Calculating Fractal Dimension from Vector Images

Kelly Ran

What Is A Fractal?

Fractal Dimension

• Lines: D=1• Squares: D=2• Fractals: Non-integer D• Calculated using multiple methods• Box-counting dimension

Fractal Dimension Applications• Texture classification• Protein molecule analysis• Plant growth analysis• Model: plants, particle motion

Vector Graphics

• Raster graphics: pixel-by-pixel• Research applications calculate D from raster

graphics• Vector graphics: Objects like paths and shapes• Scalable Vector Graphics (SVG) based on XML• Paths

SVG

• <?xml version="1.0" standalone="no"?>• <desc>Example rect01 - rectangle with sharp

corners</desc>• <rect x="1" y="1" width="1198" height="398"• fill="none" stroke="blue" stroke-

width="2"/>• </svg>

Goals

• Is it possible to calculate fractal dimension from vector images?

• Goals:• Calculate D from SVG images• Show calculations on a display screen• Compare raster v. vector• Time? Accuracy?

Procedures

• Create SVG files• Inskcape Software

Sierpinski Triangle, 3 iterations

Sierpinski Triangle, 6 iterations

Procedures

• Processing language• Import SVG• Array of objects• D calculation: size N of grid, number s of grids• Hash table: N is key, s is value

Testing

• Line, square• Fractals with known fractal dimensions

To Be Done

• Improve upon program• Gather more data• Implement raster method• Linear regression function: linear algebra least

squares method