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Image-enhanced single-pixel imaging using fractional calculus

Xiao Zhang, Rui Li, Jiaying Hong, Xi Zhou, Nian Xin, and Qin Li

Open Access Open Access

Abstract

Recent years, image enhancement for single-pixel imaging has developed rapidly and provides an image-free way for extracting image information. However, the conventional image enhancement approaches for single-pixel imaging are still based on the discontinuously adjustable operations such as integer-order derivatives, which are frequently used in edge detection but sensitive to the image noise. Therefore, how to balance between two conflicting demands, i.e. edge detection and noise suppression, is a new challenge. To address this issue, we introduce arbitrary-order fractional operations into single-pixel imaging. In experiment, the proposed technique has the capacity to detect image edges with high quality. Compared with integer-order derivative method which amplifies noise significantly while extracting edges, it offers a nice tradeoff between image SNR and performance of edge enhancement. In addition, it also shows good performance of image smoothing and improvement of image quality, if fractional order is negative. The proposed technique provides the adjustable fractional order as a new degree of freedom for edge extraction and image de-noising and therefore makes up for the shortcomings of traditional method for image enhancement.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. The simulation results of fractional/first-order derivatives and fractional integrals of sigmoid function (a) without noise (b) with noise.

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Fig. 2.
Fig. 2. Experimental set-up of image-enhanced single-pixel imaging.

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Fig. 3.
Fig. 3. (a) The measured Fourier spectrum of the object image. (b) The object image recovered from the measured Fourier spectrum. (c)-(f) Typical two-dimensional Fourier-domain operators of fractional calculus at θ=0, π/4, π/2 and 3π/4, when n=0.9.

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Fig. 4.
Fig. 4. The experimental results of fractional derivatives for object in four directions of θ=0, π/4, π/2 and 3π/4, when n=0.2, 0.4, 0.6 and 0.8 respectively.

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Fig. 5.
Fig. 5. The experimental results achieved by image-enhanced single-pixel imaging when n>0.

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Fig. 6.
Fig. 6. The experimental results of fractional integrals for object in four directions of θ=0, π/4, π/2 and 3π/4 under different n.

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Fig. 7.
Fig. 7. The experimental results achieved by image-enhanced single-pixel imaging when n<0.

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Fig. 8.
Fig. 8. The experimental relationship of fractional order n and image SNR of image-enhanced single-pixel imaging. Note that the image SNRs are calculated by Eq. (7).

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Fig. 9.
Fig. 9. The amplitude of Fourier-domain operator of one-dimensional fractional calculus under different n.

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

u ^ ( k ) = ∫ − ∞ ∞ e − i k x u ( x ) d x ,
u ( x ) = 1 / 2 π ⋅ ∫ − ∞ ∞ e i k x u ^ ( k ) d k .
D n u ( x ) ⇄ F T I F T Ω n ( k ) ⋅ u ^ ( k ) = ( i k ) n ⋅ u ^ ( k ) , n ∈ N ∗ .
D n u ( x ) ⇄ F T I F T Ω n ( k ) ⋅ u ^ ( k ) = ( i k ) n ⋅ u ^ ( k ) , n ∈ R ,
D n , θ u ( x , y ) ⇄ 2 D F T 2 D I F T Ω n , θ ( k x , k y ) ⋅ u ^ ( k x , k y ) , n ∈ R ,
Ω n , 0 ( k x , k y )  =  ( i k x ) n Ω n , π / 4 ( k x , k y )  =  ( i k x / 2  +  i k y / 2 ) n Ω n , π / 2 ( k x , k y )  =  ( i k y ) n Ω n , 3 π / 4 ( k x , k y )  =  ( − i k x / 2  +  i k y / 2 ) n .
SNR = 20 ⋅ log 10 ⁡ ( A signal / s t d noise ) ,
D n , θ u ( x , y ) = F 2D − 1 [ Ω n , θ ( k x , k y ) ⋅ u ^ ( k x , k y ) ] , n ∈ R ,
u ( x , y ) = F 2D − 1 [ u ^ ( k x , k y ) ] .
E n ( x , y ) = [ D n , 0 u ( x , y ) ] 2  +  [ D n , π / 4 u ( x , y ) ] 2  +  [ D n , π / 2 u ( x , y ) ] 2  +  [ D n , 3 π / 4 u ( x , y ) ] 2 .
| Ω n ( k ) | = | k | n .
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