3d Mesh Compression NN

Applied Soft Computing Manuscript Draft Manuscript Number: ASOC-D-11-00203 Title: Robust Digital Watermarking for Compressed 3D Models Based on Polygonal Representation Article Type: Full Length Article Keywords: Rubost Watermarking, Spherical Wavelet Transformation, Artificial Intelligent, Multi-layer Feed Forward Neural Network, Attacks, Butterfuly Transform method, Lossy Compression, BER Abstract: Multimedia has recently played an increasingly important role in various domains, including Web applications, movies, video game and medical visualization. The rapid growth of digital media data over the Internet, on the other hand, makes it for everybody to access, copy, edit and distribute digital contents such as electronic documents, images, sounds and videos. Motivated by this, much research work has been dedicated to develop methods for digital data copyright protection, tracing the ownership and preventing illegal duplication or tampering. This paper introduces a methodology of robust digital watermarking that applied on compressed polygonal 3D models, which is based on the well-known spherical wavelet transformation after producing the compressed 3D model using neural network, and we proved in this work that applying watermarking algorithm on compressed domain of 3D object more effective, efficient and robustness than working on normal domain.

1

Robust Digital Watermarking for

Compressed 3D Models Based on Polygonal

Representation

Khalaf F.Khatatneh1, Nadine B. Abu Rumman

2

1Information Technology School, Al-Balqa Applied University, Salt- Jordan

2Department of Computer Graphics and Animation, Princess Sumaya University for Technology, Amman, Jordan

Abstract Multimedia has recently played an increasingly important role in various domains, including

Web applications, movies, video game and medical

visualization. The rapid growth of digital media data over

the Internet, on the other hand, makes it for everybody to

access, copy, edit and distribute digital contents such as

electronic documents, images, sounds and videos. Motivated

by this, much research work has been dedicated to develop

methods for digital data copyright protection, tracing the

ownership and preventing illegal duplication or tampering.

This paper introduces a methodology of robust digital

watermarking that applied on compressed polygonal

3D models, which is based on the well-known

spherical wavelet transformation after producing the

compressed 3D model using neural network, and we

proved in this work that applying watermarking

algorithm on compressed domain of 3D object more

effective, efficient and robustness than working on

normal domain.

Index Terms Rubost Watermarking, Spherical Wavelet Transformation, Artificial Intelligent, Multi-layer Feed

Forward Neural Network, Attacks, Butterfuly Transform

method, Lossy Compression, BER.

I. INTRODUCTION

Multimedia has recently played an increasingly

important role in various domains, including Web

applications, movies, video game and medical

visualization. The rapid growth of digital media data over

the Internet, on the other hand, makes it easy for

everybody to access, copy, edit and distribute digital

contents, such as electronic documents, images, sounds

and videos [1]. Motivated by this, much research work

has been dedicated to develop methods for digital data

copyright protection, tracing the ownership and

preventing illegal duplication or tampering. One of the

most effective techniques for the copyright protection of

digital media data is a process, in which hidden specified

signal (watermark) is embedded in digital data. The

existing efforts in the literature on digital watermarking

have been concentrated on media data such as audio,

images, and video, but there are no effective ways for the

copyright protection of three dimensional (3D) models

against attacks, especially when the copyright of the

models is distributed over the Internet. The watermarking

technique should allow people to permanently mark their

documents, and thereby prove claims of authenticity or

ownership.

The problem of 3D models watermarking is that the

3D model watermarking has received less attention from

researchers due to the fact that the technologies that have

emerged for watermarking images, videos, and audio

cannot be easily adapted to work for arbitrary surfaces or

polygons. Therefore, there are some important points that

should be kept in mind when a watermarking mechanism

is to be adopted for 3D objects. The most important of

which are:

The scheme of the 3D object representation, which is more complex to handle than other

multimedia data, where different kinds of

representation are available for 3D models; an object

can be represented on a 3D grid like a digital image,

or in a 3D Euclidean space.

The precision and size of a 3D object should be in focus since a high precision leads to an increased

amount of data and huge size of the 3D object in the

storage. Furthermore, it should be taken into account

that embedded watermark doesn't obviously affect

the objects size.

The watermarking scheme is so important for producing robust watermarks that must be able to

survive from a variety of attacks, including resizing,

cropping, lossy compression, filtering, or attempts to

remove the watermark. The secret watermarks

should be robust to more complex transformations.

The robustness is thus evaluated by measuring the

survival rate of watermarks after attacks.

This paper provides the necessary information

related to the watermarking methods for 3D objects. First

we will introduce the background information for

watermarking methodologies, next proposed of

compression methodology using one of the most

important artificial intelligent tools, which is Multi-layer

feed forward neural network. After that the output from

compression which compressed object will be the target

for watermarking algorithm that applies the watermark in

spherical wavelet transformation on compressed data set

for polygonal mesh that is used to decompose the original

mesh into some detailed parts and an approximation part

by using spherical wavelet transformation [2]. The

watermarked model and wavelet coefficients are used to

remove the watermark, which can be a secret key or

*ManuscriptClick here to view linked References

2

image, in order to get the original model which makes the

watermarking algorithm more robust. Finally the

watermarked 3D compressed model is evaluating from

two points of view; the effectiveness of the watermarking

algorithm applied on the 3D model and the robustness of

the embedded watermark, and we will present our result

on six 3D models as sample for our algorithm and

evaluation the algorithm against various types of attacks

geometric transformation, smoothing, lossy compression.

