THESIS

GENERATION OF TERRAIN TEXTURES USING NEURAL NETWORKS

Submitted by

Santiago Alvarez

Department of Computer Science

In partial fulllment of the requirements

for the degree of Master of Science

Colorado State University

Fort Collins Colorado Fall

COLORADO STATE UNIVERSITY

September

WE HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER OUR

SUPERVISION BY SANTIAGO ALVAREZ ENTITLED GENERATION OF TERRAIN

TEXTURES USING NEURAL NETWORKS BE ACCEPTED AS FULFILLING IN

PART REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

Committee on Graduate Work

Adviser

Department Head ii

ABSTRACT OF THESIS

GENERATION OF TERRAIN TEXTURES USING NEURAL NETWORKS

Realistic visualization of terrain can b e achieved by combining color and top ographic

information Threedimensional terrain mo dels obtained from Digital Elevation Mo dels

DEMs can b e rendered by mapping aerial photographs on top of them However there

may b e terrain mo dels for which a texture photograph is not available In those cases it is

useful to have some technique which generates articial textures using terrain information

of similar regions

This pro ject explores the use of neural networks MultiLayer Perceptrons MLPs

in particular to generate articial terrain textures from DEMs This type of neural

network has b een referred as a universal function approximator In this case the network

approximates the mapping b etween elevation samples and pixel colors

Three main issues were address in this pro ject rst the quality of the textures

pro duce by the MLP second the eect of dierent input representations on the tex

ture quality in particular the eect of including slop e information and a neighborho o d

of elevation samples in the input vector and third the accuracy of the trained MLP

in rendering unseen terrain with similar characteristics as those used during training

Santiago Alvarez

Department of Computer Science

Colorado State University

Fort Collins Colorado

Fall iii

ACKNOWLEDGEMENTS

I wish to thank my advisor Dr Mike Goss for his guidance and teachings throughout

the completion of this degree I am also grateful to Dr Chuck Anderson who served

in my committee and contributed to this work with numerous discussions and reviews

Thanks also go to Dr Denis Dean for serving in my committee providing me with useful

information and reading and commenting on the initial manuscript

I wish to thank Dr Mario MejaNavarro for all the material he shared with me and

the time he devoted to help me I am grateful to Alicia C Lizarraga for providing me

with imp ortant references I have also b eneted from other friends and colleagues who

contributed to this work in many dierent ways

I express here my gratitude to my parents and sister whose encouragement has always

b een essential to my work Sin su ayuda no hubiera alcanzado esta y muchas otras metas iv

DEDICATION

A todos los que en veintiseis a nos han hecho de mi lo que soy Inclusive a aquel los

que fueron un obstaculo v

CONTENTS

Introduction

Data Description

Neural Network Mo del

Data Preparation

Normalization

Training Validation and Test Sets

Exp eriment Description

Parameter Search and Percentage of Data Sampled

Including the Surface Normal Vector

Using the HSV Color Mo del

Network Generalization

Results

Parameter Search and Percentage of Data Sampled

Including the Surface Normal Vector

Using the HSV Color Mo del

Network Generalization

Discussion

REFERENCES vi

LIST OF FIGURES

Example of a blo ck diagram of a DEM and its aerial photograph

A twolayer p erceptron showing the notation for units and weights

Two common activation functions used for MLPs

Lo calization of cam cam cam and cam

Registered aerial photographs

Blo ck diagrams of the registered digital elevation mo dels

DEM and photograph RGB histograms

DEM and photograph HSV histograms

Test and training errors for training sets with dierent sizes

Results obtained using a element neighborho o d of elevation samples

Results obtained using a element neighborho o d of elevation samples and the

surface normal vector

Slop e discontinuities in the DEMs that aect the elevationtocolor mapping

Color distributions of the target and output textures for cam

Results obtained using the single elevation sample and the surface normal vector

Results obtained using a element neighborho o d of elevation samples and the

surface normal vector HSV outputs

Network approximation of cam cam and cam from cam data

RMSE b etween target and output textures using network trained with cam

data vii

LIST OF TABLES

Lo cation of the data sets in Universal Transverse Mercator UTM co ordinates

Distribution parameters of the combined DEMs and the combined aerial pho

tographs

Elevation ranges of the DEMs used viii

Chapter

INTRODUCTION

Terrain textures are essential for rendering highquality terrain images Realistic vi

sualization can b e obtained by combining color and top ographic information Cohen and

Gotsman Miller Taylor and Barrett Terrain color can b e obtained

from aerial or satellite images These photographs usually cover a small p ortion of ter

rain from a vertical view angle Topographic information can b e obtained from Digital

Elevation Mo dels DEMs A DEM contains a number of elevation samples that can b e

extracted from stereoscopic terrain photographs

The rendering of terrain mo dels using computer graphics has multiple applications

It is used by the military for ight simulation simulation of the displays of some electro

optical weapons and mission planning This technology is also used in the civilian market

for urban and rural planning the generation of animations and many other applications

The most common metho d to render realistic images of actual terrain is to obtain a

D p olygonal mo del of the terrain from a DEM Fowler McCullagh Tarvy

das Scarlatos This p olygonalization reduces the complexity of the terrain

representation allowing faster access and pro cessing Once the terrain mo del has b een

created the terrain texture is mapp ed onto it and pro jected according to the desired

view The nal realism of the rendered image dep ends on the resolution and quality of

the D mo del generated and the realism of the terrain texture mapp ed onto the mo del

However terrain textures are not always available In those cases where a terrain

texture cannot b e obtained it would b e useful to have some technique that would generate

terrain textures with an acceptable quality using elevation and land cover information In

such cases an articially generated mo del can b e rendered Such a mo del could b e the

result of a simulation pro cess that tries to predict some terrain parameters or could also

b e an arbitrary mo del built for visualization purp oses

This pro ject explored the use of neural networks to generate articial terrain textures

from terrain mo dels A simplied mo del structure containing only elevation data was used

