The Fractal Dimension of Wilderness

The Fractal Dimension of Wilderness

Agnès Patuano PhD Landscape ArchitectureEdinburgh College of Art Newcastle University

26-28 March 2015

Agnès Patuano – PhD Landscape Architecture March 2015


The fractal analysis of landscapes• Mathematical theory• In practice: Calculation of the fractal dimension of complex images

• Results of the comparison of methods

The fractal dimension of Wilderness• Landscape preference and naturalness• The preferred dimension• Limitations: The visual differentiation between naturalness and wilderness


The Fractal Analysis of Landscapes Definition of fractals

Fractal geometry: the geometry of nature

The fractal dimension

• A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.” (Mandelbrot, 1982)

Fractal Dimension• Measures “the object’s degree of irregularity and break” (Mandelbrot, Fractal Objects, 1975)

• Line=1; Plane =2; Volume = 3• Fractal dimensions: non integer, strictly exceeds topological dimension

Sierpinski GasketDimension: log(3)/log(2) = 1.585

Credits: Taylor, 2006

Credits: Taylor, 2006


The Fractal Analysis of Landscapes Calculating the fractal dimension

The Box-counting methodAmount of the plane filled by the pattern compared at different magnifications• N(d)=1/d D

• N(d) : number of boxes• d: size of the boxes

• logN(d) plotted against logd. • line slope = - D

Richardson plot


Concerns

Physical fractals do not exist: For natural fractals, D can not be measured, it can only be estimated.• Fractals are complex mathematical constructs (equations) with an infinity of self-similar details on an infinity of scales.

• Natural objects can show fractal-like attributes but cannot behave like real fractals because of their finite quality.

Landscapes are more likely multifractals

Most non-mathematical studies use the Box counting method • Dependent on image resolution and information content

• Even within this method, results can differ for a particular image if the image is turned upside down or rotated

• The location of the offset of the grid influences results

Mathematical fractals ≠ physical fractals

The limitations of the box-counting method


Methodology

Fractal Analysis• Picture selection• Software tests• Image analysis

- Greyscale- Silhouette outlines- Extracted edges

Images from the Forestry Commission• Comparison between two types of landscape• Similar resolution and size• No building, landmark or water

• Softwares:• HarFA 5.5: (Harmonic and Fractal Image Analyser)

compiled by the Brno University of Technology• BENOITTM 1.3: Trusoft Int’l Inc, most commonly used

commercial fractal analysis software in academic research.


Preparing the images

Extracted edges

Image Analysis

Thresholded image

Find Edges

Original image Greyscale Removed sky


Silhouette Outlines

Image Analysis

Method 1: Find Edges

Method 2: Stroke


Image Analysis Greyscale analysis

Thresholding

Only one of the software carries out greyscale analysis: HarFA

• Fractal spectrum: Fractal dimension over intensity levels

• Thresholding: • i the intensity level chosen• Each pixel displaying an intensity < i goes white• Each pixel displaying an intensity < i goes black


Image Analysis Each image transformed yields a different value of D

Each software calculates a different value of D for the same pattern

1 image = 2 x 5 x D


Comparison of the different methods

Software results highly correlated (τ = 0.925, p<0.01)

Results• Correlation between methods

Correlations

DOUTLINE AVG DEDGEAVG DGREYSCALEMINAVG DGREYSCALEMAXAVG DGREYSCALEAVG

Kendall's

tau_b

DOUTLINE AVG

Correlation

Coefficient

1.000 .145 .609** -.151 .127

Sig. (2-tailed) . .123 .000 .107 .174

N 54 54 54 54 54

DEDGEAVG

Correlation

Coefficient

.145 1.000 .141 .202* .324**

Sig. (2-tailed) .123 . .135 .032 .001

N 54 54 54 54 54

DGREYSCALEMINAVG

Correlation

Coefficient

.609** .141 1.000 -.247** .134

Sig. (2-tailed) .000 .135 . .009 .156

N 54 54 54 54 54

DGREYSCALEMAXAVG

Correlation

Coefficient

-.151 .202* -.247** 1.000 .155

Sig. (2-tailed) .107 .032 .009 . .099

N 54 54 54 54 54

DGREYSCALEAVG

Correlation

Coefficient

.127 .324** .134 .155 1.000

Sig. (2-tailed) .174 .001 .156 .099 .

N 54 54 54 54 54

**. Correlation is significant at the 0.01 level (2-tailed).

*. Correlation is significant at the 0.05 level (2-tailed).


Comparison of the different methods Results

• Ability to distinguish between landscape types

Comparison between extracted edges,

silhouette outlines, and average fractal

dimension for images of forest (a) and

meadows (b)

(a)

(b)



Rationalizing the value of the experience of a landscape

Kaplan&Kaplan, 1989, Attention Restoration Theory• the degree to which a scene is natural or manmade• the extent of topographic variation• the presence or absence of water • the scale and openness of the scene

“Perceived naturalness”• Subjective concept• Linked with vegetation (quantity, type, quality) • Linked with perceived complexity

Complexity linked to concepts of mystery, legibility and coherence

Preference• Complex environments (dense forests) less preferred because

of illegibility and vulnerability to predators • Low complexity environments less preferred because

uninteresting

What is naturalness

What is complexity


The Fractal Dimension of Wilderness The preferred D Main claims and hypothesis:• Universal aesthetic value of fractal patterns

• Preferred value D= [1.3;1.5]

• mid-range D and perceived naturalness

Aks and Sprott, 1996, Quantifying aesthetic preference for chaotic patterns• Most objects in Nature D =1.3

Haggerhall, Purcell & Taylor, 2004, Fractal dimension of landscape silhouette outlines as a predictor of landscape preference • Link between landscape preference and fractal properties

• Use of the silhouette outline as fractal image

Silhouettes used in Hagerhall, Purcell and Taylor's study (2004)Fractal Dimension a) D = 1.14; b) D = 1.32; c) D = 1.51; d) D = 1.70

• Short, 1991, the Aesthetic Value of Fractal Images• Nature -> Art• Resonance to fractals• Universal preference

• Sprott, 1993, Automatic Generation of Strange Attractors• Preferred D = 1.3



Fractal Geometry is used for• observing spatial distribution, • calculating available habitat space• modelling behaviour of communities• Etc

Krummel et al, 1987, Landscape patterns in a disturbed environment.• Perimeter-area method• Comparison of patterns of forest patches• Small areas = low D

Low D considered a sign of human intervention

High D characteristic of superior naturalness

Use of fractal geometry in Ecology


The Fractal Dimension of Wilderness Limitations: Can we visually differentiate between naturalness and wildness?

Conclusion

Naturalness and Wildness: no or little human intervention• Conservation• Biodiversity

Link between D and perceived naturalness but not actual naturalness

The Box Counting method applied to digital images is not sensitive enough to discriminate

D = 1.27

Thank [email protected]

The influence of fractal dimension on landscape preference

Agnès Patuano PhD Landscape ArchitectureEdinburgh College of Art

The Fractal Dimension of Wilderness

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