The experimental runs of the watermarking

algorithms under study were performed on a TOSHIBA

Satellite A200 personal computer (Intel (R) Core (TM)

Duo, CPU T2450 (2 CPUs ~2.0GHz)), with 2550 MB

RAM, display card Mobile Intel(R) 945GM Express

Chipset Family and 110 GB disk storage.

II. BACK GROUND

Digital watermarks provide a mechanism for

copyright protection of digital media by allowing people

to permanently mark their documents and thereby prove

claims of authenticity or ownership. 3D watermarking

aims at hiding, in an invisible way, information inside 3D

object. This information can then be recovered at any

time, even if the 3D object was altered by one or more

non-destructive attacks, either malicious or not. Watermarks can be fragile or robust. Fragile watermarks

are used for authentication of modifications localization. Their goal is to detect the slightest change to the

document. Robust watermarks are designed to survive

through most attacks, where the robustness is the ability

to recover the watermark even if the watermarked 3D

object has been manipulated.

Watermarking is proposed as the most active

solution to prompted unauthorized manipulations and

malicious attacks of valuable contents of the 3D object

media by changing some of their lower significant bits.

Watermarking schemes can be classified also into private,

public and semi-public [3].

A private watermarking scheme needs the original model and watermarks to extract the

embedded watermarks.

A public watermarking scheme can extract watermarks in the absence of the original model and

the watermarks, which is also called blind so that all

fragile watermarking schemas are also public.

A semi-public watermarking scheme does not need the original model in the watermark extraction

stage, but the original watermarks are necessary for

comparing with the extracted watermarks.

Watermarks (input) are the amount of information

that is embedded in a data object; this information might

be image or secret key. The way of embedding should not

affect on the objects viewing or the general form of a 3D object. Watermark is called embedding or watermarking.

In this paper, the information encoded in the object by the

objects owner is image or a secret key (such as ID or symbol-sequence). The 3D objects (target) that host the

watermark to be embedded in using an embedding

algorithm are called cover of watermark, which are in this

paper, polygonal models. The output is then a

watermarked 3D object. Based on this, the 3D object is

the host and because 3D object has three components:

geometry, connectivity and texturing, then the

watermarking might be: 1. Geometry watermarking. 2.

Connectivity watermarking. 3. Texture watermarking.

In this research, the 3D object is used without

texturing. Thus, the watermarking, in this paper, is based

on connectivity and geometry watermarking. In addition,

working with connectivity and geometry watermarking is

more robust than texture watermarking because they

protect their components, which are vertices and faces

from mesh operation, attacks like scaling, smoothing

compression and geometry transformation.

The watermarking is based on 3D object attributes,

which are geometry and topology that make embedding

watermarking primitives either geometrical embedding

primitives or topological embedding primitives, which is

making watermarking methods either geometry-based

watermarking methods or topology based watermarking methods each of these methods has its own characteristic

that will be discussed in the following points [4].

1. Geometry Based Watermarking Methods:

The method that focuses on the geometric feature of

3D object which is the objects vertices, so embedding the watermark may modify the position of vertices in

order to insert the watermark, changing length of a line,

area of polygon or ratio of volumes of two polygons. One

of the most of simplest examples of this type is

embedding watermark directly onto vertex coordinates

[5] that works as in the following steps:

Modify the coordinates of the vertex by modulating the watermark signal with global

scaling factor and masking weight.

The masking weight for each vertex is the average of differences between of the

connected vertices to that vertex.

Adding the watermark coordinate values.

2. Topology Based Watermarking Methods:

This is the method that focuses on the topological feature

of 3D object which is connectivity of mesh vertices. So,

embedding the watermark changes the topology of a

model, where the side effect of this is a change in

geometry also. Usually, working with topology is more

robust for the watermarking, where the topology is

redefined to encode one or more bits. One of the most

famous examples of this type is encoding binary bits in

triangulating a quadrilateral way [6]. Many methods

combine geometric methods with topological methods,

which the method that will be use in this thesis

watermarking algorithm.

III. RELATED WORK

The pioneer works of watermarking 3D models

were performed by Ohbuchi [7], who introduced several

schemes for watermarking polygonal models. One

3

scheme embeds information using groups of four adjacent

triangles, while another scheme proposed using ratios of

tetrahedral volumes. The tetrahedral are formed by the

three vertices of each face and a common vertex that is

computed by averaging a few fixed mesh vertices.

Moreover, a way of visually embedding information into

polygonal mesh data is proposed by modifying the vertex

coordinates, the vertex topology, or both. Ohbuchi et al.

[8] also proposed a frequency domain approach to

watermark 3D shapes, where the mesh is segmented first

into some patches, and then for each patch, a spectral

analysis is conducted, and the watermark information is

finally embedded into the frequency domain at the

modulation step.

The approach of Guillaume [9] is quite different;

Guillaume presented a digital watermark embedded on

3D compressed meshes based on subdivision surface,

which chooses a 3D object segmented into surface

patches as a target, and then hides the watermark in the

compressed object.

Praun [10] provided a robust watermarking scheme

suitable for proving ownership claims on triangle meshes

representing surfaces in a 3D model by converting the

original triangle mesh into a multiresolution format,

consisting of a coarse base mesh and a sequence of

refinement operations. Next, a scalar basis function is

defined over its corresponding neighborhood in the

original mesh. A watermark is then inserted as follows:

each basis function is multiplied by a coefficient, and

added to the 3D coordinates of the mesh vertices. Each

basis function has a scalar effect at each vertex and a

global displacement direction, where this process is

applied as a matrix multiplication for each of the three

spatial coordinates x, y, and z.