The goal was to b e able to generate terrain textures from elevation data using a neural

network trained with available terrain mo dels and photographs Sp ecically it was of

most interest to determine the quality of the terrain textures that could b e obtained from

elevation data and how the input representation could aect the results Additionally

the generality of this approach was addressed

The simplied terrain mo del was exp ected to pro duce enough information to address

the texture generation problem The app earance of terrain at a given p oint is highly

inuenced by its altitude and slop e This relationship should b e similar within a limited

geographical area If such a close relationship exists a neural network can b e used to nd

an interpolation of the mapping b etween elevation and color

Even though elevation is not the only factor that determines the terrain texture it

should play an imp ortant role Other factors such as light exp osure type of soil and

weather have a ma jor impact on the kind of vegetation and ro cks found on some terrain

However their inuence on the app earance of the terrain was assumed constant or closely

related to elevation or slop e for this pro ject

We can overcome the lack of a general mo del that relates elevation with terrain

texture by using neural networks One feature of neural networks is that they can b e

trained on a relatively small number of examples of a relationship Once trained the

network can induce a complete relationship that interpolates in a sensible way The use of

this characteristic should give some insight ab out how close a relationship there is b etween

elevation and texture and how realistic the results pro duced by a trained network are

Data Description

Two basic types of data were considered in this pro ject Digital Elevations Mo dels

DEMs and aerial photographs As dened by Burrough any digital representa tion of relief over space is known as a DEM Their uses include

Elab oration of top ographic maps by government agencies

Supp ort of construction pro jects esp ecially for volume estimations

Display of image simulation mo dels for military purp oses and landscap e planning

Analysis of statistical features and comparison of dierent kinds of terrain

Elab oration of Geographic Information Systems GIS

There are two basic metho ds to represent DEMs mathematical metho ds and image

metho ds Mark The image metho ds are divided into line mo dels sp ecied usually

as contour lines and p oint mo dels which are commonly given as altitude matrices These

matrices or regular grids are the most common form of DEMs and are particularly p opular

b ecause of the ease with which they can b e handled using a computer Figure a This

type of DEM was used in this pro ject to obtain elevation information

z (elevation) 3600 3500 3400 3300 3200 3100 3000 2900 2800 500 400 0 300 100 200 y 200 300 100 x 400

500

a b

Figure a Blo ck diagram of a DEM b Aerial photograph of the same terrain taken

from one of the data sets used

Although altitude matrices are the most p opular format for DEMs they have some

imp ortant disadvantages First the terrain elevation is sampled at the same rate in areas

of high detail and areas of low detail The resolution may b e to o coarse in areas with

critical features while uniform areas may contain redundant information In addition the

orientation of the axis makes computations along the grid lines simple while computations

at other angles require trigonometric calculations to compute distances and angles

Altitude matrices are usually obtained from stereoscopic aerial photographs by using

a matching pro cess to obtain depth information Gro First a corresp ondence b e

tween pixels in the stereoscopic photographs is determined Once the image co ordinates

are known for a given pixel the world co ordinates of that p oint can b e found by trian

gulation using the camera geometry This pro cess is sensitive to distortions and errors

during the pixel corresp ondence phase Thus subpixel interpolations or multiple views

can b e used to provide higher accuracy

The US Geological Survey USGS p ossesses DEMs covering all of the contiguous

United States Hawaii and limited p ortions of Alaska US Geological Survey a

Three distinct digital elevation pro ducts are distributed for the contiguous United States

Degree DEM A by degree blo ck with a by arcsecond data spacing

Minute DEM A by minute blo ck with a by meter data spacing

Minute DEM Four by Minute blo cks with a by arcsecond data spac

ing

The other type of data used in this pro ject was obtained from terrain images that

provide color texture information Figure b These images are acquired from aerial

or satellite photographs which provide a vertical view of the terrain When an ortho

graphic pro jection of the terrain scene cannot b e assumed a geometric recitication of

the image is required Jensen This pro cess recties the original image to a map

co ordinate system Initially a spatial interpolation is p erformed by identifying ground

control p oints GCPs in the original image and on a reference map Finally an intensity

color interpolation is p erformed since there is not a onetoone mapping b etween the

input pixel and output pixel lo cations

Neural Network Mo del

The MultiLayer Perceptron MLP was the neural network mo del considered for this

pro ject This network is an extension of the Simple Perceptron which consists of a single

layer of pro cessing units An MLP overcomes the limitations of the single layer p erceptron

which can only b e used to classify inputs that are linearly separable That is the training

algorithm for a single p erceptron converges to a solution where the connection weights

sp ecify a hyperplane that divides the input space into two parts

An MLP is a feedforward neural network with one or more layers of pro cessing units

no des b etween the inputs and the output layer Figure The units that directly

pro duce the network output are called output units whereas the other units are called

hidden units Each unit p erforms a simple linear combination of its inputs and then

applies a function to obtain an activation value for the unit The linear combination of

the inputs is computed using the connection weights asso ciated with each input

y y 1 2 yk Output Units

w'11 w'42

x'1 x'4 x'j Hidden Units

w11 w34

xi Inputs

x1 x2 x3

Figure A twolayer p erceptron showing the notation for units and weights

The MLP unlike the simple p erceptron uses a continuous activation function This

allows the network to b e used not only for classication tasks but also for function ap

proximation problems in general This pro ject exploited this feature to nd a mapping

b etween elevation and color MLPs have b een referred to as universal approximators Hornik Chen and Jain

This type of neural network was not used in the past b ecause of the lack of an eective

training algorithm to determine adequate connection weights The invention of the back

propagation learning algorithm showed how to make an MLP learn a particular function

It was invented indep endently by Bryson and Ho Werbos Parker

and Rumelhart Hinton and Williams a b

The backpropagation algorithm is based on a gradient descent technique that min

imizes the output error Using the notation given in Figure the error function to b e

optimized is given by

X X

2

y E w d

l l

l

where d is the desired target output value for unit l y is the actual unit output and

l l

is an index into the set of training vectors Since a target or desired output is required

for every input during training backpropagation is classied as a sup ervised learning

algorithm

Given the cost error function the gradient descent rule yields the following

equation for the inputtohidden connections

E

w

ij

w

ij

Similarly we obtain for the hiddentooutput connections

E

0

w

j k

0

w

j k

where is called the learning rate which is a scale factor that indicates how far to move

in the direction of the gradient The learning rate may b e dierent for each layer A

momentum term can also b e included to improve the results obtained with equations

and Hertz Krogh and Palmer This term helps attenuate the oscillation

b ehavior that can take place during learning

The activation function has to b e dierentiable in order to b e able to apply this

gradient descent technique It is common to use a sigmoid function for the activation

function The functions generally used are dierentiable and generate b ounded outputs Figure shows two common activation functions

