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Journals >Journal of Infrared and Millimeter Waves >Volume 40 >Issue 1 >Page 108 > Article

Journal of Infrared and Millimeter Waves
Vol. 40, Issue 1, 108 (2021)

GPU accelerated level set model solving by lattice boltzmann method with application to image segmentation

Wen-Jun SHI¹, Deng-Wei WANG^2、3、*, Wan-Suo LIU¹, and Da-Gang JIANG^2、3

Author Affiliations

¹Aviation Maintenance School for NCO，Air Force Engineering University，Xinyang 464000，China

²School of Aeronautics and Astronautics，University of Electronic Science and Technology of China，Chengdu 611731，China

³Aircraft Swarm Intelligent Sensing and Cooperative Control Key Laboratory of Sichuan Province，Chengdu 611731，China

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DOI: 10.11972/j.issn.1001-9014.2021.01.016 Cite this Article

Wen-Jun SHI, Deng-Wei WANG, Wan-Suo LIU, Da-Gang JIANG. GPU accelerated level set model solving by lattice boltzmann method with application to image segmentation[J]. Journal of Infrared and Millimeter Waves, 2021, 40(1): 108 Copy Citation Text

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Abstract

A novel Graphics Processing Units (GPU) accelerated level set model which organically combines the global fitting energy and the local fitting energy from different models and the weighting coefficient of the global fitting term can be adaptively adjusted, is proposed to image segmentation. The proposed model can efficiently segment images with intensity inhomogeneity regardless of where the initial contour lies in the image. In its numerical implementation, an efficient numerical scheme called Lattice Boltzmann Method (LBM) is used to break the restrictions on time step. In addition, the proposed LBM is implemented by using a NVIDIA GPU to fully utilize the characteristics of LBM method with high parallelism. The extensive and promising experimental results from synthetic and real images demonstrate the effectiveness and efficiency of the proposed method.In addition, the factors that can have a key impact on segmentation performance are also analyzed in depth.

Keywords

graphics processing unit Intensity inhomogeneity Lattice Boltzmann method level set method segmentation

\begin{array}{l} E^{C V} (ϕ) = λ_{1}^{C V} {\int_{Ω} (I (x) - M_{i n})}^{2} H (ϕ (x)) d x + \\ λ_{2}^{C V} {\int_{Ω} (I (x) - M_{o u t})}^{2} (1 - H (ϕ (x))) d x + \\ μ \cdot \int_{Ω} δ (ϕ (x)) |\nabla ϕ (x)| d x + ν \cdot \int_{Ω} H (ϕ (x)) d x \end{array}

,（1）

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\frac{\partial ϕ}{\partial t} = δ (ϕ) [μ d i v (\frac{\nabla ϕ}{|\nabla ϕ|}) - ν - λ_{1}^{C V} {(I - M_{i n})}^{2} + λ_{2}^{C V} {(I - M_{o u t})}^{2}]

,（2）

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\{\begin{array}{l} M_{i n} (ϕ) = \frac{\int_{Ω} I (x) H (ϕ (x)) d x}{\int_{Ω} H (ϕ (x)) d x d y} \\ M_{o u t} (ϕ) = \frac{\int_{Ω} I (x) (1 - H (ϕ (x))) d x}{\int_{Ω} (1 - H (ϕ (x))) d x} \end{array}

,（3）

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\begin{array}{l} E_{x} = λ_{1}^{L B F} \int_{i n s i d e (C)} K_{σ} (x - y) |I (y) - f_{1} {(x)}^{2}| d y d x + \\ λ_{2}^{L B F} \int_{o u t s i d e (C)} K_{σ} (x - y) |I (y) - f_{2} {(x)}^{2}| d y d x \end{array}

