Limits and Errors: Optimising Image Pre-Processing

ARCHITECTURE

SCIENCE, No.7, pp.1-19, June 2013

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis Michael J. Ostwald! I Professor, 2

Josephine Vaughan 2*

Department of Architecture, University of Newcastle, Australia Research Academic, University of Newcastle, Australia

* Corresponding

author Email:[email protected];

Tel:+61-2-49854186;

Fax:+61-2-49216913

(Received Nov. 30,2012; Accepted May 17,2013)

ABSTRACT The box-counting variation of fractal analysis is the most common approach to calculating the visual complexity of buildings, cities and landscapes. This method derives a numerical value from an elevation or plan of a building or space, which reflects the amount of detail present in that image across multiple scales of observation. As a way of analysing urban plans and buildings, a range of scholars and designers have employed However,

the box-counting

despite the volume of this past research,

method over the last eighteen years.

the methodological

limits - including the

magnitude of potential error rates that are caused by variations in the images used - are yet to be quantified. As a consequence, mathematical

often widely varying results have been produced using the same

method and, ostensibly at least, the same image. In response to this situation, the

present paper tests four factors that are associated with image pre-processing

standards for the

box-counting method and which, it has been theorised, have an impact on the results. For these four factors, multiple permutations

of each of seven different test images are analysed in this paper in

order to determine the limits or sensitivities associated with each factor. The results of this analysis are used to understand the impact of variations in each of these data preparation factors. Thereafter, they are used collectively to identify an optimal range of standards for the initial architectural image data.

KEYWORDS:

Fractal Analysis, Computational Analysis, Visual Complexity, Methodological Limits

ARCHITECTURE

SCIENCE. No.7. June 2013

1 Introduction

(1: 100, 1:500 etc.) because the method calculates the visual complexity of the representation

Fractal geometry

of the building,

is used to describe irregular or regardless of its size. The algorithmic process involves

complex lines, planes and volumes that exist between overlaying whole number integer dimensions.

multiple

different

scale

grids

on

the

Thus, instead of an architectural

image

and

calculating

how

much

object having a dimension (D) of 1, 2, or 3, a fractal information

is contained in each square of each grid.

analysis of an irregular image or object might reveal a D Once the complete set of data has been collected in this of 1.51, 1.93 or 2.74 (Mandelbrot, 1977). The process of way, determining

the fractal dimension

the

results

are

mathematically

processed

to

of an object was calculate the fractal dimension of the image.

initially

demonstrated

in mathematics

in the

1980s Past

(Mandelbrot,

1982;

Voss,

1988)

and

it

research

seemingly commonly

has

noted

that

despite

being

a

became straightforward

process,

the results of the

used in the physical and medical sciences box-counting method are sensitive to variations in both

throughout the 1990s (Ashkenazy,

1999; Cross, 1997). the raw data (the "starting image") and the algorithmic

While there are multiple methods for determining

the process used to extract information from the data (Bovill,

fractal dimension of an object, Voss's

"box-counting" 1996; Lorenz, 2011; Ostwald, et aI., 2008). While the

approach (1986) is the most widely used and it was the algorithmic

process

has recently

been optimized

for

first adopted for architectural and urban analysis in the architectural analysis (Ostwald, 2013), the impact of the 1990s (Batty & Longley,

1994). Since that time, this image properties

remains

unknown.

Because

of this

method has been used for the analysis of the fractal problem past researchers using this method have relied dimension of a number of canonical buildings, ranging on a range of theorized "best-practice" from ancient

structures

to twentieth

(Bovill, 1996; Burkle-Elizondo&

century

standards for the

designs

Valdez-Cepeda,

preparation

of images

preparation

protocols

(such

as consistent

drawing

2006; and representational

systems) to

Md Rian, et aI., 2007; Ostwald, et aI., 2008). A stable moderate the potential negative impact of inconsistencies computational version of the method has also been used (Cooper&

Oskrochi,

2008).

While

such

image

to conduct a series of studies of the formal properties of pre-processing famous

houses

(Ostwald

&

Vaughan,

2009),

standards

inconsistencies, longitudinal

study of the architecture

may

assist

to control

the

a their

actual

impact

has never

been

of Frank Lloyd quantified. Therefore, the purpose of the present paper is

Wright

(Vaughan

&

Ostwald,

2011)

and

of

the to quantify

relationship

between

environmental

geometry

the error rates that might

arise in the

and box-counting

method

as a result

of diverse

Image

designed geometry (Vaughan & Ostwald, 2010). standards There

are

three

major

method;

raw

components

to

the

(typically

an

process

for

and to confirm the practical

limits within

which the method will provide stable results. box-counting

data

There architectural

drawing),

an

algorithmic

architectural

are

four

factors

associated

with

the

drawing or image being analysed, which

extracting information from the data and a mathematical past research has suggested have a potential impact on procedure

for calculating

the fractal dimension

of the the results. These are "white space", image position, line

data. The architectural image must be a black and white thickness and image resolution. In the present paper, to line drawing; usually an elevation or plan (see figures 1quantify the impact of each of these factors a series of 7). The scale of the image on the drawing is irrelevant 2