1.5 Logistic function Hyperbolic tangent 1

0.5

0 Activation value -0.5

-1

-1.5 -10 -8 -6 -4 -2 0 2 4 6 8 10

Weighted sum of inputs (h)

Figure Two common activation functions used for MLPs the logistic function

h h

e e 1

and the hyperb olic tangent g h tanhh g h

h h h

1+e e +e

The iterative application of the weight up date rules needs to b e stopp ed at an appro

priate stage If that pro cess is p erformed for to o long the network will pro duce excellent

results for the inputs used for training but p o or results for new inputs unless the network

is simple enough to not overt At that p oint the network has lost its generalization

feature This ability to generalize is highly desirable since the training data is usually a

relative small group of samples of the input space

A crossvalidation mechanism can b e used to avoid losing the generalization feature

of the network during training The inputtarget pairs available for training are divided

into three sets a training set a validation test and a test set The pairs in the training

set are actually used for training The inputs and targets in the validation set are used

at every ep o ch to measure the error that the network makes with data that is not b eing

used for training The Ro ot Mean Square Error RMSE b etween each single output and

target is used as an error measure The desired set of connection weights is found when

the lowest error of the validation set pairs is detected At that p oint the test set is used

to compute an estimation of the network error on a totally new data set that has not b een

used either for training or validation

There are several features that make MLPs an app ealing technique They can op erate

with noisy data Each unit relies only on lo cal information to compute its activation value

The large number of connections provides a high degree of redundancy that makes the

network faulttolerant Only a subset of the p ossible inputs is needed to train the network

in many cases Once it is trained the network interpolates for unseen input vectors

However there are some limitations and disadvantages that need to b e considered

The input and output data may need to b e prepro cessed normalized scaled translated

etc in order to exploit the information it can provide and obtain the desired results The

training phase is an iterative pro cess that can b e computationally exp ensive That is

the training data may have to b e pro cessed by the network a number of times b efore a

satisfactory solution is reached Moreover the learning algorithm is not guaranteed to

converge to a global optimal solution Although this is not a critical problem several

training sessions may b e needed to nd a go o d set of weights for a given architecture

Besides there is not a clear metho d to know a priori the right architecture and parameters

for a sp ecic problem

Some successful applications of MLPs that have b een published include sp eech recog

nition Lippman handwritten ZIP co de recognition Le Cun et al car

navigation Pomerleau sonar target recognition Gorman and Sejnowski a

b phoneme generation from text Sejnowski and Rosenberg and signal prediction and forecasting Lap edes and Farber Farmer and Sidorowich

Chapter

DATA PREPARATION

This pro ject used four DEMphotograph pairs cam cam cam and cam They

contain information for a small area in the Colorado State Forest next to the northwest

b order of the Ro cky Mountain National Park in Northern Colorado Some of the natural

features covered include Michigan River Diamond Peaks Seven Utes Mountain Mount

Mahler Lake Agnes Nokhu Crags Snow Lake Static Peak Mount Richthofen and Tepee

Mountain Table shows the exact lo cation of the data sets in Universal Transverse

Mercator UTM co ordinates

The relative lo cation of the four data sets is presented in Figure Hidrographic

information and the contour lines of the area are also displayed The elevation data used

was obtained from minute Digital Elevation Mo dels created by the US Geological

Survey Most of the samples are inside of the DEM Mt Richthofen No The

four data sets partially overlap In particular Lake Agnes the lake with a small island

in the middle is included in all data sets Some p ortions of the adjacent DEMs Fall

River Pass No Chambers Lake No and Clark Peak No

are also covered These four minute DEMs can b e obtained from the US Geological

Survey httpwwwusgsgov

Each DEMphotograph pair was previously registered so b oth the DEM and the aerial

photograph had the same lo cation area and sampling rate That is each sample on the

altitude matrix corresp onds to the pixel in the same lo cation on the aerial photograph

and vice versa The original minute DEMs had a data spacing of by meters

After the registration pro cess b oth the DEMs and the aerial photographs had a by

meter data spacing This resampling pro cess was p erformed using a linear interpolation of the original elevation samples in the minute DEMs

Table DEMPhotograph Lo cation in Universal Transverse Mercator UTM co ordi

nates

UTM Co ordinates cam cam cam cam

North

South

East

West

Figure Relative lo cation of cam cam cam and cam The DEM Mount Richthofen

No is outlined as a reference

Figure shows the four aerial photographs Cam and cam are rather dominated

by a single type of terrain texture Cam includes the highest elevations and therefore

covers terrain with scarce vegetation Cam on the other hand contains the lowest eleva

tions which have little ro ck texture Cam and cam show a greater degree of variation in

color Lake Agnes can b e easily identied in cam cam and cam but is o cluded in cam

lower right corner Cam was selected as the primary data set for exp erimentation The

other three were left to analyze the ability of the network to generalize using new terrain

cam cam

cam cam

Figure Registered aerial photographs

Figure shows the cam cam cam and cam DEMs They are shown as blo ck

diagrams by creating a square grid with adjacent samples The resolution of the diagrams

is thirty times lower than the original DEMs All four data sets have elevation samples

within similar ranges Cam and cam cover adjacent fragments of the valley north of

Lake Agnes and Snow Lake Cam covers the wavy area to the left of Mount Richthofen