,（4）

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\begin{array}{l} E^{L B F} = \int_{Ω} E_{x} (ϕ, f_{1} (x), f_{2} (x)) d x \\ = λ_{1}^{L B F} \int_{Ω} [\int_{Ω} K_{σ} (x - y) {|I (y) - f_{1} (x)|}^{2} H (ϕ (y)) d y] d x \\ + λ_{2}^{L B F} \int_{Ω} [\int_{Ω} K_{σ} (x - y) {|I (y) - f_{2} (x)|}^{2} (1 - H (ϕ (y))) d y] d x \end{array}

,（5）

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\frac{\partial ϕ}{\partial t} = - δ (ϕ) (λ_{1}^{L B F} e_{1} - λ_{2}^{L B F} e_{2})

,（6）

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\{\begin{array}{l} e_{1} (x) = \int_{Ω} K_{σ} (y - x) {|I (x) - f_{1} (y)|}^{2} d y \\ e_{2} (x) = \int_{Ω} K_{σ} (y - x) {|I (x) - f_{2} (y)|}^{2} d y \end{array}

,（7）

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\{\begin{array}{l} f_{1} (x) = \frac{K_{σ} * [H_{ε} (ϕ) I (x)]}{K_{σ} * H_{ε} (ϕ)} \\ f_{2} (x) = \frac{K_{σ} * [1 - H_{ε} (ϕ) I (x)]}{K_{σ} * [1 - H_{ε} (ϕ)]} \end{array}

.（8）

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\{\begin{array}{l} H_{ε} (z) = \frac{1}{2} [1 + \frac{2}{π} a r c t a n (\frac{z}{ε})] \\ δ_{ε} (z) = \frac{1}{π} \cdot \frac{ε}{ε^{2} + z^{2}}, z \in R \end{array}

.（9）

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\begin{array}{l} E^{l o c a l} (ϕ) = λ_{1}^{C V} \int_{Ω} [\int_{Ω} K_{σ} (x - y) {|I (y) - f_{1} (x)|}^{2} H (ϕ (y)) d y] d x + \\ λ_{2}^{C V} \int_{Ω} [\int_{Ω} K_{σ} (x - y) {|I (y) - f_{2} (x)|}^{2} (1 - H (ϕ (y))) d y] d x \end{array}

.（10）

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\begin{array}{l} E^{g l o b a l} (ϕ) = λ_{1}^{L B F} {\int_{Ω} (I (x) - M_{i n})}^{2} H (ϕ (x)) d x + \\ λ_{2}^{L B F} {\int_{Ω} (I (x) - M_{o u t})}^{2} (1 - H (ϕ (x))) d x \end{array}

.（11）

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E (ϕ) = E^{l o c a l} (ϕ) + w \cdot E^{g l o b a l} (ϕ)

,（12）

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L C R_{N} = \frac{V_{m a x} - V_{m i n}}{L}

,（13）

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w = κ \cdot m e a n (L C R_{N}) . (1 - L C R_{N})

,（14）

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P (ϕ) = \int \frac{1}{2} {(|\nabla ϕ (x)| - 1)}^{2} d x

.（15）

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L (ϕ) = \int |\nabla H (ϕ (x))| d x

.（16）

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E^{o u r} = E + μ \cdot P (ϕ) + ν \cdot L (ϕ)

,（17）

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\begin{array}{l} \frac{\partial ϕ}{\partial t} = δ_{ε} (ϕ) (- λ_{1}^{C V} {(I - M_{i n})}^{2} + λ_{2}^{C V} {(I - M_{o u t})}^{2}) - \\ w \cdot δ_{ε} (ϕ) (λ_{1}^{L B F} e_{1} - λ_{2}^{L B F} e_{2}) + \\ μ (Δ ϕ - d i v (\frac{\nabla ϕ}{|\nabla ϕ|})) + ν δ_{ε} (ϕ) d i v (\frac{\nabla ϕ}{|\nabla ϕ|}) \end{array}