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

seven test images

are examined each

(five of which are

architectural

elevations),

with

a

number

permutations

of the relevant factor. By tabulating and

plotting the results of these permutations,

of

their impact

can be seen on each factor. For example, as the fractal

of data types for analysis. This information is important to ensure the reliability and repeatability of future results and to assist in the interpretation

and validation of the

results of past research. This paper commences

with an overview

of the

dimension of a line drawing is within a range between

box-counting method, which explains in more detail why

1.0 and 2.0, the difference between a dimension of D =

the nature of the original raw data, the architectural

1.305 and D = 1.256 (a difference

drawing, has a potential impact on the results. Thereafter

of 4.9%) can be

readily determined for comparison. Multiple test images

the method used to test and quantify this impact is

are needed

described in more detail, before the results of the four

for the study because

method

(the mathematical

several

known

the box-counting

and algorithmic

sensitivities

which

mean

core) has that

larger

sample sizes are needed to have confidence in any trends identified (Foroutan-Pour,

separate tests are outlined. Finally, the paper concludes by providing

a new, best practice

data standard

for

architectural fractal analysis.

et al., 1999; Camastra, 2003; 2 The Box-Counting Approach

Jelinek, et al., 2005). Five houses were selected for this study based on the typical range research

(Ostwald

& Vaughan,

identified 2009;

in past

Vaughan

&

The

visual

assessed

complexity

of architecture

can

in various ways, from phenomenological

be or

Ostwald, 2010; 2011). In addition, two abstract shapes

perceptual

were added to the set of images for comparative purposes.

2003;

The rationale for the complete set of images is described

approaches

later in the paper.

Several computational approaches exist for analysing the

For each of these images, a series of variations were produced which represent different permutations factor being considered.

For example,

because

understandings

Pallasmaa,

2005)

(Stamps

(Nasar, to

1989; Stamps

mathematical

III, 2005;

III,

analytical

Wong, et al., 2012).

visual complexity of architecture including distribution

of the

analysis

by Zipfs

it is

fractals

(Stamps,

law, Van der Laan septaves 1999; Crompton

& Brown,

and 2008;

theorised that the line weight in the original drawing

Crompton, 2012). This paper addresses the box-counting

might have an impact on the calculated result, for each of

method of fractal analysis, a computational

the seven

of line

used to measure the amount of detail found in an image

were then

of a building. To calculate the fractal dimension of an

thickness

test images, were prepared.

eleven

permutations

The 77 results

application

calculated and compared with the theoretical ideal setting

architectural

and any patterns in the results identified. At the end of

the process commences by superimposing

the process, the most stable data settings were identified,

over the elevation and determining if any lines from the

along with, in several cases, the limits and indications of

facade are present in each square. Those grid boxes that

error rates. In this wayan

have some detail in them are recorded. Next, a grid of

optimal range of settings are

identified for the first time in architectural

and urban

elevation using the box-counting

method,

a large grid

smaller scale is placed over the same facade and the

applications of this method. These optimal settings are

same determination

ones that either reduce the potential sensitivities in the

in the boxes of the grid. A mathematical comparison is

method or focus it into the most robust and reliable range

then constructed

is made of whether detail is present

between

the number of boxes with

3

ARCHITECTURE

SCIENCE. No.7. June 2013 The data post-processing

detail in the first grid and the number of boxes with detail in the second grid. This comparison is made by

methodological

plotting

mathematical

a log-log

diagram

for each grid size. By

settings

variables - that is, those

that

procedure

shape

is undertaken

the

way

the

- include

grid

repeating this process over multiple grids of different

disposition

scales, an estimate of the fractal dimension of the facade

grids); scaling coefficient (the ratio between grids); and

is produced (Bovill, 1996; Sala, 2007; Lorenz, 2011). In

statistical

practice,

pre-processing variables are divided into field properties

all serious

analysis

using the box-counting

(placement

divergence

and orientation

(error

of successive

handling).

The

image

method is undertaken using software. This is because the

(white space and image position) and image properties

more grid comparisons that are made, the more accurate

(line-weight and image resolution). In the field properties

the result. However, every grid comparison involves an

category, white space is the volume of area around the

increase in processing

time and effort; meaning that,

image that is processed as an integral part of the set of

while the first two scales of comparison might involve

calculations and "image position" is the distribution of

only the analysis

white space relative to the location of the image. In the

of 40 or 60 cells,

by the

12th

comparison the number will be much more. For example,

image properties category, image resolution refers to the

Ediz & Ostwald (2012) undertook a fractal analysis of a

number of "dots-per-inch"

Turkish Mosque and recorded more than 12800 pieces of

saved at (or compressed to) and line thickness is the line

information in a single facade, and calculated the fractal

weight of the drawing being analysed. While the statistical validity of the fractal analysis

dimension of the entire building using over one million results.

For the present

undertaken

paper,

using Archlmage

all calculations

(Vers.1.16),

were

all images

(dpi) the image has been

method is largely reliant on the data-post variables,

image pre-processing

processing

factors also have the

were prepared using ArchiCAD (Vers.14) and for some

potential to cause errors. It is these factors that are

permutations,

investigated in the present paper. Moreover, these factors

modified

using Adobe

Photos hop CS2

(Vers.9.0). The

are particularly perplexing for people using the method box-counting

simplicity,

method,

despite

its apparent

is actually made up of multiple, complex,

because it is possible to test four seemingly

identical

elevations, which are all derived from the same CAD file,

meaning that it is relatively

but because of the way they have each been saved,

difficult to isolate the impact of anyone factor. It is also

positioned, or set-up, will produce different results. Each

known to have particular

of the four image pre-processing factors are described in

interconnected

processes,

strengths and weaknesses

certain ranges of dimensions types (Foroutan-Pour,

and for particular

in

image

more detail hereafter.

et aI., 1999). As a result of this, 2.1 White Space and Image Position

scientists

and mathematicians

mathematical methodological combination,

refinements, and can

along

data have

have identified

quality an

with

a range

variables

impact

on

several

that,

the

of

The background on which the image being analysed

10

is placed, is called the field. This field comprises three

results

components, white space, image space and empty space.