Cam covers the area with the steep est slop es including Mount Mahler Mount Richthofen and Static Peak

z (elevation) z (elevation) 3800 3800 3600 3400 3600 3200 3400 3000 3200 800 700 700 600 600 500 0 500 0 400 100 200 400 100 300 300 y 200 300 y 400 300 500 200 400 200 600 500 700 100 600 100

x 800 x 700

cam cam

z (elevation) z (elevation) 3600 3600 3400 3400 3200 3200 3000 3000 800 800 700 700 600 600 0 500 0 500 100 100 200 400 200 400 300 y 300 y 400 300 400 300 500 500 600 200 600 200 700 700 100

x 800 100 x 800

cam cam

Figure Blo ck diagrams of the registered digital elevation mo dels

Normalization

Initial tests using MLPs to nd the elevationtocolor mapping were p erformed with

small pieces of the cam terrain The input vector was built using a nineelement neigh

b orho o d for each elevation sample The output vector contained three color comp onents

at each given pixel Despite the increase in the dimensionality of the input vector using

a neighborho o d was exp ected to pro duce b etter results The single elevation value was

b elieved to provide insucient information for a satisfactory estimation of the terrain

color at that p oint Two p oints at the same altitude can have extremly dierent colors

dep ending on vegetation the presence of water and terrain features like slop e The use

of a neighborho o d was exp ected to help cop e with this problem

Due to the homogeneity present in the small fragments of terrain the network mapp ed

all inputs to the same outputs The nal result was an image with the same gray value

in all its pixels This problem was exp ected to b e alleviated in part using larger pieces

of terrain with more signicant dierences in elevation However a more general solution

was selected by transforming the data so that the eect of the width of the input and

output ranges could b e further reduced

The four data sets were linearly transformed using the elevation and color means

and standard deviations Equation This normalization converts the original data

to a distribution with zero mean and unit standard deviation where x is an elevation

i

X is the sample mean and is the sample standard deviation This or color sample

X

transformation enhances the relative dierences b etween samples while preserving the

0

original distribution by translating and scaling each value x to a new sample x

i

i

x X

i

0

x

i

X

However the parameters used for normalization must b e obtained from the distri

bution asso ciated with the terrain to which the neural network is exp ected to generalize

Using a general distribution guarantees that all inputs that would eventually b e fed to

the network would b e scaled consistently In order to approximate a general distribution

for the type of terrain considered the four data sets cam cam cam and cam were

blended to obtain a combined set of elevations and a combined set of pixel colors These

two sets were used to obtain the normalization parameters

Figure shows distributions of the combined data sets Figure a shows the

elevation histogram of the samples in the four original DEMs Figures b c and d show

the combined color distributions using the RedGreenBlue RGB color mo del Table

shows the parameters of the elevation and color distributions including the mean and

standard deviation used for normalization The parameters for the color distributions in

the HueSaturationValue HSV color mo del are also included

500000 500000

450000 450000

400000 400000

350000 350000

300000 300000

250000 250000

Frequency 200000 Frequency 200000

150000 150000

100000 100000

50000 50000

0 0 2827 2939 3050 3162 3273 3385 3497 3608 3720 3831 3943 0 25 50 75 100 125 150 175 200 225 250

Elevation (meters) Intensity (Red)

a b

500000 500000

450000 450000

400000 400000

350000 350000

300000 300000

250000 250000

Frequency 200000 Frequency 200000

150000 150000

100000 100000

50000 50000

0 0 0 25 50 75 100 125 150 175 200 225 250 0 25 50 75 100 125 150 175 200 225 250

Intensity (Green) Intensity (Blue)

c d

Figure Histograms of the combined DEMs and the combined photographs using the

RGB color mo del a Elevation histogram b Red comp onent histogram c Green

comp onent histogram d Blue comp onent histogram

The HSV color mo del was additionally considered to train the network The orig

inal color information was in RGB format Therefore the color comp onent had to b e

transformed to generate the new outputs b efore training The shadows present in the

Table Distribution parameters of the combined DEMs and the combined photographs

using the RGB and the HSV color mo dels

Parameter Elevation Red Green Blue Hue Saturation Value

Lower Bound

Upp er Bound

Minimum

Maximum

Mean

Standard Dev

photographs used for training aect the network p erformance Using the HSV mo del a

shadow should aect mainly the value comp onent of the pixel Using the RGB mo del

on the other hand all three color comp onents are aected by a shadow The same nor

malization pro cedure was p erformed on the transformed color comp onents Figure

shows the distributions of the combined data sets using the HSV color mo del including

the distribution of the combined elevation samples again

The normalized inputs were directly fed to the network without any further trans

formation However the normalized output values needed to b e mapp ed to the prop er

output range of the network The exact parameters of this transformation dep ended on

the activation function used in the output units used and the minimum and maximum

values of the color comp onents

Only a p ortion of the total activation function range was used when transforming the

color comp onents The two asymptotic ends of the sigmoid functions see Section

are hardly reached by a network A extremely high or low weighted sum is required to

pro duce an output close enough to the resp ective asymptote The center of the range

was actually used when mapping the outputs in order to accelerate the network training

and improve p erformance

Training Validation and Test Sets

Cam was used as the primary data set for exp erimentation It contains approxi

mately elevation and color samples A training set a validation set and a test

500000 500000

450000 450000

400000 400000

350000 350000

300000 300000

250000 250000

Frequency 200000 Frequency 200000

150000 150000

100000 100000

50000 50000

0 0 2827 2939 3050 3162 3273 3385 3497 3608 3720 3831 3943 0 36 72 108 144 180 216 252 288 324

Elevation (meters) Intensity (Hue)

a b

500000 500000

450000 450000

400000 400000

350000 350000

300000 300000

250000 250000

Frequency 200000 Frequency 200000

150000 150000

100000 100000

50000 50000

0 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Intensity (Saturation) Intensity (Value)

c d

Figure Histograms of the combined DEMs and the combined photographs using the

HSV color mo del a Elevation histogram b Hue comp onent histogram c Saturation

comp onent histogram d Value comp onent histogram

set were created by sampling the data set at random These three sets are used in the

crossvalidation mechanism describ ed in chapter

The elevationcolor pairs were divided in dierent prop ortions to create the three

sets The validation set was created with of the sample pairs Dierent sizes were

used for the training set Preliminary tests showed that the validation error decreased

asymptotically with the p ercentage of pairs used for training That is there is little

dierence in the results for high p ercentages of samples Therefore a reduced number of

elevationcolor pairs could b e used to train the network with a satisfactory p erformance

Based on these observations the p ercentage of samples taken to build the training set

was sub ject to exp erimentation The actual values used are describ ed in the next chapter

