Skeletonization Based on Wavelet Transform Outline Introduction How to construct wavelet function according to its application in practice Some new characteristics.

Skeletonization Based on Wavelet Transform

OutlineIntroductionHow to construct wavelet function according to its application in practiceSome new characteristics of new wavelet function Implementation of wavelet transform in the discrete domainExtraction of wavelet skeletonSome sets of schemes for modifying artifacts of primary skeletonsResults of experiments Future works

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What is the skeleton of a shape?The skeleton is defined as a smooth curve that follows the shape of a character equidistantly from its contours.

(Pixel-Based Methods)The skeleton of a shape is referred to as the locus of the symmetric points or symmetry axes of the local symmetries of the shape.(Non-Pixel-Based Methods)

Skeleton

Shape

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Pixel-Based Methods

Methods Based on Thinning Techniques

Methods Based on Distance Transform These methods suffer from the following drawbacks:

A skeleton is not helpful for recognizing the underlying shape since the generated skeletons are in discrete forms;

The resulting skeleton may not be centred inside the underlying shape;

The computation complexity is high since all foreground pixels are used for computation skeletons.

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Non-Pixel-Based Methods Different local symmetry analysis maybe result in

different symmetric points, and hence different skeletons and skeletonization methods are produced. Namely:

Blum’s Symmetric Axis TransforBrady’s Smoothed LocaL SymmetryLeyton’s Process-Inferring Symmetry Analysis

Their Main shortcomings

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Blum’s Symmetric Axis Transfor (SAT)In Blum’s Symmetric Axis Transform, the symmetric point of the local symmetry formed by A and B is defined as the centre of the maximal inscribed symmetric circle

Symmetry point

Symmetry circleB

A

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Brady’s Smoothed LocaL Symmetry (SLS)

Brady defines the symmetric points as the midpoint of a straight line segment AB

Symmetry point A

B

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Symmetry segment

Leyton’s Process-Inferring Symmetry Analysis (PISA)In Leyton’s Process-Inferring Symmetry Analysis, the symmetric pints is defined as the midpoint of the arc

Symmetry point

Symmetry arc

A

B

BA

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Main Shortcomings of Methods Based on above Symmetry Analyses :

For SAT and PISA, a skeleton segment may lie in a perceptually distinct part of the underlying shape;

For SLS, Some perceptually irrelevant symmetric axes may be created;

In the discrete domain, It is generally difficult to determine the symmetric points from boundary curves which are used the above symmetry analyses.

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Main drawbacks of existing more than 300 algorithm of skeletonization proposed

It may take a long time to skeletonize a high-resolution image.Skeletons may not contain sufficient information to reconstruct the original shapes;A skeletons may not be centred inside the underlying shape;Skeletons obtained are sensitive to noise and shape variations such as rotation and scaling;A shape and its skeleton may have a different number of connected components;Skeletons may contain artifacts such as noisy spurs and spurious short branch between split junction points;Skeleton branches may be serious erode;a lot of methods for extraction skeleton are limited within the shapes of only binary image and are invalid for a great deal of gray images.

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Three basic geometric structures of edges with Lipschitz exponents

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The step-Structure Edge

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The Roof-Structure Edge

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The Dirac-Structure Edge

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Some Concepts on Wavelet Function

Wavelet Function

If a 2-D function satisfies:

)( 22 RL

R R

dxdyyx 0),(

Scale Wavelet Transform

For and scale , the scale wavelet transform of is defined by

)( 22 RLf 0s),( yxf

dudvs

uy

s

ux

svufyxfyxfW

R R

ss ),(1

),(),)((),(2

Where ).,(1

),(2 s

v

s

u

svus

RRdxdyyx0),(

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The local properties of wavelet transformOn the time-domain

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On the frequency-domain

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How to construct wavelet function

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Which functions are selected as ?Gaussian Function

Gaussian function is not always the best one for all applications. Especially, it is not the best candidate for characterizing some structure edge.

Quadratic Spline Function Quadratic Spline Function is better than

Gaussian Function, but it is not suitable for Dirac-structure edge. For example

Which function is the best one ?

),( yx

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Major problems based on Quadratic Spline Function

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New Wavelet Function Constructed

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Where

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The Graphical Descriptions of New Wavelet Function (1)

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The Graphical Descriptions of New Wavelet Function (2)

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The Graphical Descriptions of New Wavelet Function (3)

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Wavelet Transform Based on New Wavelet Function

Wavelet Transform

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the Gradient direction of the Wavelet

Transform

Corresponding the Amplitude

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Some new characteristics of new wavelet function

Gray-level invariant: the local maximum moduli of the wavelet transform with respect to a Dirac-structure takes place at the same points when the images with different gray-levels are to be processed.

Slope invariant: the local maximum moduli of the wavelet transform of a Dirac-structure is independent on the slope of the shape.Width Invariant: For various widths of the Dirac-structure in an image, the location of maximum moduli depend on the scale of wavelet transform rather than its width under certain circumstance.

Symmetry: The two new lines formed by maximum moduli of wavelet transform is symmetric with respect to its central line of the shape.

