spectrum is obtained.

Result

2

We take a typical typing curve

i.e.

the von Koch curve

as an example and use the Fourier and Franklin methods to perform orthogonal decomposition and reconstruction. The characteristics of the fractal curve expression algorithm based on the Franklin function are verified. Moreover

compared with other orthogonal function systems

it emphasizes the advantage of the Franklin function system in the frequency domain representation of fractal curves. Based on the von Koch curve

Sierpinski square curve

and Hilbert curve

we use different resolutions and parameters to conduct comparative experiments. The experiments verify the superiority of the Franklin function system in the frequency domain expression of fractal curves. By freely adjusting the Franklin spectrum

many new fractal curves with different shapes can be easily generated. According to the properties of the Franklin function system (orthogonality and multiresolution)

the Franklin spectrum describes the fractal curve from the frequency domain perspective and achieves the optimum multilevel hierarchical approximation of the entire fractal curve. Low-frequency components focus on the contour information of the fractal curve

whereas high-frequency components describe its detailed information. When the Franklin spectral coefficient reaches 2

n

+1 terms

an accurate reconstruction of the fractal curve can be achieved; ordinary orthogonal function systems cannot achieve such accuracy.

Conclusion

2

We take the fractal curves of the study as an example. Classical fractal objects

such as von Koch snowflake

Siepinski carpet

and Hilbert curve

have an infinitely small-scale hierarchical structure. However

when they are stored in a computer and drawn

they can only present approximation results within a certain scale. Therefore

they are continuous but unsmooth polylines. Afterward

choosing Franklin orthogonal function systems can make proper orthogonal expressions for such objects. The Franklin function can be accurately expressed with limited orthogonal basis functions. On the one hand

Franklin functions are not smooth functions and thus can express fractal graphs well with polyline segments. On the other hand

Franklin functions are not discontinuous orthogonal function systems

and continuous functions can be expressed with a few term basis functions. This expression does not have fragility; in other words

the fractal orthogonal expression using the Franklin function does not distort the fractal curve. In general

the Franklin spectrum can not only achieve limited and accurate reconstruction of fractal curves but also describe the morphological characteristics of fractals on different scales. The fractal generation method based on Franklin spectrum adjustment provides new ideas and solutions for fractal generation.