(Camastra, 2003; Jelinek, et ai, 2005). These variables

The

can

surrounding the image, "image space" refers to the lines

be

broadly

divided

into

two

post-processing and image pre-processing.

4

types;

data

descriptor

"white

space"

refers

to the region

that make up the image and "empty space" is any region

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

enclosed by the lines. The image space, and the empty

because it is the first determinant of the practical limits

space within the image are effectively fixed quantities,

of the analytical process (Buczkowski, et al., 1998). The

but the initial amount of white space is determined when

larger the field, the more grid comparisons

the image is positioned on a "page" prior to analysis.

constructed and the more accurate the result. However,

Why is this significant?

increasing the image size increases the processing power

Because,

hypothetically,

the

more white space there is around an image the more the

needed and the practical

results are skewed by factors that are not intrinsic to the

around 3 Megabytes.

elevation (being the combination

may be

limit for current software is

of image space and

The line thickness factor relates to the width of the

if there is almost no white

lines in the image being analysed. If the image is drawn

space (that is, the image is tightly cropped), then the first

in a heavily weighted line, it is thought that the method

few grid comparisons may be statistically biased because

will incorrectly

every cell may have information in it.

thickened lines will be counted twice (or more), with

empty space). Alternatively,

Just as the volume of white space surrounding the

calculate

the dimension

because

the

each smaller grid scales, leading to a high error rate & Taylor,

image is thought to have an impact on the result, so too is

(Taylor

the location of the white space relative to the image

practical

space. If, for example, the field is twice as large as the

should be pre-processed with edge detection software to

image on it, then the image could be positioned

convert every element in the image into one-pixel-width

range of alternative

in a

locations on that field. If it was

1991; Chen,

et al., 2010).

solution to this problem

The

is that all images

lines (Chalup, et ai., 2009). Alternatively,

it has been

placed to the left side of the field, a large amount of

suggested

white space will appear to the right. However,

CAD software to choose the finest practical line that it

if the

that all images should be pre-processed

image space is primarily on the top right of the field, the

can produce.

white space on the lower left will be counted

additional

in a

Line weight

reason.

As

is also relevant

previously

stated,

in

for one the

scale

different iteration of the box-counting process, with both

(meaning 1: 100, or 1:50) of the image representation

architectural

irrelevant for the method, but for any useful comparison

images, essentially the same, but possibly

resulting in different fractal dimensions.

is

to be made between different buildings, the architectural drawings must display a similar level of detail and using

2.2 Image Resolution

and Line Thickness similar representational

Too often in architectural

(which include line

analysis,

weight). This is also why most past uses of the method

the size of the image being analysed is given as a metric

have compared houses with other houses, or urban plans

measure;

with other urban plans,

description

for

example,

is often

computational

standards

"200mm

meaningless

x

100mm".

because

This

it is the

because

a similar

level of

resolution must be represented.

resolution of the image - its dpi - and its size in pixels that is relevant, not its physical size. The same image, printed at the same physical size, will be very blurry at

3 Research Method Four image-processing

factors are examined in this

75dpi but very sharp at 500dpi. Thus, the field size of a

paper using seven standard test images. Between five and

digital

eleven permutations

image must be understood

as its length and

breadth measured in pixels. The field size is important

to the particular

of each of the test images, tailored factor

being

considered,

are each

5

ARCHITECTURE

SCIENCE. No.7. June 2013 a fractal dimension result (DEst.),

artificial "elevations" were added to the set. The first of

which was in tum, after an initial review of results,

these is an empty square (like a blank facade) which was

compared with a target result (DTarg).

expected

processed

producing

While it is possible that an architectural could contain some level of representation

drawing

of the colour

to have the lowest result. Indeed, for some

permutations

of the analytical process the square was

repeatedly measured as having a fractal dimension

of

or materiality of a building, or a sense of its inhabitation

around 0.998, which means that it is so minimal that it is

(entourage

(plants and

no longer an image, but has become a "dust" of points

trees) and time of day (shadows), these features are not

(Mandelbrot, 1982). The second artificial image added to

conventionally represented in the images used for fractal

the

analysis. For the present paper the test images are all

(suggesting a highly detailed facade) which was intended

black and white, computer-produced

line drawings. The

to be within the higher part of the range, and in practice,

process for deciding which architectural elements should

always measured as the second highest result (Figs 1 & 2).

elements)

be included "significant

the local vegetation

in the drawing

involves

the logic

set

being

tested

was

a densely

packed

grid

of

lines" (Ostwald & Vaughan, 2012); that is,

the level of detail present in the building representation which is significant for the purpose of the study. The levels selected for the test images in this paper are limited to the outline, primary form and secondary form of the design. This includes the silhouette of the building