Finally the test set was constructed with all the pairs that were not used for the training

set or the validation set

Chapter

EXPERIMENT DESCRIPTION

Parameter Search and Percentage of Data Sampled

One of the limitations of most neural networks is the lack of a clear theoretical base

to select adequate values for their parameters Useful values are generally exp ected within

certain ranges but sp ecic values in those ranges can generate networks with imp ortant

dierences in p erformance Moreover adequate values are generally problem dep endent

and previous exp eriences are not always useful Therefore some exploration of the pa

rameter space is required to ensure that an acceptable p erformance is achieved

However the number of scenarios dened by the combination of several parameter

values easily b ecomes to o exp ensive Therefore the number of parameters to b e explored

and the number of values that each of them can take need to b e reduced This reduction

of the parameter space do es not guarantee an optimal conguration but certainly makes

more manageable the selection of parameter values

An initial approach to the selection of appropriate parameter values was p erformed

on a no de CNAPS Server I I from Adaptive Solutions Inc Even though the training

time is greatly reduced by this machine the lack of oatingp oint arithmetic aects the

results This limitation should not b e imp ortant for classication applications but it is

prejudicial for function approximation purp oses This constraint led to the use of a single

pro cessor workstation reducing the number of scenarios to b e studied due to the lower

computing capacity

Four basic network parameters were xed number of layers momentum term acti

vation function and ratio b etween the learning rate in the hidden layer and the learning

rate in the output layer A twolayer network was used since it is the minimum number of

layers required to overcome the limitations of a single p erceptron The parameter values

selected for exp erimentation were chosen based on the preliminary results obtained with

the CNAPS Server The momentum term was set to zero and the ratio b etween the hid

den layer learning rate and the output layer learning rate was set to four The traditional

logistic function was used as the activation function for b oth layers

g h

h

e

The number of units in the hidden layer and the learning rate were the two parameters

explored Three dierent numbers of hidden units were considered and Two

learning rates were used and These values dene six dierent scenarios that

were explored

Each exp eriment dened by the combination of hidden units and learning rate was

replicated ve times Exp eriment replications allow computation of a b etter estimation

of the error measures obtained during training Replications also give some idea of the

variability of the results This replicated parameter search was p erformed with ve dif

ferent training sets All of them were taken from cam but contain dierent amounts of

training information as explained in the next section

Preliminary exp eriments showed the necessity for reducing the number of elevation

color pairs used for training The execution time dramatically increased in prop ortion to

the number of pairs However the test error did not decrease prop ortionally Therefore

it was desirable to use as few pairs for training as p ossible but using enough samples to

nd a prop er elevationtocolor mapping

Five dierent p ercentages of sample pairs from cam were used to p erform the pa

rameter search and That is when of the pairs were used for

training each sample was selected with probability The eight closest neighbors of

each selected sample were used to build an input vector with nine comp onents This input

vector was exp ected to pro duce b etter results since it provides more global information

that just the single sample

Including the Surface Normal Vector

A DEM describ es a piece of terrain as a twodimensional elevation function which

can also b e visualized as a threedimensional surface The normal vector is one of the

attributes of a surface This vector gives information ab out the surface orientation at a

given p oint The eect of adding the normal vector to the input vectors was studied

This inclusion increased the length of the input vector by four comp onents three normal

comp onents and the magnitude value of the vector

The inclusion of the normal vector in the input vector may lo ok redundant As

mentioned b efore the input vector was initially built using a element neighborho o d

the elevation sample and its eight adjacent neighbors Therefore the normal could b e

estimated from these nine inputs by the network if it is imp ortant to estimate the outputs

However the use of additional information should yield b etter results since it implies some

amount of prepro cessing

The normal vector was estimated using the elevation gradient vector G hg g i

x y

The normal vector n is computed by taking the cross pro duct of the two gradient com

p onents equation

n g g

x y

The gradient comp onents were computed using an adjustable gradient lter prop osed

by Goss This lter represents an improvement over the xedresp onse lters

commonly used to approximate the gradient vector The lter is generated by truncating

limiting in space an ideal gradient lter and multiplying it by a window function Kaiser

window The width of the window function can b e adjusted using a parameter which

controls how quickly it tap ers o to zero at the edges Dierent values pro duce dierent

levels of smo othing in the gradient

For a lter with an even number of nonzero elements M M element to the right

and left of the center element zero the convolution with the elevation data x that

ij

estimates the gradient vector hg g i is given by

x y

M 2

m

X

I

0

g x

x i+mj

I m

0

m=M 2

M 2

m

X

I

0

x g

ij +m y

I m

0

m=M 2

where I is the order zero mo died Bessel function of and is calculated using

0

m and M as follows

s

2

m

M

A lter with elements and an value of four was used to compute the gradient

information lter values This

conguration preserves the high frequency detail in the elevation grid much b etter than

the traditional centerdierence metho d used for gradient estimation Goss

Using the HSV Color Mo del

Shadows are one of the problems that may aect the generation of terrain textures

Aerial photographs are taken under sp ecial atmospheric conditions and during sp ecic

p erio ds of time However shadows have to b e tolerated to some extent A small amount of

clouds could still b e present or the sun may not b e right on its zenith The transformation

of the output color values to the HSV color mo del is exp ected to reduce the eects that

shadows can have on the neural network Shadows act as a very strong source of noise

since p oints at similar elevations can have imp ortant color dierences due to shadows

The HSV HueSaturationValue is a color mo del that is based on the intuitive app eal

of the artists mo del of tint shade and tone The value of H ranges from zero to

The values of S and V go from zero to one A hue value with saturation and value

one represents the pure pigment used as starting p oint in mixing colors Decreasing S

corresp onds to adding white to the original pigment Decreasing V on the other hand

corresp onds to adding black to the pigment

The V comp onent is exp ected to b e the main dierence b etween two p oints with

similar terrain cover but one covered with shadow The presence of shadows on cam is

esp ecially evident on the highest p eaks In most p eaks one of the slop es was receiving

light directly from the sun while the other was receiving reected light Two of the three

color comp onents hue and saturation are exp ected to b e similar for p oints in either of

the two slop es In the RGB color mo del the three color comp onents are exp ected to b e

dierent for those p oints

A set of exp eriments were run using the HSV color mo del The training pairs were

constructed with the transformed color values Foley et al describ e the algorithms

required to convert from the RGB color mo del to the HSV mo del The same structure

of the input vector describ ed in Section was used a element neighborho o d plus the

normal vector information of the samples in cam was taken at random for training

and the same parameter search for the learning rate and the number of hidden units was

p erformed

Network Generalization

Once trained a neural network is exp ected to generate a sensible approximation of

