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## Interferenceless and motionless method for recording digital holograms of coherently illuminated 3D objects by coded aperture correlation holography system

Equations (9). Equations on this page are rendered with MathJax. Learn more. (1) H O B J ( r ¯ ) = ∫ I O B J ( r ¯ , z ) ∗ t ( r ¯ , z ) d z , (2) I I M G ( r ¯ , z j ) = H O B J ( r ¯ ) ⊗ R ( r ¯ , z j ) , (3) A I n ( r ¯ ) ∗ f ( r ¯ ) 2 ≈ A I n ( r ¯ ) 2 ∗ q ( r ¯ ) , (4) A I n ( r ¯ ) ∗ ∑ k a k δ ( r ¯ − r ¯ k ) 2 = ∑ k a k A I n ( r ¯ − r ¯ k ) 2 = ∑ k a k 2 A I n ( r ¯ − r ¯ k ) 2 = A I n ( r ¯ ) 2 ∗ ∑ k a k 2 δ ( r ¯ − r ¯ k ) , (5) I k = O ( r ¯ R / r ¯ R M T M T ) 2 ∗ ∑ i = 1 N a i 2 δ ( r ¯ R − r ¯ k , i ) , (6) I I M G = H O B J ⊗ H P S H ′ = O ( r ¯ R / r ¯ R M T M T ) ∗ H P S F 2 ⊗ H P S H ′ = O ( r ¯ R / r ¯ R M T M T ) ∗ [ ∑ i = 1 N a i δ ( r ¯ R − r ¯ 1 , i ) − ∑ i = 1 N b i δ ( r ¯ R − r ¯ 2 , i ) ] 2 ⊗ H P S H ′ = O ( r ¯ R / r ¯ R M T M T ) 2 ∗ [ ∑ i = 1 N a i 2 δ ( r ¯ R − r ¯ 1 , i ) − ∑ i = 1 N b i 2 δ ( r ¯ R − r ¯ 2 , i ) ] ⊗ H P S H ′ = F − 1 { F [ O ( r ¯ R / r ¯ R M T M T ) 2 ] F [ ∑ i = 1 N a i 2 δ ( r ¯ R − r ¯ 1 , i ) − ∑ i = 1 N b i 2 δ ( r ¯ R − r ¯ 2 , i ) ] exp ⁡ ( − j φ ) } = F − 1 { F [ O ( r ¯ R / r ¯ R M T M T ) 2 ] h exp ⁡ ( j φ ) exp ⁡ ( − j φ ) } = O ( r ¯ R / r ¯ R M T M T ) 2 ∗ F − 1 { h } ≈ O ( r ¯ R / r ¯ R M T M T ) 2 , (7) F { ∑ i = 1 N a i 2 δ ( r ¯ R − r ¯ 1 , i ) − ∑ i = 1 N b i 2 δ ( r ¯ R − r ¯ 2 , i ) } = h exp ⁡ ( j φ ) . (8) H O B J = ∑ q = 1 P { O q ( r ¯ R / r ¯ R M T M T ) 2 ∗ ∑ p = 1 P [ ∑ i = 1 N a p , i g q , p ( r ¯ R − r ¯ 1 , p , i ) − ∑ i = 1 N b p , i g q , p ( r ¯ R − r ¯ 2 , p , i ) ] } , (9) I I M G = H O B J ⊗ H P S H , l ′ = ∑ q = 1 P { O q ( r ¯ R / r ¯ R M T M T ) 2 ∗ ∑ p = 1 P [ ∑ i = 1 N a p , i g q , p ( r ¯ R − r ¯ 1 , p , i ) − ∑ i = 1 N b p , i g q , p ( r ¯ R − r ¯ 2 , p , i ) ] } ⊗ H P S H , l ′ = ∑ q = 1 P O q ( r ¯ R / r ¯ R M T M T ) 2 ∗ [ ∑ i = 1 N a q , i δ ( r ¯ R − r ¯ 1 , q , i ) − ∑ i = 1 N b q , i δ ( r ¯ R − r ¯ 2 , q , i ) ] ⊗ H P S H , l ′ + ∑ q = 1 P ∑ p ≠ q P O q ( r ¯ R / r ¯ R M T M T ) 2 ∗ [ ∑ i = 1 N a p , i g q , p ( r ¯ R − r ¯ 1 , p , i ) − ∑ i = 1 N b p , i g q , p ( r ¯ R − r ¯ 2 , p , i ) ] ⊗ H P S H , l ′ ≈ O l ( r ¯ R / r ¯ R M T M T ) 2 + ∑ q ≠ l P O q ( r ¯ R / r ¯ R M T M T ) 2 ∗ g q , l ( r ¯ R ) . .

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