Figure 1 "Square" and any changes in form and material represented by a single line separating surfaces. Basic mullions in doors and windows, stair treads and other elemental projections of a similar scale are included. These representational standards have been used for the fractal analysis of Le Corbusier's Villa Savoye (Bovill, 1996) and of an urban district in Istanbul (Cagdas, et aI., 2005). The seven test images include five elevations

of

works by well-known architects and two artificial shapes. The elevations

were selected because, based on past

published results (Bovill, 1996; Lorenz, 2011; Ostwald & Vaughan, 2009), they represent a range of D values spanning from a relatively

simple facade composition

through to a very complex one. The D results for the elevations typically fall between l.3 and l.6, called the "architecture range", because most buildings have levels of visual complexity that are in the mid range. However, to further test the limits of the method, two additional

6

Figure 2 "Grid" After the square, the image with the lowest D result is the south elevation of Kazuyo Sejima's House in a Plum Grove (2003); a typically minimalist facade from this Japanese architect (Fig 3). The next pair of facades, which have similar levels of visual complexity, are the north elevation of Eileen Gray's Tempe

a Pailla

(1934)

(Fig 4) and the north elevation of Robert Venturi and Denise Scott-Brown's

Vana Venturi House (1964) (Fig

5). Thereafter, the north elevation of Le Corbusier's

Villa

Savoye (1928) (Fig 6) is the next most complex and

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

finally, the most complex facade tested is the south

image-processing

factors. For white space (Fig 8), nine

elevation of Frank Lloyd Wright's Robie House (1910)

permutations

of each test image were prepared.

(Fig 7).

permutation

involved

gradually

adding

Each

a controlled

amount of white space around the same image (growing

D

CD

the field). The first image tested had only a minimal amount of white space (that is, it was cropped very close to the elevation)

and the incremental

growth

was

determined for each test image by calculating the number of pixels equivalent to a given percentage of the shortest Figure 3 House in a Plum Grove, South elevation, Kazuyo Sejima

image dimension (I) and adding half of this value to each side of the image, creating the final field (Fig 9). This provides a consistent area, relative to the image space, for all of the test images. The percentage

increments

used are 0, 10, 20, 40, 50, 60, 70, 80, 90 and 100%. For the image position factor (Fig 10), the same image was placed on the same field but in one of nine Figure 4 Tempe

a Pail/a,

North elevation, Eileen Gray

different

positions

creating

different

relationships

between the white space and the image. The field size was determined for the initial image by adding 100% of the length of the shortest side of the image (I) to each side. Then, within this oversized field, the image was located Figure 5 Vana Venturi House, North elevation, Venturi and Scott-Brown

in nine different

combination

positions;

designated

by a

of left, centre or right and top, centre or

base. The impact of line thickness

was examined

by

producing eleven permutations of each of the test images

u

with different weights and the results calculated (Fig 11). For this last process, the standard edge detection and

Figure 6 Villa Savoye, North elevation, Le Corbusier

reduction settings in the software were disabled. The line weights tested are 1pt, 1Opt, 20pt, 30pt, 40pt, 50pt, 60pt, 70pt, 80pt, 90pt and 100pt width. It is also possible to express these weights as a proportion of the length of the shortest side of the starting image (I). For example, if 1 =

Figure 7 Robie House, South elevation, Frank Lloyd Wright

1000 pixels, then a Ipt line is 0.0011 and a 30 pt line in the same image would be 0.03/.

For each of the seven test images, permutations

were

prepared

to

examine

a series of the

four

However, because the

present research is comparing a set of images that do not possess a consistent I, this approach is not used.

7

ARCHITECTURE

SCIENCE. No.7. June 2013

CD

1

CD

I

~DJ

~ml

!

Figure 8 White space incremental

01i

growth at 0, 50% and 100%

Figure 9 Robie House, white space increase " , ..... #

•. t.;_ ~

~

l-

a~l-l

i;

II .

:.

~

.

t

'•

,

,-!

1

..

~ h

t' .

H ;

J=t

'! •.• ~••• L

"If

,

,; #'

..

.~

Figure 10 Image position for the grid: left base, left centre, left top, centre top, right centre, right base

Figurell

Line thickness, Vanna Venturi House 1pt, SOpt, 100pt

..

Figure 12 Image resolution, Tempe it Pailla, Sdpi (left) and 17Sdpi (right)

8

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

For the final factor, image resolution (Fig 12), the

The DEst results were tabulated and charted. Using

a)

test images

were

of

this data, and informed by past, theorised ideal

compression.

This was done by starting with a 175dp

standards, a "target" permutation was chosen for

then, each test image was resampled (bicubic) reducing

comparison for each of the four factors. The target

the resolution from 175, to 150, 125, 100 and finally 75.

is the setting which past research has suggested -

Resampling

either based on logic, observation or experience -

the

saved

at five different

image

kept

the

levels

same

physical

would be the most stable one against which others

dimensions but changed its pixel dimensions. In total, using seven test images to examine between

could be compared. Through such a comparison

five and eleven permutations of four factors, 245 results

any trends in the results can be quantified, and the

were produced (from almost 2000 calculations) (Table l).

suitability (or not) of the various permutations can

In order to ensure that each factor being tested was

be examined.

isolated from other variables, and its impact able to be

theorised target setting was confirmed by the test

measured,

all other factors

results as being optimal. In the other two tests, the

standards.

For example,

were set to a range of

except

In two of the four cases

results were less differentiated

for the test of line

the

and a range of

weights, all other line weights being analysed were set at

stable results were identified for comparison.

