Skeletonization Based on Wavelet Transform

Outline Introduction How to construct wavelet function according to its application in practice Some new characteristics of new wavelet function Implementation of wavelet transform in the discrete domain Extraction of wavelet skeleton Some sets of schemes for modifying artifacts of primary skeletons Results of experiments Future works 2

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 3

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. 4

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 Transfor Brady’s Smoothed LocaL Symmetry Leyton’s Process-Inferring Symmetry Analysis Their Main shortcomings 5

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 circle B A 6

Brady’s Smoothed LocaL Symmetry (SLS) Brady defines the symmetric points as the midpoint of a straight line segment AB Symmetry point A B 7 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 arcLeyton’s Process-Inferring Symmetry Analysis (PISA) Symmetry point Symmetry arc A B 8

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. 9

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. 10

Three basic geometric structures of edges with Lipschitz exponents 12

The step-Structure Edgestep-Structure 13

The Roof-Structure EdgeRoof-Structure 14

The Dirac-Structure Edge 15

Some Concepts on Wavelet Function Wavelet Function If a 2-D function satisfies: Scale Wavelet Transform For and scale, the scale wavelet transform of is defined by Where 16

The local properties of wavelet transform On the time-domain 17

On the frequency-domain 18

How to construct wavelet function 19

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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 exampleFor example Which function is the best one ?? 21

Major problems based on Quadratic Spline Function 22

New Wavelet Function Constructed 23

Where 24

The Graphical Descriptions of New Wavelet Function (1) 25

The Graphical Descriptions of New Wavelet Function (2) 26

The Graphical Descriptions of New Wavelet Function (3) 27

Wavelet Transform Based on New Wavelet Function Wavelet Transform 28

the Gradient direction of the Wavelet Transform Corresponding the Amplitude 29

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. 30

Implementation of wavelet transform in the discrete domain Wavelet transform formula in the discrete domain Wavelet coefficients and its calculation 39

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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 42  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 shape By locating the local maxima of wavelet transform, we can detect the boundary of the shape 41

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” 43

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. 44

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? ) 45

Two Examples Original Image Maximum Moduli Primary Skeleton Some points disappear in the junction. How to modify ? 46

Six Typical Junctions return 47

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. junction area and intersection area 48

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.four modifying criteria 49 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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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. 53

. Method based on intersecting Points of Joining Branches

..... Method based on Minimum Distance- Square Error 55

Computing maximum moduli Extracting Primary skeleton Modifying Primary skeleton 56

Extracting Primary skeleton Computing maximum moduli Modifying Primary skeleton Dec

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Future Works So 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. 63

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