目的 分形几何学的理论研究与应用实践方兴未艾，在分形的计算机生成领域，传统方法是在空间域中，通过对生成元的迭代操作而形成。为了扩展分形的生成方法，本文将频谱分析引入到分形几何中。方法 正交函数系是频谱分析的核心问题之一。考虑到分形曲线是一类连续而不光滑的折线型信号，通常的三角函数（Fourier变换）、连续小波变换仅适用于光滑的对象，否则会出现所谓“Gibbs现象”；另一方面，以V-系统为代表的正交分段多项式函数系适用于表达包含间断性的对象，否则会出现信息冗余。因此，通常的正交函数系均不适合分形的频谱表达与分析。针对分形曲线的特点，本文将其视为一次样条函数，通过引入一类正交样条函数系-Franklin函数系，实现了对分形曲线的有限项精确正交表达，得到Franklin频谱，从而完成分形的时频变换。然后，对Franklin频谱系数在不同尺度上进行修改。最后，通过正交重构得到新的分形。结果 对比实验验证了Franklin函数系在分形曲线频域表达方面的优越之处，它既能通过最小项数实现分形的正交表达，而且不会出现Gibbs现象。本文以von Koch曲线、Sierpinski square曲线和Hilbert曲线这3个经典分形为例，通过对Franklin谱在不同尺度上的自由调节，能够方便地生成大量形态各异的新的分形曲线。结论 Franklin谱不仅能够实现对分形曲线的有限精确重构，而且还能在不同尺度上刻画分形的形态特征。基于Franklin频谱调节实现的分形生成方法，只要修改频谱就可以得到大量的新型分形曲线，而且这些分形的样式千变万化，几乎不可预测，这种分形生成方式为分形设计带来了巨大的自由空间，为分形的生成提供了新的思路与方案。

Objective 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 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 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 2n+1 terms, an accurate reconstruction of the fractal curve can be achieved; ordinary orthogonal function systems cannot achieve such accuracy. Conclusion 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.