1pt thickness. Next, except for the examination

this situation, the central setting in the range was

of the

impact of image positions, all other images were centred

chosen as the "target"

on their fields. Finally, both the line weight and image

interpretation

resolution tests were conducted with a consistent volume

absolute indicator.

of white space around them which was determined by

The

b)

In

and used to assist the

of the results, rather than as an

difference,

expressed

as

a percentage,

calculating the shortest dimension of each image in the

between the DEst and DTarg results were recorded.

set and adding 20% of this length to each side of the

The

image (defining the field). The data processing settings

calculated as well as its correlation

used for all calculations

(R2). The R2 value is an indicator of degree to

were scaling

1.41421, edge growth (Top-Left)

coefficient

grid disposition

of and

average

of these

differences

was

then

co-efficient

which one body of data may be efficaciously compared to another. In this case, the paper charts

with no correction for statistical divergence.

the results of the permutations

(DEst) against the

3.1 Data Analysis Method target For each of the four factors following

processes

were

followed

fractal

dimension

equal

For a

the

perfect

to interpret

the

lower the result below one, the less consistent the

results.

R2 should

(DTarg).

tested,

being

correlation,

value

1.0. The

correlation. Despite this, because all of the tests in Table 1 Summary

Factors Description of permutations examined White Space Increase white space: 0-100% Image Position Various image positions, 1 field Line thickness Saved at line weights 1-100pt Resolution Saved at 75-175 dpi Total number of results

of Tests

Target for com~arison 50% increase Centre field 1 t

Test images

{#} 7 7 7 7

Permutations

{#} 10 9 11 5

Results

{#} 70 63 77 35 245 9

ARCHITECTURE

SCIENCE. No.7. June 2013

this paper are comparing

variations

of similar

particular factor and are not universal.

images, even the worst of the R2 results, 0.84, is

4 Results and Discussion

relatively high. c)

The tabulated data was analysed using a scatter graph

of D results

and a distribution

4.1 White Space

graph, The theorised impact of white space has been one of

charting results against the percentage gap (DEs! the more contentious issues in fractal analysis with many DTarg) to identify patterns and to quantify limits. authors ignoring the issue and others suggesting various This process clarifies the range of divergences approaches

to handling

it (Bovill,

1996; Cooper

&

from the target figure and assists to identify trends Oskrochi 2008; Ostwald, et al., 2009). There is a broad and quantify the average magnitude of errors. For agreement, these charts, linear and polynomial

amongst

those who have considered

the

trend-lines question, that some white space around the image is

were used to assist the analysis. necessary, but that too much will undermine the veracity The result with the highest percentage

difference of the method. An initial review of the results (Table 2,

was identified;

this is effectively

the worst result or Fig

13) confirmed

this,

suggesting

that

the

most

highest error. For evaluation purposes, anything with less consistent

set were in the central part of the graph

than 20% of this level of difference was considered as (between 30% and 60% white space) and so the 50% indicating

either a reasonable

level of accuracy

or a result was selected as the target (Fig 9). In this zone there

stable or robust zone in the results. This also means that was typically less than a 1.58% average variation caused the limits used to analyse each factor are relative to that Table 2 Results of white space analysis White Space 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

10

Square

Sejima

VSB

Gray

Le Corbo

Grid

Wright

1.038

1.217

1.241

1.305

1.315

1.398

1.528

% diff

5.00%

0.80%

0.70%

4.90%

1.50%

2.90%

0.00%

1.017

1.242

1.237

1.286

1.304

1.401

1.506

% diff

2.90%

3.30%

0.30%

3.00%

2.60%

3.20%

2.20%

0.977

1.173

1.254

1.279

1.397

1.382

1.508

1.10%

3.60%

2.00%

2.30%

6.70%

1.30%

2.00%

0.991

1.196

1.259

1.283

1.328

1.388

1.557

%diff

0.30%

1.30%

2.50%

2.70%

0.20%

1.90%

2.90%

1.004

1.172

1.241

1.261

1.34

1.38

1.545

%diff

1.60%

3.70%

0.70%

0.50%

1.00%

1.10%

l.70%

0.988

1.209

1.234

1.256

1.33

1.369

1.528

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.998

1.196

1.255

1.278

1.333

l.403

l.53

%diff

1.00%

1.30%

2.10%

2.20%

0.30%

3.40%

0.20%

0.992

1.179

l.27

1.276

1.337

l.404

l.549

% diff

0.40%

3.00%

3.60%

2.00%

0.70%

3.50%

2.10%

0.974

1.237

1.268

1.237

1.354

1.387

l.52

% diff

1.40%

2.80%

3.40%

l.90%

2.40%

1.80%

0.80%

0.941

1.148

1.262

1.236

1.344

1.373

1.507

% diff

4.70%

6.10%

2.80%

2.00%

1.40%

0.40%

2.10%

1.036

1.248

1.261

1.242

1.331

1.463

1.488

4.80%

3.90%

2.70%

1.40%

0.10%

9.40%

0.40%

% diff

%diff

% diff

Av.

% diff 0.9818 1.80% 0.9535 2.43% 0.9620 2.98% 0.9929 1.92% 0.9884 1.45% 1.000 Target 0.9912 1.58% 0.9854 2.48% 0.9912 2.18% 0.9712 2.47% 0.8427 2.98%

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

10.00%

•

•

Square

•

Sejlma

A

VSB

X

Gray

o

LeCorb .

•

Grid

+

Wright

8.00%

o a. ro

•

6.00%

o "g 0

--

4.00%

Poly. (Square)

••••••••• Poly. (Sejlma)

2.00%

0.00%

---.-

Poly. (VSB)

-.-.-

.• Poly. (Gray)

---.