In 3D model represented as cloud of vertices and list

of edges corresponding Kundur [11] provided a new

method based on finding and synchronizing particular

areas used to embed the message by using data hiding

that relies on modifying the topology of the edges in

chosen area.

A wavelet-based multiresolution analysis is used for

polygonal models proposed by Wan-Hyun Cho [12].

First, generate the simple mesh model and wavelet

coefficient vectors by applying a multiresolution analysis

to a given mesh model. Then, watermark embedding is

processed by perturbing the vertex of chosen mesh at a

low resolution according to the order of norms of wavelet

coefficient vectors using a look-up table. The watermark

extraction procedure is to take binary digits from the

embedded mesh using a look-up table and similarity test

between the embedded watermark and the extracted one

follows.

JIN Jian-qiu et al [13] proposed a robust

watermarking for 3D mesh. The algorithm is based on

spherical wavelet transformation, where the basic idea is

to decompose the original mesh of details at different

scales by using spherical wavelet; the watermark is then

embedded into the different levels of details. The

embedding process includes: global sphere

parameterization, spherical uniform sampling, spherical

wavelet forward transformation, embedding watermark,

spherical wavelet inverse transformation, and at last

resampling the watermarked mesh to recover the

topological connectivity of the original model.

Adrian G.Bors [14] also proposed public

watermarking algorithm that is applied on various 3D

models and does not require the original object in the

detection stage using a key to generate a binary code, a

set of vertices and their neighborhoods are selected and

ordered according to a minimized distortion visibility

threshold. The embedding consists of local geometrical

changes of the selected vertices according to the

geometry of their neighborhoods.

The proposed robust watermarking algorithm

should meet the following technical requirements:

1. Direct Embedding: The watermark should be directly embedded into the compressed

geometry data or topology data of the polygonal

model.

2. Invisible: The embedded watermark must be perceptually invisible within the model and

unnoticeable for the user.

3. Small geometric error: The geometric error of the polygon data caused by the embedding must

be small enough in order not to disturb the

application use.

4. Robustness: The embedded watermark must be unchanged or difficult to be destroyed under the

possible 3D geometric operations done on the

3D polygonal model.

5. Capacity: The amount of the watermark which can be embedded in the model is large enough

to record the information needed for the

application.

6. Efficient Space: A data embedding method should be able to embed a non-trivial amount of

information into model.

IV. COMPRESSTION 3D OBJECT ALGORITHM

The neural network employed in this paper is MLFF

neural network to provide lossy compression method,

where neural network tool used for this algorithm is

Mathworks tool (Neural Network Toolboxs with Multi-layer Feed Forward Architecture) the object is created

manually (modeling them using Autodesk Maya 2008)

and before entering the data in neural, the pre-processing

should be applied on these data.

4.1 The Pre-Process Data Set

Before the inputs are presented to the MLFF, the

data should be pre-processed. Accuracy of the outputs of

neural network depends on the data pre-processing step.

The following are the steps that should be done in data

pre-processing stage:

Normalization

Extract main features of the dataset

The supervised learning problem is divided into

parametric and nonparametric models. The problem here

4

lies in the nonparametric model because there is no prior

knowledge of the form of the function being estimated.

Therefore, neural network learning by example is

required. The learning process will be performed by a

learning algorithm. The objective of this algorithm is to

change the synaptic weight of the network to attain a

desired design objective, which is the compressed object.

Once the network has been trained, it is capable of

generalization. The examples that used in neural network

as training set are ten 3D different models that are created

manually using Maya. Our target is to create desired

outputs. These outputs are generated by external package

for geometry compression using Java 3D package.

Standard representations of a polygon mesh uses:

An array of floats to specify the positions, and

An array of integers containing indices into the position array to specify the

polygons

.

A similar scheme is used to specify the various

properties and how they are attached to the mesh. For

large and detailed models, this representation results in

large substantial size, which makes their storage

expensive and their transmission very slow.

All the positions of vertices should be normalization

between 0.0 and 7.0. This step helps neural network to

focus on the aim of reducing the number of vertices and

reconstructing the faces. All vertices value should be

normalization between [0 -7] to make all neural network

work concentrate on compression. The minimum value

for all vertices coordination of all ten created models

should be 0 and the maximum values should be 7, the

normalization process is doing using Maya to make all

vertices coordination between 0 and 7.There are ten 3D

objects are samples that used in the training in MLFF

neural network as input object. But first before entering

these models in neural network we should have also

target object which is compressed object using Java 3D

compression geometry package. we considered ten

models are enough for training neural network because

the compression of 3D models with same attributes as

representation and with normalization between [0 - 7] for

all those models. Also from our experiments we noticed

that training neural network takes a lot of time and after

ten models the neural network became capable of

generalization.

4.2 Java 3D Geometry Compression

There are three main aims for the geometry

compression technique; efficient rendering, progressive

transmission, and maximum compression to save disk

space [15].

