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Tracing and c ompressing digital curves. András Hajdu Department of Computer Graphics and Image Processing Faculty of Informatics, University of Debrecen , Hungary 17th Summer School on Image Processing , Ju ly 2-11 2009, Debrecen . Overview of the problem. Overview

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Tracing and compressingdigital curves

András Hajdu

Department of Computer Graphics and Image Processing

Faculty of Informatics, University of Debrecen, Hungary

17th Summer School on Image Processing, July 2-11 2009, Debrecen.


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Overview of the problem


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Overview

of the proposed approach

  • Assign an underlying graph to the curve

  • Find an Euler decomposition of the graph

  • Trace the Euler components with respect to a linear optimality criterion at junctions

  • Compress the curve components using an alphabet of line segments (polygon approximation)


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Assigning an underlyinggraph to the curve - Basics

  • A graphG = (V,E) V: set of vertices, E=V×V= {{v1, v2} | v1, v2V}: set of edges.

  • G is undirected, loops and multiple edges allowed.

  • The degree of a vertex is the number of edges containing it.

  • A path is a list of edges: {v1,v2}, {v2,v3}, . . . , {vn-1,vn}.

  • A route is a closed path with v1= vn.

  • A path containing all the edges exactly once is an Euler path (route)


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Assigning an underlyinggraph to the curve - Basics

  • An Euler decomposition of G is the union of its edge-disjoint Euler subgraphs Gi (i=1,...,n).

  • Results on Euler graphs and decompositions:

  • Every Euler graph is connected,

  • A connected graph has Euler route iff all of its vertices have even degree,

  • A connected graph has Euler path iff at most two vertices have odd degree,

  • Every connected graph has an Euler decomposition.


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Assigning an underlyinggraph to the curve

  • The curve tracing (CT) algorithm:

  • Assign a graph GC = ( VC, EC) to the curve C, where VC contains all the end and junction points of C. The edges are the curve segments connecting these vertices.

  • Create an Euler decomposition  Ciof C based on GC.

  • Trace all the Ci’s separately through their Euler paths.

n

i=1


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Realization of CT algorithm -preprocessing

  • The input can be any one-pixel wide self intersecting object (curve) for graph assignment.

  • To guarantee one-pixel wideness, thinning can be applied first to remove undesired thickening pixels.


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Realization of CT algorithm -locating vertices

  • Vertices are assigned to curve junctions and end points

Junctions: curve points with more than two 8-neighbors,

End points: curve points with exactly one 8-neighbor.


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Realization of CT algorithm -extracting edges

  • Non-vertex 8-neighbors of vertices with exactly two 8-neighbors are marked as edge end points.

  • Edges are 8-connected paths connecting edge end points.

edge end point

edge point

junction point


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Realization of CT algorithm -the underlying graph

end points:

1,10

junction points:

2,3,4,5,6,7,8,9,11,12,13


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Realization of CT algorithm -junctions of special type

  • Junctions degenerate if tangents of curve segments almost coincide.

  • Junction points closer than a threshold K are found, and the pixels between them are merged.

degenaratedintersection

edge end point

edge point

junction point

K = 2

K = 9


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Realization of CT algorithm -tracing the components

  • The first step to trace an Euler curve is to locate a starting vertex having odd degree.

  • Then, we take an edge from the starting vertex to initialize the tracer.

possible starting vertices:

1 (finishing at 10)

10 (finishing at 1)

possible starting edges:

{1,4} (finishing with {8,10})

{10,8} (finishing with {4,1})


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Realization of CT algorithm -tracing through junctions

  • More Euler paths may exist. Which edge to take next at junctions?

  • For better polygon approximation, go on straight:

  • calculate the centre of the junction having E1, . . ., Ekedge end points (we arrive at E0)

  • calculate the change of direction for all Ei:

  • select the most linear direction towards El with:


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Realization of CT algorithm - tracing through junctions

we arrive at E0 ...

... we go on with E2


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Realization of CT algorithm -restrictions for tracing

The most straightdirection cannot be always selected

freely at junctions.

We must obey the graph traversal (Fleury’s) algorithm

to extract an Euler path:

  • Always leave one edge available to get back to the starting vertex or to the other odd vertex.

  • Do not use an edge to go to a vertex unless there is another edge available to leave it.


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Realization of CT algorithm - complete tracing

Euler path:

{1,4,5,2,4,8,13,11,7,5,2,3,6,7,9,13,12,6,3,12,11,9,8,10}

Start

End


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Realization of CT algorithm -complete tracing of the curve


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Compressing the curve -

steps of compression

  • A finite alphabet of digital line segments is defined.

  • The traced curve is compressed online to replace curve segments with letters from the alphabet.

  • Complementary variable length (Huffmann) coding is applied.


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Compressing the curve -

alphabet of digital line segments

Λ is a finite alphabet of Bresenham line segmentsof all orientations having length at most T.