Poly. (Le Corb.)

-.-.-

Poly. (Grid)

- .. - . Poly. (Wright)

White Space

Figure 13 Results, impact of white space by the differing quantities of white space (Fig 8). Outside

such outliers.

of this zone, while not consistent, the average growth in 4.2 Image Position variation was in the order of 3.98%, with isolated results up to 9.4%. After

The results for the image position test were the least considering

image-processing space

with

best

consistent of any of the four sets examined in this paper

setting was either 40% or 50% white

(as reflected in the R2 results) (Table 3, Fig 14). With no

a typical

these

of

less

the

±0.74%

clear pattern in the results, the centre-centre position was

divergence. The magnitude of errors caused by too little

adopted as the target value for the analysis. However,

white space was relatively

in the

with the highest difference result being 11%, the optimal

± 1.2% range, but for larger amounts of white space this

zone was therefore any average difference result of less

grew to around ±1.49%, with higher trends suggested

than 2.2%; significantly, a range which none of the other

beyond that (> ±2%). However, despite this and taking

permutations consistently fell within. If then, the results

into account the R2 results, it is also clear that within the

by position are considered relative to the centre, the only

30% to 60% range white space has less impact on the

position which had a similarly "low" set of error rates

results

was the centre-base position (±1.25%) with the top-left

than previously

range

results,

than

similar, commonly

suggested

with none of the

average differences in that range being above ±0.86%. In this set of results one test image showed a much higher sensitivity to the changes than any other. The grid

being the worst, (±2.14%). In combination though, the magnitude of the error rates, regardless of position, was relatively minor.

had a low result of 1.1% difference and a high of 9.4%,

Once curiosity in this test relates to the Villa Savoye

more than double the typical range for the test images.

elevation which had a wide range of results from a low

There is no way to explain this anomaly using the

of 1.1% difference to a high of 11%. For the remainder

present results although a larger range of test images,

of the test images, a much smaller range of between

including a carefully graduated

1.3% and 4% was more common.

series of regular grid

patterns, could be examined in a future study to consider

11

ARCHITECTURE

SCIENCE No.7

June 2013

Table 3 Results of image position analysis Image Position

Square

Sejima

DEs!

1.036

% diff DEs!

VSB

Gray

1.179

1.279

1.288

1.342

1.462

1.524

0.00%

6.90%

1.80%

1.50%

1.10%

0.10%

3.60%

1.048

1.201

1.281

1.251

1.402

1.488

1.516

% diff

1.20%

4.70%

2.00%

2.20%

7.10%

2.50%

2.80%

o-;

1.036

1.248

1.261

1.273

1.331

1.463

1.488

%diff

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Centre-Base Centre· Top Centre . Centre Left· Base

Le Corbo

Grid

DEs!

1.062

1.208

1.303

1.321

1.387

1.478

1.535

% diff

2.60%

4.00%

4.20%

4.80%

5.60%

1.50%

4.70%

Left- Centre

DEs!

1.062

1.186

1.287

1.276

1.367

1.452

1.502

% diff

2.60%

6.20%

2.60%

0.30%

3.60%

1.10%

1.40%

Left-To P

DEs!

1.077

1.229

1.287

1.295

1.441

1.504

1.527

% diff

4.10%

1.90%

2.60%

2.20%

11.00%

4.10%

3.90%

Right -Base

DEs!

1.05

1.202

1.294

1.304

1.359

1.46

1.534

% diff

1.40%

4.60%

3.30%

3.10%

2.80%

0.03%

4.60%

Right -Centre

DEs!

1.05

1.19

1.285

1.261

1.343

1.433

1.5

% diff

1.40%

5.80%

2.40%

1.20%

1.20%

3.00%

1.20%

DEs!

1.062

1.221

1.287

1.289

1.41

1.477

1.523

% diff

2.60%

2.70%

2.60%

1.60%

7.90%

1.40%

3.50%

Right -Top

2.50% 0.9598 3.55% 1.000 Target 0.9625 4.13% 0.9694 2.53% 0.9465 4.28% 0.9142 3.07% 0.962 2.47% 0.9673 3.28%

o 10.00%

u

o

a. TO

t.9

•

6.00%

"if'.

DEs!

0.9645

12.00%

8.00%

R2 DTargto

Av. % diff

Wright

•

Square

•

Sejima

...

VSB

X

Gray

o

Le Corbo

•

Grid

+

Wright

.............Poly. (Square) --

4.00%

-

Poly. (Sejima) -

- Poly. (YSB) . Poly. (Gray)

2.00%

0.00%

- - -

Poly. (le Corb.)

-

. -

Poly. (Grid)

-

-

Poly. (Wright)

tmage Position

Figure 14 Results, impact of image position analysis of the same box-counting

grid; a lpt line is

4.3 Line thickness either emphatically The clearest trend in any of the results was for the line weight

factor.