similar inputs Cam cam and cam were used to test the capacity of the neural network

to generalize terrain textures These exp eriments attempted to determine if the elevation

tocolor mapping found by the neural network can b e applied to nearby terrain It is

imp ortant to determine this generalization capacity since this would b e the intended use

of this technique The trained network should b e used to generate an approximation of a

texture from a terrain mo del that do es not necessarily exist Previous exp eriments used

cam to train and test the neural network

The input vector describ ed in Section was used for these DEMs This vector has

the highest dimensionality of the ones tried and it contains the most information ab out

the data sets The network used was the one that gave the lowest test error after b eing

trained with data from cam Section discusses the architecture and parameters of

this network

The neural network is exp ected to color unseen elevation samples by interpolating

from the pairs that were used for training However in some cases the network had to

color elevations that were not in the range of the training data Table shows the

elevation ranges of the DEMs used Cam has higher elevations and cam has lower

elevations than the ones from cam used for training Furthermore cam has steep er

slop es that the one in cam In those cases the neural network also had to extrap olate

p oints

Table Elevation ranges of the DEMs used

DEM Minimum Elevation Maximum Elevation

cam m m

cam m m

cam m m

cam m m

Chapter

RESULTS

This chapter contains the results of the exp eriments describ ed in the previous chapter

Each section matches one section in chapter where all the details of the exp eriments are

presented Section contains the results of the parameter search p erformed with an

input vector built with a element neighborho o d The results obtained by including the

normal vector and its magnitude in the input vector are presented in section Section

presents the results of transforming the color information to the HSV color mo del

Finally the results of the generalization exp eriments are included in section

Parameter Search and Percentage of Data Sampled

Figure presents the test and training errors for the two learning rates

and and the ve dierent p ercentages of samples and

considered All exp eriments used a element neighborho o d of elevation samples as input

vector and the color outputs in RBG formatThe results obtained with Figures

a and b are similar to the ones observed in preliminary exp eriments Both the test

and the training error decrease as the p ercentage of samples used increases However the

b ehavior of the test error for was signicantly dierent In this case the test error

shows a minimum around That minimum has the lowest test error of all the scenarios

considered

The b ehavior in Figure b may b e a symptom of overtting As the p ercentage of

samples increases the network is trained with a larger number of pairs b efore checking

for overtting If the addition of more samples is not providing new information there

is a greater chance that the network overts the training pairs and loses its ability to generalize

0.21 25 hidden units 0.21 25 hidden units 35 hidden units 35 hidden units 45 hidden units 45 hidden units 0.2 0.2

0.19 0.19

0.18 0.18

0.17 0.17 RMSE RMSE 0.16 0.16

0.15 0.15

0.14 0.14

0.13 0.13

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

Percentage of samples Percentage of samples

a b

0.21 25 hidden units 0.21 25 hidden units 35 hidden units 35 hidden units 45 hidden units 45 hidden units 0.2 0.2

0.19 0.19

0.18 0.18

0.17 0.17 RMSE RMSE 0.16 0.16

0.15 0.15

0.14 0.14

0.13 0.13

0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40

Percentage of samples Percentage of samples

c d

Figure Test and training errors RMSE for training sets with dierent sizes a

Test error with b Test error with c Training error with d

Training error with

Based on the previous result the p ercentage of samples used in training was xed

to for all the exp eriments This might not necessarily b e the b est value since the

parameter search was not exhaustive However this p ercentage yielded the b est results

and help ed reduce the training time to reasonable values Training using of the sam

ples typically required b etween three and ve hours dep ending on the actual network

parameters and architecture and the load on the HP workstations used

The search for go o d values for the learning rate and the number of hidden units was

p erformed indep endently for each of the dierent input vectors considered The network

conguration that pro duced the b est results for the nineelement neighborho o d considered

at this p oint had and hidden units These values were consistent with the

number of hidden units and the learning rate that were found b est for most of the other

scenarios The test error during training for this scenario was

Figure shows the results obtained with this input vector nineelement neighbor

ho o d of elevation samples The output texture the target texture cam and an error

image are displayed as a at image and mapp ed onto the terrain The error image cor

resp onds to the distance b etween pixels in RGB space A darker gray means a greater

distance b etween pixels ie a higher error magnitude

The regions that have a homogeneous land cover in the output and the target texture

have a close resemblance The rst dierence that can b e noted is the lack of high

frequency detail The output texture do es not show the sp ots at low elevations that are

not covered with vegetation There is also a smo oth transition b etween ro ck and vegetation

that is much sharp er in the target texture Moreover the network seems to dene several

thresholds on the elevation where the terrain color changes smo othly The smo othing

eect observed in the output is directly asso ciated with the noise removal prop erties of

neural networks and the use of a neighborho o d as an input vector

The colors in the target and the output textures also have a close resemblance es

p ecially in the areas covered with vegetation These areas have a bluish app earance on

b oth textures but the red comp onent has lower values in the target texture The bluish

app earance of the target texture can b e explained by the multiple factors that inuence

p erception of distant vegetation In particular there are atmospheric eects light scat

tering and absorption that distort color

The color approximation of lake Agnes is esp ecially p o or There is not a sharp edge

b etween water and ground and the lake shore is totally missing in some parts of the lake

Besides the vegetation and water color are similar However these two colors are also

very alike in the target texture In this case the lake is mainly recognized by its shap e

Moreover the small island inside the lake is not present in the output This situation is

caused by the lack of elevation samples of that feature in the original minute DEM

As shown on the error image Figures e and f the areas with the highest error

are along the tree line These errors are exp ected since the tree line is not dened by

a b

c d

e f

Figure Results obtained using a element neighborho o d of elevation samples a

output image b Output image mapp ed onto the terrain c Original photograph target

image d Original photograph mapp ed onto the terrain e Error image dierence b etween output and target images f Error image mapp ed onto the terrain

a hard elevation threshold That is there are p oints with similar elevation that fall in

areas covered with vegetation while others are covered by ro ck This problem is directly

asso ciated with the fact that the color should b e a function of many variables not just

elevation

Including the Surface Normal Vector

The b est results obtained with this extended input vector element neighborho o d of

elevations samples three normal vector comp onents and normal magnitude are presented

in Figure The network conguration used to get this result had and hidden

units all other parameters remaining the same The test error was

The output texture has some artifacts that give it a blo cky app earance The regu

larity of those artifacts made very unlikely that they were caused by the neural network