,（18）

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\begin{array}{l} f_{i} (\vec{r} + \vec{e_{i}} Δ t, t + Δ t) = f_{i} (\vec{r}, t) + \frac{1}{τ} [f_{i}^{e q} (\vec{r}, t) - f_{i} (\vec{r}, t)] \\ + \frac{D}{b c^{2}} \cdot \vec{F} \cdot \vec{e_{i}} \end{array}

,（19）

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f_{i}^{e q} (ρ, \vec{u}) = ρ (A_{i} + B_{i} (\vec{e_{i}} \cdot \vec{u}) + C_{i} (\vec{e_{i}} \cdot \vec{u}) + D_{i} {(\vec{u})}^{2})

,（20）

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\{\begin{array}{l} ρ = \sum_{i} f_{i} \\ \vec{u} = \frac{1}{ρ} \sum_{i} f_{i} \vec{e_{i}} \end{array}

,（21）

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f_{i}^{e q} (ρ, \vec{u}) = ρ A_{i}

.（22）

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\begin{array}{l} f_{i}^{e q} = ρ α_{i} (1 + 3 (\vec{e_{i}} \cdot \vec{u}) + \frac{9}{2} {(\vec{e_{i}} \cdot \vec{u})}^{2} - \frac{3}{2} {(\vec{u})}^{2}), \\ i = 0,1, \dots, 8 \end{array}

,（23）

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γ = \frac{2}{9} (2 τ - 1)

.（24）

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\frac{\partial ρ}{\partial t} = γ \nabla \cdot \nabla ρ + F

.（25）

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\frac{\partial ρ}{\partial t} = γ \nabla \cdot \nabla ρ + F = γ d i v (\frac{\nabla ϕ}{|\nabla ϕ|}) + F

,（26）

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\begin{array}{l} f_{i} (\vec{r} + \vec{e_{i}} Δ t, t + Δ t) = f_{i} (\vec{r}, t) + \frac{1}{τ} [f_{i}^{e q} (\vec{r}, t) - f_{i} (\vec{r}, t)] + \\ \frac{D}{b c^{2}} \cdot (w \cdot δ_{ε} (ϕ) (- λ_{1} {(I - M_{i n})}^{2} + λ_{2} {(I - M_{o u t})}^{2}) - \\ δ_{ε} (ϕ) (I - I^{L I F}) (u_{1} - u_{2}) + μ \cdot Δ ϕ) \end{array}

.（27）

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|\begin{array}{l} (1) i m a g e_{G P U} = g p u A r r a y (i m a g e); \\ (2) M o d e l P a r a m e t e r S t r u c t u r e_{G P U} = g p u A r r a y \\ (M o d e l P a r a m e t e r S t r u c t u r e); \\ (3) F_{G P U} = a r r a y f u n \\ (@ b o d y_f o r c e, i m a g e_{G P U}, M o d e l P a r a m e t e r S t r u c t u r e); \\ (4) F = g a t h e r (F_{G P U}) . \end{array}

(28)

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J S = \frac{N (S_{r e f e r e n c e} ⋂ S_{t e s t})}{N (S_{r e f e r e n c e} ⋃ S_{t e s t})}

,（28）

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D S C = \frac{2 N (S_{r e f e r e n c e} ⋂ S_{t e s t})}{N (S_{r e f e r e n c e}) + N (S_{t e s t})}

,（29）

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ϕ_{i n i t i a l} (x, y) = \{\begin{array}{l} - c, (x, y) \in Ω_{i n i t i a l} - \partial Ω_{i n i t i a l} \\ 0, (x, y) \in \partial Ω_{i n i t i a l} \\ c, (x, y) \in Ω - Ω_{i n i t i a l} \end{array}

,（30）

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Wen-Jun SHI, Deng-Wei WANG, Wan-Suo LIU, Da-Gang JIANG. GPU accelerated level set model solving by lattice boltzmann method with application to image segmentation[J]. Journal of Infrared and Millimeter Waves, 2021, 40(1): 108

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