It was readily

apparent

in the

preliminary analysis that, as the line weight increases, so

inside or outside a 1pt grid line,

whereas a 20pt line can be partially inside (say, 8pts) and partially

outside (12pts) which means that it will be

counted twice at that scale.

too does the calculated result (Table 4). The target line

With the thinnest line as DTarg, the DEst, value grows

weight for this comparison was therefore the thinnest,

relatively steeply in the chart as the lines thicken and as

1pt, as this cannot be counted multiple times in an

confirmed by the R2 results (Fig 15). However, both of

12

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

the first two permutations, the line weights of I and 10,

remam

each have identical results. Beyond the 10 point line

increases to 30point whereupon the error rate increases

thickness, the more abstract test images - including the

slightly, remaining at around 3.4% (20% of 17) until the

Square, Grid and Sejima's

50 point permutation

elevation - display a rapid

rate of increasing errors. Most of the other architectural

largely

unchanged

until

the

is reached

line

thickness

and the error rate

escalates.

images -by Gray, Le Corbusier and Wright - tend to Table 4 Results of line thickness analysis Line Thickness Ipt. 10 pt. 20 pt. 30 pt. 40 pt. 50 pt. 60 pt. 70 pt. 80 pt. 90 pt. 100 pt.

Square

Sejima

Gray

VSB

Le Corbo

Grid

o-;

0.977

1.196

1.279

1.279

1.397

1.382

1.51

%diff

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

DEs!

0.977

1.196

1.279

1.279

1.397

1.382

1.513

% diff

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

DEs!

0.977

1.216

1.279

1.279

1.397

1.382

1.513

% diff

0.00%

2.00%

0.00%

0.00%

0.00%

0.00%

0.00%

1.279

1.298

1.397

1.434

1.513

0.00%

5.20%

0.00%

DEs!

0.998

1.237

% diff

2.10%

4.10%

0.00%

1.90%

DEs!

1.015

1.25

1.279

1.311

1.413

1.458

1.513

% diff

3.80%

5.40%

0.00%

3.20%

1.60%

7.60%

0.00%

DEs!

0.998

1.258

1.279

1.326

1.419

1.475

1.513

% diff

2.10%

6.20%

0.00%

4.70%

2.20%

9.30%

0.00%

DEs!

1.068

1.285

1.28

1.328

1.43

1.496

1.543

% diff

9.10%

8.90%

0.10%

4.90%

3.30%

11.40%

3.00%

DEs!

1.062

1.285

1.284

1.34

1.443

1.509

1.547

% diff

8.50%

8.90%

0.50%

6.10%

4.60%

12.70%

3.40%

DEs!

1.062

1.303

1.286

1.351

1.442

1.522

1.551

% diff

8.50%

10.70%

0.70%

7.20%

4.50%

14.00%

3.80%

DEs!

1.072

1.309

1.303

1.362

1.45

1.535

1.558

% diff

9.50%

11.30%

2.40%

8.30%

5.30%

15.30%

4.50%

DEs!

1.072

1.309

1.303

1.362

1.45

1.535

1.558

% diff

9.50%

11.30%

2.40%

8.30%

5.30%

15.30%

4.50%

15.00%

13.00%

... //

Av. % diff

Wright

2

R DTarg

to DEs!

Target 0.00% 0.9981 0.29% 0.9853 1.90% 0.9736 3.09% 0.9633 3.50% 0.9453 5.81% 0.9449 6.39% 0.9315 7.06% 0.9336 8.09% 0.9336 8.09%

•

square

•

sejima

.,

Gray

X

VSB

o

leCorb.

•

Grid

+

Wright

•

L-----------------------~"7'...:..-------~

Poly. (square) --------.

Poly. (sejima)

_____ . Poly. (Gray) _.

Poly. (VSB)

---

10

20

30

40 Line Thickness

so

-

60

70

80

90

_ Poly. -

(t.e

Cor b.)

- Poly. (Grid) -

Poly. (Wright)

100

(points)

Figure 15 Results, impact of line thickness

13

ARCHITECTURE

SCIENCE. No.7. June 2013

When the complete set of line thickness results is considered

two

things

are

apparent.

First,

for

test images showed errors of over 6% magnitude for the

a

thicker permutations;

sufficiently large starting image (say, 1MB), as long as

values which are higher than for

the other factors considered in this paper.

the lines being analysed are very thin (less than l Opt in 4.4 Image Resolution this case), the impact on the results is negligible. Thus, the standard procedure of retracing every architectural elevation,

a time-consuming

analysis of the results of the image

may not be as

resolution test shows that the higher the resolution, the

critical as previously thought provided the software or

more convergent the results (Table 5). However, this is

computational method is powerful enough to examine a

not a clear linear trend, like that of the line thickness test,

much larger starting image. Second, once lines become

although it does broadly conform to the theorised ideal

marginally thicker, they have a heightened capacity to

standard (Fig 16). For this reason, the 175pdi version

produce quite substantial errors. In particular, five of the

was selected as the target. Moreover, there is another

Image Resolution

75 dpi

100 dpi

125 dpi

150 dpi

175 dpi

Square

process,

A preliminary

Table 5 Results of image resolution Le Sejima Gray VSB Corbo

analysis Grid

Wright

DEs!

0.953

1.169

1.226

1.230

1.343

1.353

1.465

%diff g-c

4.50%

5.00%

3.70%

3.6%

1.10%

0.60%

3.80%

14

14

14

17

16

14

12

o-:

0.998

1.178

1.213

1.273

1.34

1.376

1.51

%diff g-c

0.00%

4.10%

5.00%

0.70%

1.40%

2.90%

0.70%

15

15

15

17

17

15

12

DEs!

0.982

1.207

1.246

1.27

1.368

1.353

1.515

% diff

1.60%

1.20%

1.70%

0.40%

1.40%

0.60%

1.20%

g-c

16

15

15

18

17

16

13

DEs!