The geometry compression using Java 3D package

can achieve (lossy) compression ratios between 10:6 to

one object, depending on the original representation

format and the desired quality of the final level. The

compression proceeds in four stages. The first stage is the

conversion of triangle data into a generalized triangle

mesh form. The second is the quantization of individual

positions, colors, and normals. Quantization of normals

includes a novel translation to non- rectilinear

representation. In the third stage, the quantized values are

delta encoded between neighbors. The final stage

performs a Huffman tag-based variable-length encoding

of these deltas. Decompression is the reverse of this

process. The decompressed stream of triangle data is then

passed to a traditional rendering pipeline, where it is

processed in full floating point accuracy. The

improvement in this package by adding optimization

compression makes the lossy in detail of the 3D object

much more little. Figure 3.21 displays the pseudo-code

that describes part of this work. The geometry

compression algorithm steps for Java 3D package are as

follows [15]:

1. Input explicit bag of triangles to be compressed, along with quantization

thresholds for positions, normals, and

colors.

2. Topologically analyze connectivity, mark hard edges in normals.

3. Create vertex traversal order & mesh buffer references.

4. Histogram position, normal, and color deltas.

5. Assign variable length Huffman tag codes for deltas, based on histograms, separately

for positions, normals.

6. Generate binary output stream by first outputting Huffman table initializations,

then traversing the vertices in order,

outputting appropriate tag and delta for all

values.

Also, there are some definitions that have been

added to identify the critical vertices so that removing

those critical vertices can be controlled such that the

number of vertices remains correspondent to these edges

which are never used by the compression algorithm. The

following are the definitions of those vertices depending

on invariant vertex identification that is provided by [16].

1. Boundary vertices of the 3D model are the vertices that cannot be used by the compression

algorithms because these are critical vertices.

These are defined as vertices which influence

the shape of the 3D model. These vertices can be

used for watermarking later on.

2. Neighboring vertices to split vertex will never be used by the compression algorithms.

3. Vertices of edges which do not form simple triangle a simple triangle, so it cannot be collapsed.

That can be calculated from the data of 3D models by

storing all the vertices and faces according to the label

of vertices, and then check every two consecutive faces.

If any two consecutive triangles have two of its vertices

common so that two vertices form the complex triangle.

In this way, this pair of vertices cannot be used by the

5

compression algorithm. The complexity of invariant

vertex selection is analyzed as follows according to

[17]:

1. The complexity of selecting boundary vertices of the 3D object by computing convex hull takes O

(n log n) using quick hull algorithm [18].

2. The neighboring vertices to split, which are computed after each refinement. If p is the

number of split vertices in a refinement and d is

the maximum degree for a vertex, then the

complexity for processing these set of vertices is

O (p*d).

3. Computing the vertices of edges which are not simple triangles. First, sort all the faces

according to the label of vertices which takes O

(n log n). Then, checking between two

consecutive faces takes O (n) time. So, the

overall complexity for computing this set of

vertices is O (n log n).

T (n) = n log n + n log n + n

T (n) = 2n log n + n (1) The overall worst time complexity of the algorithm is

also O (n log n).

where T (n) is time complexity and n is the number of

vertices.

The overall complexity of remesh algorithm using

Java 3D geometry compression in addition to invariant

vertex selection algorithm is as follows:

1. The invariant vertex selection algorithm complexity is O (n log n).

2. The remesh algorithm complexity is O (n).

T (n) = (2n log n +n) * (15n + 4)

T (n) = 30n2 log n +15 n

2 +8 n log n

+4n (2) where T (n) is time complexity and n is the number of

vertices.

The overall worst time complexity of the algorithm is

also O (n2 log n).

4.3 The Structure of MLFF Neural Network

The neural network structure contains input layer,

one hidden layer and output layer, all nodes are fully

connected, the network takes x, y and z coordinates of

vertices as input, the activation function is sigmoid

function with as learning rate which is equal to 0.9.

F(x, )=1/(1+exp(- x)) (4.3.1)

Sigmoid functions are also prized because their

derivatives are easy to calculate, which is helpful for

calculating the weight updates in certain training

algorithms. The derivative is given by:

F'(x, )=- F(x, )(1-F( ,x)) (4.3.2)

The number of neurons in input layer is 4, where the

first three input vectors are the x, y and z vertices

coordinates and the fourth input is max face ratio which

indicates that the maximum face must remain as it is;

number of neurons in hidden layer is between 3 and 4

because compression process overall depends on hidden

layer so the number of neuron should be absolutely less

than number of neuron in input layer to do compression.

For higher accuracy, the number of neurons in hidden

layer should be increased but this reduces compression

process. A two-layer feed-forward network with sigmoid

hidden neurons and linear output neurons can fit multi-

dimensional mapping problems arbitrarily well, given

consistent data and enough neurons in its hidden layer.

Figure 4.1 displays the neural network structure

with a given 3D model object sample for input object and

target object

Figure 4.1: One hidden layer feed forward neural

network structure.

Because 3D Java geometry compression package is

concentrated on triangle 3D object and because training

the proposed algorithm on neural network is based on

examples from this package so network here also

concentrates on triangle 3D object.

4.4 The Training Samples

The network trains 1000 times with the training set

until MSE (error function) is small that is the difference

between the output objects and desired objects. Training

automatically stops when generalization stops improving,

as indicated by an increase in the mean square error of the

validation samples, adjustment all weight that produces

from hidden layer in each time the neural network

training on training samples, which helps to learn neural

network how to compress other samples. The network

will be trained with gradient-descent back propagation

algorithm with adaptive learning rate. Training time for

each model takes approximately 5 hours and 4 minutes.

For all the ten models, it takes 55 hours and 40 minutes.

The complexity of the overall algorithm that

proposed in this thesis is analyzed as follows:

1. The remesh algorithm used in section 4.2 takes O

(n2 log n).