T=6, first octant


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Compressing the curve -

alphabet of digital line segments

all orientations


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Compressing the curve -

alphabet of digital line segments

  • Cardinality of Λ:

  • Number of bits to code a letter:

To keep cardinality small, we consider unique (Bresenham) segments to connect two points.


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Compressing the curve -

replacing curve segments

  • Online curve compression is used:during tracing the curve, current curve segment is replaced from Λ if it violates linearity:

the segment should stay between two parallel lines [Rosenfeld and Klette].

Information loss is allowed as Bresenhamsegments differ, but linearity is common criterion.


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Compressing the curve -

comparing with JBEAM

  • JBEAM [Huo and Chen] is a state-of-the-art algorithm for the compresson of one-pixel wide objects.

  • It divides binary (curve) imageusing quadtree decomposition till having single linear curve segments incells to be substituted by beamlets.


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Compressing the curve -

comparing with JBEAM


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Compressing the curve -

comparing with JBEAM

~50% improvement

With tracing, the proposed CT algorithm avoids subdivision.


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Compressing the curve -

comparing with MPEG-4

  • In MPEG-4 a vertex-based shape approximation was developed to code the outline ofshapes.

  • The boundary is approximated by a polygon, with a recursive splitting process that starts with the longest axis (diameter).

  • If the distance of the curve and polygon segment is > dmax the segment is splitted into two new parts.


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Compressing the curve -

comparing with MPEG-4


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Compressing the curve -

comparing with MPEG-4


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Compressing the curve -

comparing with MPEG-4

~similar performance with MPEG-4 for dmax=1

(dmax=1 violates the linearity criterion for MPEG-4)


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Compressing the curve -

comparing with MPEG-4


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Compressing the curve -

comparing with MPEG-4

MPEG-4 recommends the application of a complementary

variable length coding, e.g. Huffmann coding

~50% improvement in compression performance.


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Compressing the curve -

more MPEG-4 improvements

Some more minor recommendations from MPEG-4:

  • For lossy shape coding the selection of the vertices on the object boundary is not optimal. Therefore, the vertices can be shifted by 1 pixel within a neighborhood of size 3 × 3.

  • The maximal length of the alphabet elements can be fixed dynamically as the length of the longest polygon segment. In this case, this length information should be transmitted, as well.


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Alternative approaches –

weighted graphs

  • The same finite alphabet of line segments and online curve compression is considered.

  • Weighted graph approaches can be considered with a natural weight function for the edges:

  • The number of letters needed:


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Alternative approaches –

weighted graphs

number of letters needed to compress edges (weights)


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Alternative approaches –

compressing edges separately

  • Compression can be executed at edge level without tracing.

  • Thissimple approach also avoids curve subdivision, but has drawbacks:

  • the coordinates of the start pixel of each curve segment for appropriate geometricpositioning.

  • lack of junction information leads to distortion at junctions during decompression (the curve segments must be connected ad-hoc).


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Alternative approaches –

compressing edges separately

Optimal tracing performs well for curves having more straight sections (e.g. General, Lines)


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Alternative approaches –

avoiding Euler decomposition

  • Our basic approach is to decompose every underlying graph into Euler subgraphs.

  • Decomposition can be avoided with taking some edges more than once (Chinese Postman Problem, CPP).

  • Advantage is that we can avoid graph decomposition.

  • Disadvantage is the redundant compression of edges.

  • Algorithmic solution is known for the CPP problem:we must try to select the redundant edges with keepingthe extra weight sum minimal.


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Alternative approaches –

avoiding Euler decomposition

  • Vertices 1,2,3,12,16,17 are of odd degree (non-Euler)

  • Allowing edges {2,3} and {16,17} twice gives minimal weight sum for an Euler traversal.


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Alternative approaches –

avoiding Euler decomposition

Start

End

Path with minimal weight sum:

{1,6,7,4,6,10,15,13,9,7,4,3,2,3,5,8,9,11,15,16,17,16,14,8,5,14,13,11,10,12}


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Discussion,

applications

  • To ”untie” curves can have impact in curve watermarking (with providing the data in terms of few large blocks)

  • Tracing the curve according to its ”natural” direction looks feasible to reconstruct hand-written text or figures, or to classify vessel types (arterial/venous):

(image is from Rothaus et al. 2007)


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Related publications

  • A. Hajdu and I. Pitas: Piecewise linear digital curve representation and compression using graph theory and a line segment alphabet, IEEE Trans. on Image Processing17/2 (February 2008), 126-133.

  • A. Hajdu and I. Pitas: Compression optimized tracing of digital curves using graph theory, IEEE International Conference on Image Processing (ICIP 2007), San Antonio, Texas, USA, Vol. VI, 453-456.

ACKNOWLEDGEMENT

This work was partially supported by the project SHARE: Mobile Support for Rescue Forces, Integrating Multiple Modes of Interaction, EU FP6 Information Society Technologies, Contract Number FP6-004218.