Relative Fuzzy Connectedness on Directed Graphs and its Application in a Hybrid Method for Interactive Image Segmentation

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Relative Fuzzy Connectedness on Directed Graphs and its Application in a Hybrid Method for Interactive

Image Segmentation

Hans Harley Ccacyahuillca Bejar

To cite this version:

Hans Harley Ccacyahuillca Bejar. Relative Fuzzy Connectedness on Directed Graphs and its Applica- tion in a Hybrid Method for Interactive Image Segmentation. LatinX in AI Research at ICML 2019, Jun 2019, Long Beach, California, United States. �hal-02244966�

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Relative Fuzzy Connectedness on Directed Graphs and its Application in a Hybrid Method for Interactive Image Segmentation

Hans Harley Ccacyahuillca Bejar hans@ime.usp.br

Advisor: Prof. Dr. Paulo A.V. Miranda pmiranda@ime.usp.br

Department of Computer Science — Institute of Mathematics and Statistics — University of São Paulo (USP)

Abstract

Original OIFT ORFC

Image segmentation consists of dividing an image into its compo- sing regions or objects, for example, to isolate the pixels of a target object of a given application. In segmentation of medical images, the object of interest commonly presents transitions at its border predominantly from bright to dark or dark to bright. Traditio- nal region-based methods of image segmentation, such as Relative Fuzzy Connectedness (RFC), do not distinguish well between si- milar boundaries with opposite orientations. The specification of the boundary polarity can help to alleviate this problem but this requires a mathematical formulation on directed graphs.

A discussion on how to incorporate this property in the RFC framework is presented in this work. A theoretical proof of the optimality of the new algorithm, called Oriented Relative Fuzzy Connectedness (ORFC), in terms of an energy function on directed graphs subject to seed constraints is presented, and its application in powerful hybrid segmentation methods. The hybrid method proposed ORFC & Graph Cut preserves the robustness of ORFC respect to the seed choice, avoiding the shrinking problem of Graph Cut (GC), and keeps the strong con- trol of the GC in the contour delination of irregular image boundaries. The proposed methods are evaluated using medical images of MRI and CT images of the human brain and thoracic studies.

Results

In this work, we introduced the ORF C technique and showed that it can effectively exploit the boundary polarity improving the results in relation to its predecessor RF C. We also presented a powerful hybrid approach, which outperforms the previous works [KCC13, KCC12C].

A conference paper was published in SIBGRAPI [HHCB14], and one journal paper was published in the EURASIP Journal on Image and Video Processing [HHCB15].

Theorical definition of ORFC ORFC as a directed cut in the digraph

ORFC is supported by a graph cut optimality crite- rion, which encompasses RFC as a particular case. In the case of directed graphs, we have two possible sets of cuts (inner and outer). Let’s consider the inner cut:

C_in(x) = {ha, bi ∈ E : x(a) = 0 ∧ x(b) = 1}

So we have the following formulation for the energy func- tional of the ε_∞-minimizing problem.

εⁱⁿ_∞(x) = max

ha,bi∈C_in w(a, b)

Let εⁱⁿ_∞↓ be the minimum value of the energy εⁱⁿ_∞(x), that is:

εⁱⁿ_∞↓ = min{εⁱⁿ_∞(x) : x ∈ X(S_o, S_b)}

Therefore, we have the following set of solutions:

X_∞ⁱⁿ(S_o, S_b) = {x ∈ X(S_o, S_b) : εⁱⁿ_∞(x) = εⁱⁿ_∞↓}

The ORF C algorithms on digraphs have the following defi- nitions based on cut in graph:

For the inner cut “in” with one internal seed s₁, A^in,_{ORF C}^" ({s₁}, S_b) = χ

O ∈ X_∞ⁱⁿ({s₁}, S_b) : |O| = min {|P| : χ

P ∈ X_∞ⁱⁿ({s₁}, S_b)}

Hybrid segmentation: ORFC & GC

IRFC ORFC+GC

Algorithm to compute A^out,_{ORF C}^" _+GC(S_o, S_b):

1 Compute P : χ

P = A^out,_{ORF C}^" (S_o, S_b).

2 Compute Q: χ

Q = A^in,_{ORF C}^" (S_b, S_o).

3 Compute and return A^out_OGC(P, Q).

Example of A

^in,_{ORF C}^"

(S

_o

= {s

_i

}, S

_b

)

(a) G = (V, E, w) (b) Initialization f_min^S^b

(c) V_b(s_i) = 1 (d) G_≤

(e) G^T_≤ (f) Result

Example of A

^out,_{ORF C}^" _+GC

(S

_o

, S

_b

)

ORFC algorithm

RFC RFC+GC ORFC+OGC

Algorithm to compute A^in,_{ORF C}^" ({s_i}, S_b):

1 Compute the value of the map V_b(s_i) for the function f_min^S^b .

2 Remove from the graph G all edges with weight ≤ εⁱⁿ_∞↓ = V_b(s_i), obtaining a new graph G_≤.

3 Assign to the object the pixels that belong to the

directed connected component of basepoint s_i in the transpose graph of G_≤ (i.e., A^in,_{ORF C}^" ({s_i}, S_b) = χ

O : O = DCC_G_T

≤(s_i)).

Algorithm to compute A^out,_{ORF C}^" ({s_i}, S_b):

1 Compute the value of the map V_b^6k(s_i) for the function f_min^{6k S}^b. 2 Remove from the graph G all edges with

weight ≤ ε^out_∞↓ = V_b^6k(s_i), obtaining a new graph G_≤.

3 Assign to the object the pixels that belong to the directed connected component of basepoint s_i in the graph G_≤

(i.e., A^out,_{ORF C}^" ({s_i}, S_b) = χ

O : O = DCC_G_≤(s_i)).

Experimental Results

Calcaneus

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

0 5 10 15 20

Dice coefficient

Erosion radius (pixels)

IRFC RFC O-IFT

RFC+GC O-RFC+GC O-RFC

O-GC

0 0.05 0.1 0.15 0.2 0.25 0.3

0 5 10 15 20

Normalized false positive

IRFC RFC O-IFT

O-GC

Brain

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

0 2 4 6 8 10 12 14 16

Dice coefficient

IRFC RFC O-IFT

O-GC

0 0.05 0.1 0.15 0.2

0 2 4 6 8 10 12 14 16

Normalized false positive

IRFC RFC O-IFT

O-GC

References

[HHCB15] Bejar, Hans H C. and Miranda, Paulo A V. Oriented relative fuzzy connectedness:

theory, algorithms, and its applications in hybrid image segmentation methods. EURASIP Journal on Image and Video Processing, 2015(1), 2015.

[HHCB14] Hans H. C. Bejar and Paulo A.V. Miranda. Oriented Relative Fuzzy Connectedness: Theory, Algorithms, and Applications in Image Segmentation.

27th SIBGRAPI Conference on Graphics, Patterns and Images, 304–311, 2014.

[KCC13] Krzysztof Chris Ciesielski and P.A.V. Miranda and A.X. Falcão and Jayaram K. Udupa. Joint graph cut and relative fuzzy connectedness image segmentation algorithm. Medical Image Analysis, 17(8):1046–1057, 2013.

[KCC12C] , K.C. Ciesielski and P.A.V. Miranda and J.K. Udupa and A.X. Falcão.

Image segmentation by combining the strengths of relative fuzzy connectedness and graph cut. Proc. of the International Conference on Image Processing, 2005–2008, 2012.