6

2. The MLFF neural network algorithm takes O (n3).

Theorem (1): The worst time complexity of

overall of compression algorithm using MLFF neural

network that proposed in this chapter is O (n3).

Proof:

T(n) =2n log n + n (1)

Worst time complexity for invariant vertex selection

algorithm.

T (n) =15n + 4

Worst time complexity for remesh algorithm.

T (n) = 30n2 log n + 15 n

2 + 8 n log n + 4n ..(2)

Worst time complexity for remesh algorithm in

addition identify for invariant vertex.

T (n) =10n (3)

Worst time complexity for pre- process data set.

T (n) =n3 .(4)

Worst time complexity for MLFF neural network.

Thus by (1), (2), (3) and (4)

T (n) = 30n2 log n + 10 n + n

3

where T (n) is time complexity and n is the number of

vertices.

The overall time complexity is O (n3).

4.5 The Results

By using MLFF neural network algorithm, the

performance of the 3D Java geometry compression

increases. The compression ratio is between 5.5 and 5:10

of the original object. The noise ratio depends on the

MSE, which provides minimum noise for the visual eye.

Figures bellow illustrate the result for the two 3D test

models before and after compression.

Compression using MLFF algorithm is an adaptive

one. This means that it can iteratively change the values

of its parameters (i.e., the synaptic weights). These

changes are made according to the learning rules. By

inserting new models different than the 10 models used to

train the MLFF neural network, the execution time and

accuracy for any new model becomes very fast and more

precise, respectively. Below are the figures and tables

that show the result.

(a) Original object

237018 vertices

474048 polygons

8532792 Byte Size on

disk

(b) Compressed Object

71101vertices

142214 polygons

2559780 Byte Size on disk

Figure 4.1: The shaded angel 3D object model before

and after compression.

(a) Original object

237018 vertices

474048 polygons


(b) Compressed Object

71101vertices

142214 polygons


Figure 4.2: The point cloud angel 3D object model before and

after compression.

(a) Original object

(b) Compressed object

7

543652 vertices

1087716 polygons


disk

108556 vertices

217542 polygons


Figure 4.3: The shaded happy 3D object model before


(a) Original object

543652 vertices

1087716 polygons


disk


108556 vertices

217542 polygons


Figure 4.5: The point cloud happy 3D object model

before and after compression.

(a) Original object

48485 vertices

96966 polygons


disk


14546 vertices

29088 polygons


Figure 4.6: The shaded horse 3D object model before


(a) Original object

48485 vertices

96966 polygons


disk


14546 vertices

29088 polygons


Figure 4.6: The point cloud horse 3D object model before and after compression.

(a) Original object

2904 vertices

5804 polygons



872 vertices

1740 polygons


Figure 4.7: The shaded cow 3D object model before and after compression.

(a) Original object

2904 vertices

5804 polygons



872 vertices

1740 polygons


Figure 4.8: The point cloud cow 3D object model before


Table 4.1 shows the results achieved by the proposed

algorithm for the six new models entered to MLFF neural

network with the following max face ratio:

Models Samples

/Performance

Metrics

Angel

Model

Happy

Model

Horse

Model

Cow

Model

Max face ratio 0.30000 0.20000 0.30000 0.30000

Edges collapsed 165917 435087 33939 2032

No of final edges 213321 326313 43632 2610

Compression ratio 3.33304 5.05457 3.33343 3.33384

Mean Square Error 0.69465 0.82077 0.79666 0.76822

Vertex signal to noise ratio 0.24736 0.20456 0.00527 0.18737

*Execution Time

As CPU Time 76.74 191.65 15.35 1.10

Table 4.1: The Compression results of six new models.

*CPU Time returns the total CPU time (in seconds) used by MATLAB application from the time it was started. This number can overflow the internal representation and wrap around.

8

Figure 4.9 shows the difference between the

proposed algorithms work using the MLFF neural network and the Java 3D geometry compression package,

by fixing the compression ratio.

3 3.5 4 4.5 5 5.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Compression Ratio

Nois

e R

atio

Multi layer feed forward Neural Network

3D java Geomtery Compression Package

Figure 4.9: Comparing between the compression

algorithm in thesis and 3D Java geometry compression.

V. WATERMARING ALGORITHM FOR COMPRESSED 3D OBJECT

5.1 The Watermarking Algorithm

The output result from the compression algorithm

mentioned in section 4, which is the compressed 3D

model, will be the input of the watermarking algorithm

proposed in this section. The proposed watermarking

algorithm is based on spherical wavelet transform which

is considered among the most robust watermarking

methods. The watermarking algorithm in this thesis is

based on the method in [2], which is performing the

efficient spherical wavelet function depending on

spherical wavelet presented in [19].

Phase 1: Generate the Compressed 3D Model using

MLFF Neural Network from Original Mesh

The compression algorithm mentioned in section 4

is applied. Moreover, the critical vertices are identified so

that removing those critical vertices can be controlled

such that a part of the watermark can be embedded in

these vertices. The watermarking algorithm is applied on

compressed 3D model after the object is being

compressed, and the invariant vertex selection, in the

compression algorithm, is identified.

All related details have been explained before. So, a

now compressed 3D model is produced and ready to

work on. This means that all the following work will be

done in compression domain not on the normal domain of

3D object which distinguishes this algorithm from the

other work.