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Implementation of wavelet transform in the discrete domain

Wavelet transform formula in the discrete domain

Wavelet coefficients and its calculation

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Some Application of New Wavelet Function Constructed in Image Processing

Detection of Edge

Recognizing Different Structures Edges

Extract central line of shape, such as skeletons of Ribbon-like Shapes

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Based on The properties of wavelet transform, New symmetry analysis, which is different from foregoing three symmetry analysis, is proposed.

The local maxima moduli of wavelet transform and the boundary of a shapeBy locating the local maxima of wavelet transform, we can

detect the boundary of the shape

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Symmetry Analysis Based on Maxima Moduli of Wavelet Transform

Central line

Location of maximum moduli of wavelet transform

Original boundary of a segment of Ribbon-like

shape

This distance equals to the scale “s”

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Maxima Moduli Symmetry and Wavelet Skeleton

Maxima Moduli Symmetry (MMS) For wavelet transform with the scale “s” which be

bigger than or equal to the width of ribbon-like shape. the points of it's maxima moduli form the two new lines which locate in the edge periphery of a shape, and they are local symmetrical with respect to the central line of a shape.This symmetry be called maxima moduli symmetry.

Wavelet Skeleton (WS) The wavelet skeleton of a Ribbon-like shape is defined

as the connective of all central line of location of symmetrical maximal moduli of a shape.

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Algorithm of Extracting Wavelet Skeleton

1.Select the suitable scale for wavelet transform according to the width of ribbon-like shape;2.Calculate all the wavelet transforms ;3.Calculate the local maxima of image contains Ribbon-like shapes and the gradient direction; 4.For each point with local maximum, search the point whose distance along the gradient direction from the point is s. If it is a point of local maxima, the center point is detected;5.The primary skeletons formed by all the points detected in Step 4 are what we need;

6.Modify the primary skeletons.(Why? )

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Two Examples

OriginalImage

MaximumModuli

PrimarySkeleton

Some points disappear in the junction.

How to modify ?46

Six Typical Junctions

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Two new scheme proposed to improve the structural quality of the skeleton

Depend on Gradient Direction Code of Wavelet Transform and its maximum moduli points;Based on the corner points of the

edge lines.

For most methods based on contour analysis (such as symmetric analysis etc.) How to extract the skeleton of junction area and intersection area of shape is still puzzling many researcher all over the world . Here, depending on wavelet transform, we try to propose the following two schemes. Experiments show that they perform relatively well on extracting the skeleton of shapes with some junction areas.

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After finishing wavelet transform, for every point in the image, we may calculate its corresponding gradient value and encode according to its gradient value.

At most four encoding values are considered and they represent four different discrete gradient direction respectively.

Based on four modifying criteria proposed by us, we can modify the primary wavelet skeletons and obtain perfect final results.

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For example

Criterion 1: If lost points in the locus of primary wavelet skeleton need to be resumed if there exists one of its the nearest points sampling (only eight ones) possess the same gradient direction (GCWT) as its ones and this points locates in the central line.Criterion 2: If lost points in the locus of primary wavelet skeleton possess the same gradient direction or GCWT as the terminal or end point of this locus and the distance from this end points to next one lies in the locus is the scale “s” or a half of its, all points need to be connected as a part of final wavelet skeleton.Criterion 3: If lost point in the locus of wavelet skeleton possess the same GCWT as the terminal or end point of this locus, and there exists single corresponding boundary point of the shape along the gradient direction or opposite direction and the distance from the point to the boundary is a half of the scale “s”, all such points need be retrieved as elements of the wavelet skeleton.Criterion 4: As long as any lost point be extended through s/2 points along its normal direction of the point gradient direction to meet the point of maximum moduli line, resuming process need to be stopped.

50 return

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22222

4 4 44 4 4

2222222 2222222

4 4 444

4

33

33 1

1

11

3311

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Modifying Scheme based on the corner points For the contour image (It can be obtained by

calculating local maximum moduli of wavelet transform), we search all corner points by the methods of finding singular point of curve(every contour line can be regard as a curve) based on wavelet transform technique.

Decide the central point of the junction or intersection area of the shape by using the following two schemes.

Method based on intersecting Points of Joining Branches;

Method based on Minimum Distance-Square Error.

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nx

n

ix 1

ny

n

iy

1

.

),( yx ii

Method based on intersecting Points of Joining Branches

..

.

.

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.

n

iiii yxdyxD

1

2),(),(

..

.

.

Method based on Minimum Distance-Square Error

),( yxd iii

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Computing

maximum moduli

Extracting

Primary skeleton

Modifying

Primary skeleton

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Extracting

Primary skeleton

Computing

maximum moduli

Modifying

Primary skeleton

Dec 19. 2001 3957

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Future WorksSo far, most algorithms of skeletonization of a shape

proposed are based on the contour of the shape, obviously, computational complexity is high and the location of central line depend completely on the edge. So we try to explore some new schemes to skeletonize directly shape independent of its contour. Recently, Some progresses have made by us.

Some other related applications in image processing of our new wavelet function may be extended as well.

Additionally, based on our experience of single wavelet applications, multiwavelet, especial non-separable wavelet with many good properties, we are trying to apply to our current field on extracting skeleton and detecting edge of images.

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