Therefore the elevation values were checked at those p oints where ma jor color discontinu

ities were detected Figure shows how signicant elevation discontinuities were found

in the DEM

The distance b etween discontinuities shows that this problem was originally in the

minute DEMs It was not created by the linear interpolation of elevation samples that

was p erformed during the registration step of the DEMs and the aerial photographs This

problem can b e approached by smo othing the elevation map Dudgeon and Mersereau

Another approach is to apply a nonlinear interpolation that guarantees continuity

Press et al

There are several features where the improvement over the previous result can b e

p erceived The tree line is much b etter dened Also the network pro duced a similar

color dierence b etween the two slop es of Mount Mahler on the left An analogous shadow

pattern is generated but the color generated on the right side of the mount is darker than

desired The error image Figures e and f shows how the dierence b etween the

output and the target textures concentrates on the highest elevations

There is an imp ortant improvement in the app earance of Lake Agnes Ab out half

of the lake shore is now clearly dened The b order of the lake seems to b e partially

a b

c d

e f

Figure Results obtained using a element neighborho o d of elevation samples and the

surface normal vector a output image b Output image mapp ed onto the terrain c

Original photograph target image d Original photograph mapp ed onto the terrain e

Error image dierence b etween output and target images f Error image mapp ed onto the terrain

z (elevation) z (elevation)

3350 3300 3325 3275 3300 3250 3275 3225 3250

35 35 30 30 25 25 0 20 0 20 5 5 10 15 y 10 15 y 15 20 10 15 10 25 5 20 5

x 30 35 0 x 25 0

Figure Slop e discontinuities in the DEMs that aect the elevationtocolor mapping

surrounded by steep slop es The inclusion of the normal vector generates this feature on

the output texture However there are other p ortions of the lake esp ecially the upp er left

side where its b oundaries are still p o orly approximated The water color and the island

inside the lake remain as problems that do not b enet from the inclusion of the normal

vector

In general the inclusion of the normal in the input vector improved the app earance

of the the output texture despite the increase in the test error from to due

to a greater color dierence in the output texture This dierence is mainly asso ciated

with the red comp onent The red values at low elevations are higher and at high elevation

are lower that they should b e

Figure shows how the network generates color distributions for the red and blue

comp onents that resemble the original distributions The lo cation of the highest p eaks are

approximated with an acceptable degree of accuracy despite the dierences in magnitude

and the dierences in color ranges However the network pro duces a p o or result for the

red comp onent The original distribution reects the great number of samples that have a

very high red comp onent which corresp ond to ro ck The output distribution on the other

hand has a p eak around the middle This error accounts for most of the color dierences in the output texture

250000 250000

200000 200000

150000 150000

Frequency 100000 Frequency 100000

50000 50000

0 0 0 25 50 75 100 125 150 175 200 225 250 0 25 50 75 100 125 150 175 200 225 250

Intensity (Red) Intensity (Red)

a b

250000 250000

200000 200000

150000 150000

Frequency 100000 Frequency 100000

50000 50000

0 0 0 25 50 75 100 125 150 175 200 225 250 0 25 50 75 100 125 150 175 200 225 250

Intensity (Green) Intensity (Green)

c d

250000 250000

200000 200000

150000 150000

Frequency 100000 Frequency 100000

50000 50000

0 0 0 25 50 75 100 125 150 175 200 225 250 0 25 50 75 100 125 150 175 200 225 250

Intensity (Blue) Intensity (Blue)

e f

Figure Color distributions of the target and output textures for cam a Red

comp onent target b Red comp onent output c Green comp onent target d Green comp onent output e Blue comp onent target f Blue comp onent output

After nding how the normal vector improves the results a reduced version of this

input vector was tried The nine input comp onents corresp onding to the nineelement

neighborho o d were reduced to only the center element The comp onents corresp onding

to the normal vector remained in the same p osition Since the neural network was not

implemented on a parallel machine the reduction of the input vector means an imp ortant

decrease in computation time

Figure shows the results obtained The output texture lo oks similar to the result

obtained using all nine elements of the neighborho o d However the blo cky app earance

is more obvious The use of a neighborho o d partially smo oths the discontinuities present

in the DEM This approach would rely much more on smo othing the elevation values

previous to pro cessing

This reduced input vector generates a greater range of terrain colors For instance

those areas with a small slop e and low elevation have a lighter color This coloration helps

visualize the terrain but do es not necessarily corresp onds to the target texture Besides

the dierence in color b etween the two slop es of the mountains do es not show up in the

output texture This feature should b e closely related to the normal vector of the terrain

but the inclusion of a neighborho o d seems to play an imp ortant role in its generation

The detail around Lake Agnes conrms the imp ortance of the normal vector in the

generation of this area The problem with the water color remains since it is related to

the close similarity in the water and the terrain colors in the target texture The absence

of island inside the lake also remains since it is caused by undersampling and similarities

in elevation

Using the HSV Color Mo del

Figure presents the b est results obtained using the HSV color mo del The network

parameters used were and hidden units Once again of the samples in cam

were used for training the learning rate for the hidden layer was four times larger and

the momentum term was set to zero The input vector was comp osed of a element

neighborho o d the three normal vector comp onents and the normal vector magnitude

The test error was

a b

c d

e f

Figure Results obtained using the single elevation sample and the surface normal

vector a output image b Output image mapp ed onto the terrain c Original

photograph target image d Original photograph mapp ed onto the terrain e Error

image dierence b etween output and target images f Error image mapp ed onto the terrain

a b

c d

e f

Figure Results obtained using a element neighborho o d of elevation samples and the

surface normal vector HSV outputs a output image b Output image mapp ed onto

the terrain c Original photograph target image d Original photograph mapp ed

onto the terrain e Error image dierence b etween output and target images f Error image mapp ed onto the terrain