Phase 2: Generate the Sphere Coordinate for Each

Vertex in 3D Mesh

Step 1: Perform Function on Sphere

This function is representing data set on the sphere;

the function used for this representation is spherical

harmonics that helps in making multiresolution

representation of 3D mesh. The point on the unit sphere

can be denoted as follows: p= (cos sin, sin sin,

cos), where is (0=0) denote the angles of longitude and latitude respectively, f

(,) is defined as the unit of sphere and the expansion of f (,) in spherical harmonics is as follow [20]:

The functions are defined on the unit sphere S2 as:

m m

Y (,)=(-1)m kl,m P (cos) *eim

(5.1) i

i

where kl,m is constant, is the associated

Legendre polynomial. Therefore, any

spherical function f: S2R can be

expanded as a linear combination of spherical harmonics:

l m f(,)= cl,m Y (,) (5.2)

l=0 m=-1

l

where the coefficients c l,m are uniquely determined by:

2 m Cl,m= f(,) Y (,) sin(,) d d (5.3)

0 0

l

Step 2: Perform Spherical Parameterization for 3D

Mesh

The parameterization of a triangle mesh onto the

sphere means assigning a 3D position on the unit to each

of the mesh vertices. Topological sphere for 3D object is

a close manifold genus mesh that means embedding its

connectivity graph on the sphere. The following two steps

explain how spherical parameterization information is

generated for 3D mesh:

a. Generating a progressive mesh representation with local parameterization information based

on the pervious equations explained in the

previous step [20].

b. Computing the subdivision of each triangle into 4 smaller triangles in 3D mesh, and then project

on the sphere whose radius is one.

Figure 5.1 shows the samples for applying spherical

parameterization on 3D model.

m

P

l

9

(a) Before (b) After

(a) Before (b) After

Figure 5.1: The samples before and after applying

spherical parameterization on it.

Phase 3: Generate the Spherical Wavelet

Transformation

Because wavelets have been proved to be powerful

bases for use in signal processing based on they only

require a small number of coefficient to represent general

functions and large data sets. Due to local support in both

spatial domain and frequency domain which are suited

for spare approximation of function, the spherical wavelet

proposed in [19] is chosen in this work and the butterfly

wavelet transformation is selected in particular. The

following is a brief description about the wavelet

transformation in general and butterfly wavelet

transformation in particular:

For general wavelet transformation analysis [19]:

Forward Wavelet

Transform (Analysis)

j,k = iK(j) j,k,i j+1,i

(5.4)

j,m= iM(j) j,m,i j+1,i

(5.5)

Scaling function

coefficient, coarse to fine

Wavelet coefficient, fine to

coarse

Inverse Wavelet Transform

(Synthesis)

j+1,i = kK(j) hj,k,i j,k + mM(j) gj,m,i j,m

(5.6)

Scaling function

coefficient, coarse to fine

When n,- ( n is finest resolution level) with forward

transform to get j,- the wavelet coefficient current level,

coarse approximation part n-i,- after performing the

decomposition i times so in the same way if we have n-i,- and j (j=n-i,n-i+1,..,n-1) we can perform the inverse

transform to n,-, different h, , g, denote different wavelet basis function. In Euclidean space we have hj, k, i

=hi-2k, gj, k, i =gi-2k, j, k, i =i-2k.

Figure 5.2: Neighbors used in spherical wavelet

transform.

M (j) and K (j) are abstract sets the index set

concrete in sphere in figure 4.5 the mesh include dashed

edges in figure is j+1 level resolution where k(j) denotes

the point of intersection points of the solid lines and M(j)

denotes the set of the intersection points of the dash lines

and the union k(j) U M(j) =K(j=1). We need now

compute j and j approximation part and detailed part, by decomposition in the neighborhood of m. in this butterfly

transform method and bellow is briefly describe.

The work that done on [2] was based on linear and

linear lifting transforms methods, where in linear

transform the scaling coefficient (approximation part) are

subsampled and kept unchanged, this basic interpolatory

form uses the stencil k K ={v1,v2}(see Figure 4.5) for analysis and synthesis:

j,m= j+1,m -1/2(j+1,v1 +j+1,v2) (5.7)

j+1,m=j,m+1/2(j,v1+j,v2) (5.8)

Respectively, note that this stencil does properly

account for the geometry provided that the m sites at

level j+1 have equal geodetic distance from the {v1,v2}

sites on their parent edge. And linear lifting update the

scaling coefficients by using the wavelet coefficients of

linear wavelet transform to assure that the wavelet has at

least one vanishing moment sj,v1,m=sj,v2,m=1/2.

In this work the Butterfly transform used to

decomposition the geometric signal of the approximation

parts and detailed parts, where Butterfly method used all

immediate neighbors (all the sites km={

v1,v2,f1,f2,e1,e2,e3,e4}. Here sv1=sv2=1/2, sf1=sf2=1/8 and

se1=se2=se3=se4=-1/16 which used construction of smooth

mesh.

10

Butterfly Transform (Analysis)

j,k = j+1,k (5.9) (4.2.9)

j,m= j+1,m - 1/2(j+1,v1+j+1,v2)

- 1/8(j+1,f1+j+1,f2 ) + 1/16(j+1,e1+j+1,e2 +

j+1,e3+j+1,e4) (5.10) (10)

Butterfly Transform (Synthesis)

j+1,k =j,k (5.11)

j+1,m= j,m + 1/2(j+1,v1+j+1,v2) +

1/8(j+1,f1+j+1,f2 ) - 1/16(j+1,e1+

j+1,e2 +j+1,e3+j+1,e4)

(5.12)

Before

After

Figure 4.6: The samples before and after applying

spherical wavelet transform on it where the colored

vertices are induction for wavelet coefficients'.

