Spotted cDNA microarray image segmentation ... - IMT Bucharest

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252 S. Ni et al.Algorithm:1. Initialize φ 0 by φ 0 , n = 0.2. Compute c 1 (φ n ) and c 2 (φ n ) by∫c 1 (φ n Ω) =µ 0(x, y)H(φ(x, y))dxdy∫H(φ(x, y))dxdyandc 2 (φ n ) =3. Solve the PDE in φ from∫ΩΩ µ 0(x, y)(1 − H(φ(x, y)))dxdy∫Ω (1 − H(φ(x, y)))dxdy .∂φ∂t = δ ε(φ)[µdiv( ∇φ|∇φ| ) − v − λ 1(u 0 − c 1 ) 2 + λ 2 (u 0 − c 2 ) 2 ] = 0 in (0, ∞) × Ω,φ(0, x, y) = φ 0 (x, y) in Ω,δ ε (φ) ∂φ|∇φ| ∂ −→ = 0 on ∂Ω,nwhere −→ n is the exterior normal to the boundary ∂Ω, and ∂φ∂ −→ is the normal derivativenof φ at the boundary.4. Reinitialize φ locally to the signed distance function to the curve (optional).5. Check whether the solution is stationary. If it is not, then n = n + 1 and repeatthe algorithm.The C-V model was implemented as in [4] using finite differences equations:φ n+1i,j− φ n i,j△t= δ h (φ n i,j)[ µ △ x +φ n+1h 2 △x i,j−.()√(△ x +φ n i,j )2 /(h 2 ) + (φ n i,j+1 − φn i,j−1 )2 /(2h) 2+ µ h 2 △y −.(√△ y +φ n+1i,j)(φ n i+1,j − φn i−1,j )2 /(2h) 2 + (△ y +φ n i,j )2 /(h 2 )− v − λ 1 (u 0,i,j − c 1 (φ n )) 2 + λ 2 (u 0,i,j − c 2 (φ n )) 2 ].The C-V method is the minimization of an energy based segmentation. For example,an image is denoted by u 0 with the boundary denoted by C 0 . The image canbe divided by two regions: the inside object denoted by u i 0 and the outside objectdenoted by u o 0 .The fitting function is defined as follows:∫F 1 (C) + F 2 (C) =inside(C)∫|u 0 (x, y) − c 1 | 2 dxdy + |u 0 (x, y) − c 2 | 2 dxdy,outside(C)
Spotted cDNA microarray image segmentation using ACWE 253where C is the variable curve, c 1 , c 2 are the constants depending on C and C 0 is theminimum of the fitting functioninfC {F 1(C) + F 2 (C)} ≈ 0 ≈ F 1 (C 0 ) + F 2 (C 0 ).In the C-V model the fitting function is minimized and some more terms areadded like the length of the curve and the area inside the C . The energy functionF (c 1 , c 2 , C) is defined as follows:F (c 1 , c 2 , C) = µ · Length(C) + v · Area(inside(C)) +∫+ λ 1 |u 0 (x, y) − c 1 | 2 dxdy+ λ 2∫inside(C)outside(C)|u 0 (x, y) − c 2 | 2 dxdy,where µ ≥ 0, v ≥ 0, λ 1 , λ 2 ≥ 0 are constant.The ACWE method minimizes the energy function to get the boundary whereφ = 0. The inside part (φ > 0) is the segmentation result needed. It means that theboundary curve has been found and the area inside the boundary is the actual objectobtained by the segmentation.The C-V model has some limitation. For example it cannot detect the textureimage. These limitations have no effect on applying the C-V model to cDNA microarrayimage segmentation. If we apply the ACWE method directly on the wholemicroarray image it would not give us the correct segmentation. Therefore we madesome adjustment for applying ACWE on the microarray image as follows:1. We used ACWE to segment each spot patch one at a time.2. We provided the grid file which gives the approximate spot locations.3. We decreased the iteration numbers and made the computing time shorter bytaking into consideration the current segmentation setting.4. We adjusted the µ value and found smaller spots.We use C-V method to compute the exact boundary of each spot. We can get theexact location of each spot beforehand by using the grid file from the database. Thegrid file gives out the location of each spot which will be printed on the microarrayduring the printing process of the microarray. We use the spot patch which is thelocation (rectangle or square) of each spot as a sub image. Since the patch outsideof every spot is the same, we only need to preset up the initial function and otherparameters once for PDE equations. After solving the equations we will get the exactboundary of the individual spot and the intensity value of that spot. To computethe background intensity value, we use the local background. The background areaconsidered in this paper is defined as the area that lies outside the boundary of
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