Despite the dierences in color the general app earance of the output texture is very

similar to the result obtained with the RGB color mo del However there is a high error in

the areas covered by ro ck Those areas do not have a redish color as in the target texture

but a bright green color instead Nevertheless the areas covered with vegetation have a

b etter approximation The error image contains dark areas high error on the highest

elevations but light gray tones low error on the lower elevations The approximation

of the vegetation texture is signicantly b etter than the previous results obtained The

target texture is clearly dominated by two hue values green and red and the network

is nding a b etter relationship b etween elevation and color for those p oints with a green

hue

This error dierence b etween the areas covered with vegetation and those covered

with ro ck was unexp ected since the use of the HSV should have yielded b etter results

for higher elevations with a redish coloration where shadows are present However the

assumption that the color of those ro ck areas covered with shadows dier mainly on the

V comp onent was found incorrect The insp ection of cam gives indication that in most

cases the opp osite holds The V comp onent remains almost unchanged while the hue

and the saturation vary This unexp ected feature should b e related to the remote sensing

pro cess of the terrain

Network Generalization

Figure shows the results obtained for cam cam and cam The original terrain

textures are also presented for comparison All textures were mapp ed onto the DEMs and

rendered as seen from the west The network used corresp onds to the one describ ed at

the b eginning of section and hidden units trained with of cam

The output textures for cam cam and cam have visually the same quality of

the texture obtained for cam Snow Lake which is east of lake Agnes is rendered with

the same color as the surrounding ro ck It is recognized by its homogeneous coloration

and its edges The lake shore has steep slop es that cause a shadow eect that give a D

app earance to the lake The two lakes can b e recognized in the output textures even

though they cannot always b e recognized in the resp ective target texture

a b

c d

e f

Figure Results obtained using a element neighborho o d of elevation samples and

the surface normal vector from cam a cam b Network approximation of cam c

cam d Network approximation of cam e cam f Network approximation of cam

Figure shows the error RMSE b etween the target and the output textures for

each of the color comp onents The error values are relatively similar for all DEMs The

error in the output texture obtained for cam is slightly higher This data set contains

terrain that is at high elevations and is mostly covered by ro ck Moreover the highest

elevations are actually ab ove the range of cam Therefore the network extrap olates at these p oints and should b e prone to pro duce outputs with a higher error

100

80

60 RMSE 40

20

0 RBRBGGGGR B R B

CAM1 CAM2 CAM3 CAM4

Figure RMSE b etween target and output textures using network trained with cam

data

The RMSE value for each data set seems to b e asso ciated with the amount of ro ck

This may b e caused by insucient training samples from those zones in cam covered with

ro ck Those areas have the highest values for the red comp onent which has the highest

RMSE Using a target texture for training with a more even distribution of terrain types should help reduce this problem

Chapter

DISCUSSION

This pro ject shows how MLPs can b e used to approximate a mapping b etween ter

rain elevation and color This type of neural network has b een proved to b e capable of

pro ducing arbitrarily accurate approximations to an arbitrary function Even though the

color of a given terrain lo cation do es not only dep end on its elevation the relationship

b etween these two is strong enough to pro duce a result that clearly resembles the original

terrain

The input representation was an imp ortant factor for the generation of results The

normal vector was found to b e essential to improve the quality of the output textures

obtained Features like shadows lakes and the tree line are signicantly sharp er when

the normal vector is included The magnitudes of the connection weights in the hidden

layer also conrm the imp ortant role that the normal plays Similar results might b e ob

tained by replacing the three normal comp onents and its magnitude with the two gradient

comp onents of the elevation

However there are several limitations asso ciated with this approach The noise re

moval feature of neural networks contributes to the smo oth app earance of the output

textures This characteristic is intrinsic to neural networks therefore image enhancement

and restoration techniques will b e required to improve the nal image Jain Gon

zalez and Woo ds In addition a trained network will only pro duce textures with

visible dierences if the elevation maps have signicant dierences

Another limitation of this approach is asso ciated with the availability of adequate

training data The DEM and the aerial photograph used for training should provide

enough samples of all the terrain features that want to b e approximated This limitation

is contributing to the color errors of ro ck and lakes in the data sets used Moreover the

training data should have an elevation range wide enough so the network do es not have

to extrap olate color values when a new DEM is pro cessed

The applicability of a trained network is also limited to terrain that has features

similar to the training data In other words a network can only generate terrain it has

b een taught A network trained with terrain that has some particular elevation range

and land cover should pro duce p o or results if used to approximate texture of terrain with

dierent features Moreover some features that are not related to elevation such as

manmade structures are not likely to app ear in the output textures

The training time can b e seen as a drawback of the backpropagation algorithm used

to train MLPs The training times ranged b etween three and ve hours for most exp er

iments as run in this pro ject However this might not b e a problem if more computing

resources are available or if a parallel machine can b e used One of the p ositive features

of this approach is its generalization capability The error measures obtained when apply

ing the trained network to new DEMs show that this technique is useful to approximate

terrain textures within the constraints previously mentioned

However some imp ortant problems remain to b e solved To remove the blo cky ap

p earance of some of the textures the DEMs should b e smo othed b efore pro cessing them

Nevertheless this eect could p ersist to some degree dep ending on the severity of the dis

continuities in the original data The error in the estimation of the red comp onent is the

main cause of color dierences This problem could b e related to an insucient number

of samples with a high red intensity used during training

Even though the use of the HSV color mo del did not proved useful for removing

shadows it help ed improve the results for areas covered with vegetation It would b e

desirable to nd a way to take advantage of this feature and obtain the ro ck approximation

generated with the RGB color mo del at the same time

An insucient number of samples is also causing the dierences in the coloration of

lakes A greater number of samples corresp onding to the lakes should b e used Alterna

tively some articial training pairs could b e used to make the network learn an appropriate color when the normal vector is zero

Finally the articial textures pro duced by the neural network might b e improved

by using additional terrain information In particular larger input vector could include

land cover information The land cover maps distributed by the USGS includes terrain

classications such as forest land water wetland barren land and tundra US Geological

Survey b Maps with more sp ecic classications level I I are also available The

inclusion of this information should sharp en the terrain texture eg the tree line and

help attenuate the blo cky app earance obtained so far

However the inclusion of this information also has some drawbacks A larger input

vector imp oses a grater number of restrictions on the terrain mo dels that can b e applied

to a trained network Also the generation of terrain mo dels to b e applied to a trained

network might b ecome a problem Some of the additional information might not b e

available or could not b e estimated Furthermore a larger input vector implies a greater amount of resources needed for storing and pro cessing the terrain mo dels

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