Although the butterfly is considered to take more

time than linear transformation but because the work is

on a compressed domain that greatly affects and then

makes butterfly and linear close in time consuming, but

butterfly is supposed to be more robust in watermarking

algorithm, in this work the level wavelet decomposition

will be to 3 level.

.

Phase 4: Embedding Watermark

Step 1: Generation Watermark

Watermark can be a secret key or image. This

algorithm is adopted to embed watermark as secret key or

image but embedding watermark by these two ways

should be sequences of binary bits, which means that by

the secret key case (all characters and number) should be

converted to a sequence of binary bits, and in the case of

image, the image should be converted to gray scale level

in order to be as a sequence of binary bits. But in all

experimental results that display thesis was just use by

image because the complexity of image more than secret

key which helps to cover the entire model.

Step 2: Compute the Wight of embedding intensity

Because this algorithm is based on the work

proposed in [2], it presents this function as the following

equation.

F(vj)= 1/j * if band j is one of detailed parts

log vj if band j is on approximation parts

(5.13)

where vj is all vertices of M and belong to j.

Step 3: Watermark embedding

The watermark embedding is done by the following

equation:

vj' =vj* F(vj)*W (5.14)

where vj' is the all vertices after the watermark is

being embedded, vj is the set of all vertices of M and

belong to j (which is produced from phase 4) , W is

watermark (logo image for example), f(vj) is function to

compute the Wight of embedding intensity that can be

computed from equation 5.13.

Phase 5: Extracting Watermark

In order to extract watermark from 3D model the

following steps have been applied:

Step 1: Mesh Registration

The mesh registration here is based on the ICP

(Iterative Closest Point) algorithm, according to the

method that proposed in [22] that applied on

watermarked mesh, which is working as follow:

The Point Set P with No_P points, model shapes M

Iterate until convergence Compute closest points

Squared Euclidian distances d(p1,p

2)= ||p1-p2||

2

Compute registration (rotation and translation)

Apply the registration

Step 2: Spherical Wavelet Forward Transformation

After produced the mesh registration the spherical

wavelet forward transform applied on two meshes

registration mesh and compressed mesh (original mesh

before applying watermarking algorithm), then

comparing the results meshes in order to extract

watermark image as sequence of binary bits.

5.2 The Results

This section presents the evaluation of the

watermarking algorithm mentioned in 5. There are three

performance metrics, which will be discussed bellow.

11

1. Maximum Geometric Error

The visual impact of the watermarking on the

protected 3D object should be as limited as possible to

measure the effect of the embedded watermark on 3D

objects. In this work, the maximum geometric error is

used.

Hausdorff distance :

Hmax (M1,M2)

H(M1,M2) = max { MaxaM1 MinbM2 d(a,b) ,MaxaM2

MinbM1 d(a,b) } (14)

2. Capacity of Watermark Embedding

Here, capacity means the amount of information

embedded in 3D object, this amount should be closely

related to the complexity of the object (number of

vertices, number of faces). It is assumed that the data

capacity of watermark should be not more than from the

complexity of 3D object depending on the number of

vertices, depending on choosing the watermark as image,

the logo image shouldnt be more 125*125 pixel (which noticed by experiments) and then converted to binary

(gray scale) which produced 16384 bits ready to embed

into 3D object

The following equation is used to determine the

capacity of watermark information embedded in 3D

object.

Data Capacity for 3D object in bits

12

Table 5.1 shows the measurements of robustness

that are achieved by applying the watermarking algorithm

in this thesis on the six models using BER. Table 5.2

shows the measurements of robustness achieved by

applying watermarking algorithm in [2] also using BER.

Form the results that appear in table 5.1 and 5.2 we

proved that applying watermarking algorithm on

compressed domain more robustness than applying

watermarking algorithm on normal domain.

D Models Samples

/ Robustness

Metrics against Attacks Angel

Model

Happy

Model

Horse

Model

Cow

Model

Lossy Compression

BER 0.0291 0.0535 0.0945 0.1764

SR 0.9709 0.9465 0.9055 0.9335

Tran2slation

(x+20,y-5, z-13)

BER 0.0018 4.2725e-

004 0.0015

2.4414e-

004

SR 0.9982 0.9996 0.9985 0.9998

Translation

(x-2, y+13, z+5)

BER 0.0011 4.2705e-

004 0.0035

2.4454e-

004

SR 0.9989 0.9996 0.9965 0.9998

Rotation

(y-coordination 30)

BER 0.0020 9.7656e-

004 0.0013

3.0518e-

004

SR 0.9980 0.9990 0.9987 0.9997

Rotation

(x- coordination 30

And

z- coordination 60)

BER 0.0026 9.7436e-

004 0.0025

3.0508e-

004

SR 0.9974 0.9990 0.9975 0.9997

Scale

(x-scale 0.6

,y-scale 2, z-scale 3)

BER 8.2393e-

004

4.5746e-

004

1.5279e-

004

7.0180e-

004

SR 0.9992 0.9995 0.9998 0.9993

Scale

(x-scale 3,

y-scale 0.5

, z-scale 0.2)

BER 8.2397e-

004

4.5776e-

004

1.5259e-

004

7.0190e-

004

SR 0.9992 0.9995 0.9998 0.9993

Smoothing

mesh with regular

subdivisions 1:4

BER 0.0183 0.0237 0.0243 0.0304

SR 0.9817 0.9763 0.9757 0.9696

Table 6.1: The robustness measurement results of BER

for watermarking algorithm in this paper.