0.998

1.207

1.253

1.235

1.346

1.372

1.496

% diff

0.00%

1.20%

1.00%

3.10%

0.80%

2.50%

0.70%

g-c

16

16

16

19

18

16

14

o-;

0.998

1.219

1.263

1.266

1.354

1.347

1.503

% diff

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

g-c

17

16

16

19

18

17

14

R2DTarg to

Av.

% diCf

DEs,

0.8414 2.20% 0.9715 1.80% 0.9974 0.90% 0.9886 1.60% 1.000 Target

•

square

•

Sejima

A

Gray

X

V58

o

le Corb.

•

Grid

+

Wright Poly. (square)

.__ ..- Poly. (5ejima) •.••••.

Poly. (Gray)

• - - -

Poly. (V58)

----

Poly.

- - -

Poly. (Grid)

-

75

100

125 Image Resolution

150 (dpi)

Figure 16 Results, impact ofImage

14

resolution

175

-

(le

Corb.)

- Poly. (Wright)

Limits and Errors: Optimising Image Pre-Processing Standards for Architectural Fractal Analysis

compelling rationale for choosing this target, the larger

However, when additional tests were undertaken using

the image, the more grid comparisons may be undertaken

the low resolutions

by the software, and the more statistically

viable the

resolutions up to 27Sdpi (Table 6), the program was not

result. Thus, for this factor test, an additional piece of

able to return results for all of the images. At the low

information,

resolutions, the images appeared blurry and the software

the number of grid comparisons

possible,

was also tabulated.

of SOdpi and 2Sdpi and higher

was unable to detect the full extent of the images at

Despite the results generally worsening with lower

SOdpi and in some cases, could not detect the image any

image resolutions, some of the indicators for the 12Sdpi

longer at 2Sdpi (shown as 'x' in Table 6). At the higher

permutation

resolutions, particularly 2S0dpi and 27Sdpi, the amount

are superficially

superior to those of the

ISOdpi version. The former has both a lower percentage

of information processing required was too high for the

difference and a higher R2 value (both suggesting a better

program and the calculations

result) although it always has a lower grid-comparison

(shown as 'xx' in Table 6). Thus, image resolution has a

value

clear upper and lower limit beyond which a result simply

than both the

lS0dpi

and the target

17Sdpi

permutations. However, as none of the results are below

could not be completed

cannot be produced.

the reduced error zone of I% (being 20% of the highest

5 Conclusion difference S), only the target value is considered to be the best option for resolution settings for image processing.

The tests conducted in this paper reveal that two of

Another potentially misleading part of this set of results

the pre-processing

options for images have clear trends;

is that, as a by-product of changing resolution, the errors

line thickness and image resolution. In both cases, even

range from 0.00% to ±2.S%, which might seem to

seemingly minor changes in image standards can result

indicate that resolution has little impact on the image.

in errors of up to ±9%; completely

undermining

the

Table 6 Complete results for Image resolution Image Resolution DEs!

25 dpi

Square

Sejima

Gray

VSB

Le Corbo

x

x

x

x

x

% diff

--

g-c

11

8

0.866

0.857

0.841

1.229

% diff

1805.00%

-86.70%

-71.50%

-86.60%

-85.70%

-83.60%

-122.50%

g-c

13

12

12

15

12

13

xx

1.357

1.501

1.00%

0.20%

17

17

14

1.351

1.339

1.493

0.40%

0.30%

0.80%

1.00%

17

19

18

15

1.264

0.70%

0.10%

g-c

17

17

17

DEs!

1.008

1.229

1.267

% diff

1.00%

1.00%

g-c DTa.,

18

17

% difl g-c

xx

1.037

1.222

1.274

1.352

1.499

3.90%

0.30%

1.10%

0.50%

0.40%

18

17

17

18

15

1.221

xx

xx

~ 275 dpi

1.343

1.212

0%

xx

xx

xx

xx

xx

0.20%

0.30%

b-e

18

15

-

51.47

181.20

0.52

0.75

1.24

1.506

% diff

---

Av.

% diff

10

l.l0%

0.998 % diff

--

250 dpi

1.014 48.90%

0.726

~

225 dpi

0.292 105.50%

0.87

--

200 dpi

Wright

-0.05

~ 50 dpi

Grid

0.25

IS

ARCHITECTURE

SCIENCE. No.7. June 2013

Table 7 Suggested settings (assuming an image size of between 2MB and 3MB at 17Sdpi) Variable White space

Reduced error zone 30-60% increase

Image position

Centre-Centre or C-C Centre-Base 1-30 Qoints 1 Qt 175dpi 175 dpi ___________

Line thickness Image resolution

veracity of fractal dimension

Optimal setting 40-50% increase

calculations

made using

in both cases

the use of

Notes A moderate (40-50%) amount of white space around a starting image produces the most consistent result. The ideal white space is determined by measuring the shortest axis of the image space, then calculating 40% or 50% of that length, and adding half of that amount to each side of the image to define a rectangular field. The more centred the image, the more consistent the results. The thinner the line, the better the result. The higher the resolution (relative to the size of t__ h_c_e_image) the better the result. _

Acknowledgements these

settings.

particular

However,

standards or settings identified in this paper

will limit possible errors to less than ±0.5%.

Discovery

The results for variations in white space and image position

are less compelling.

An ARC Fellowship

(FT0991309)

Grant (DP1094154)

and an ARC

supported the research

undertaken in this paper.

For the former case, a

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