3D Models

Samples

/ BER for

attacks

Angel

Model

Happy

Model

Horse

Model

Cow

Model

Lossy

Compression 0.4888 0.3052 0.4272 0.3709

Translation

(x+20,y-5, z-

13)

0.0012 0.0014 9.7656e-

004

6.1035e-

004

Translation

(x-2, y+13,

z+5)

0.0011 0.0017 9.7666e-

004

6.1075e-

004

Rotation (y-

coordination

30)

0.0025 5.4932e-

004 0.0031 0.0018

Rotation (x-

coordination

30 and z-

coordination

60)

0.0037 5.4911e-

004 0.0039 0.0012

Scale (x-

scale 0.6,y-

scale 2, z-

scale 3)

0.0018 0.0061 0.0055 0.0049

Scale (x-

scale 3, y-

scale 0.5, z-

scale 0.2)

0.0023 0.0049 0.0061 0.0051

Smoothing

mesh with

regular

subdivisions

1:4

0.1366 0.1831 0.3520 0.2191

Table 6.2: The robustness measurement results of BER

for the algorithm in paper [2].

Figure 6.1 and figure 6.2 show a comparison

between the performed work in this thesis and the work

in [2] from robustness of two watermarking algorithms

against the attacks. This clearly shows that performance

from BER of our watermarking algorithm is better in

most types of attacks that mention before.

13

Figure 6.1: The experimental result for the work that

proposed in [2]

Figure 6.2: The experimental result for the work in

this paper.

VII. CONCLUSIONS

Compression algorithm using MLFF neural network

produces a compressed 3D model with compression ratio

reaches 5.5 which reduces the size of 3D model on disk

with minimum losing details and vertex signal to noise

ratio, this is noticed experimentally by applying the

proposed algorithm on six different 3D models samples.

The MLFF neural network as an AI tool played an

important role in the performance of compression

algorithm making the algorithms performance better than 3D compression geometry proposed in [19], which is

the neural network that learns the compression on which

it is based.

The methodology of applying watermark on 3D

model after compression ( on compressed domain) is

proved to reduce the processing time of the watermarking

algorithm, in addition to allowing embedding the

watermark into the model without much increasing on

model size compared to original model before

compression.

Implementing the watermarking algorithm based on

spherical wavelet as butterfly transform method for

vertex bases wavelet coefficients. Although this way

costs more time than other vertex bases, but because this

work in compressed domain, this cost of time is not

affected any more. Butterfly vertex bases do not only

work on immediate neighbors, which are more suitable

for embedded robust watermark and affected on BER

smaller than work with other transform method for vertex

bases like linear vertices which is noticed from

experiments for different performance metrics.

This work proved that applying watermark on

compressed 3D model is more robust than uncompressed

3D model and also identifying invariant vertex on

compression algorithm stage as critical vertices cannot be

deleted under any condition and helps to identify vertices

to embed watermark into, in addition to vertex bases

wavelet coefficients.

References

[1] R.Ohbuchi, M.Nakazawa and T.Takei,"Retrieving

3D shapes based on their appearance", ACM SIGMM

Workshop on Multimedia Information Retrieval,

Berkeley, California, pp. 3946,2003.

[2] 3D Object Processing: Compression, Indexing and

Watermarking. Edited by J.-L. Dugelay, A. Baskurt and

M. Daoudi, John Wiley & Sons, Ltd. ISBN: 978-0-470-

06542-6, 2008.

[3] P.Besl and D.Mckay,"A method for registration of 3D

shape", IEEE Transaction on Pattern Analysis and

Machine Intelligence, 14(2):239-256,1992.

[4] Chang, A.M. and Hall, L.O. (1992). The validation of fuzzy knowledge-based systems, Fuzzy Logic for the Management of

Uncertainty, L.A. Zadeh and J. Kacprzyk, eds, John Wiley, New York,

pp. 589-604.

[5] M.Deering "Geometry compression", ACM SIGGRAPH, pp. 1320,1995.

[6] M.Isenburg, Stefan Gumhold, Out-of-Core Compression for Gigantic Polygon Meshes, Proceedings of SIGGRAPH'03, pages 935-

942, July 2003.

[7] M. Isenburg, Peter Lindstrom, Jack Snoeyink, Lossless Compression

of Floating-Point Geometry, Proceedings of CAD'3D, May 2004.

[8] M. Isenburg and J. Snoeyink, " Face Fixer Compressing Polygon

Meshes with Properties" , ACM Siggraph Conference Proc, pp. 263-270,2001.

[9] Zachi Karni and Craig Gotsman. Spectral compression of mesh geometry. ACM, SIGGRAPH 2000, New Orleans, 2000.

[10] G. Zigelman, R. Kimmel and N. Kiryati. Texture mapping using

surface flattening via multidimensional scaling. Preprint, 2000.

[11] E. Piperakis. A.ne Transformations on 3D Objects Represented

with Neural Networks. IEEE, Proceedings of the Third International Conference on 3-D Imaging and Modeling, 2001.

[12] S. Haykin. Neural Networks, A Comprehensive Foundation. Macmillan College Publishing Company, 1994.

14

AUTHORS

K. F. Khataneh is with the Information Technology

School, Al-Balqa Applied University, Salt-Jordan (e-mail: [email protected]).

N. Abu Romman is with Princess Sumaya University for Technology (PSUT), Department of Computer Graphics, Amman-Jordan (e-mail: [email protected]).

Documents

3d Mesh Compression NN