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A novel image inpainting technique based on median diffusion

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Abstract

Image inpainting is the technique of filling-in the missing regions and removing unwanted objects from an image by diffusing the pixel information from the neighbourhood pixels. Image inpainting techniques are in use over a long time for various applications like removal of scratches, restoring damaged/missing portions or removal of objects from the images, etc. In this study, we present a simple, yet unexplored (digital) image inpainting technique using median filter, one of the most popular nonlinear (order statistics) filters. The median is maximum likelihood estimate of location for the Laplacian distribution. Hence, the proposed algorithm diffuses median value of pixels from the exterior area into the inner area to be inpainted. The median filter preserves the edge which is an important property needed to inpaint edges. This technique is stable. Experimental results show remarkable improvements and works for homogeneous as well as heterogeneous background. PSNR (quantitative assessment) is used to compare inpainting results.

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Authors and Affiliations

  1. G Narayanamma Institute of Technology and Science, Hyderabad, 500008, India

    RAJKUMAR L BIRADAR

  2. Poojya Doddappa Appa College of Engineering, Gulbarga, 585104, India

    VINAYADATT V KOHIR

Authors
  1. RAJKUMAR L BIRADAR
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  2. VINAYADATT V KOHIR
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Correspondence to RAJKUMAR L BIRADAR.

Appendix A

Appendix A

Median is the maximum likelihood estimator of location for Laplacian distribution (Ioannis & Anastasios 1992).

$$ \label{eq10} f\!\left( {x;\mu ,\lambda } \right)=\frac{1}{2\lambda }e^{\frac{-\left| {x-\mu } \right|}{\lambda }}. $$
(10)

This distribution is symmetric about μ and decreases exponentially to right and left, with λ the dispersion parameter. It will be seen that the sample median \(\overline x \) is the maximum likelihood estimator of μ.

Proof: The dispersion parameter is given by

$$ \label{eq11} \lambda =\sum\limits_{i=1}^n {\frac{\left| {x_i -\overline x } \right|}{n}}. $$
(11)

The likelihood function of μ is given by

$$ \label{eq12} \phi =\frac{1}{\left( {2\lambda } \right)^n}e^{\frac{-\sum\limits_{i=1}^n {\left| {x_i -\mu } \right|} }{n}}. $$
(12)

Maximizing ϕ with respect to μ is equivalent to minimizing \(\sum\limits_{i=1}^n {\left| {x_i -\mu } \right|} \). This sum is minimum for a set of {x i } when μ is the median of the set. Furthermore, setting \({\partial \phi } \mathord{\left/ {\vphantom {{\partial \phi } {\partial \lambda =0}}} \right. \kern-\nulldelimiterspace} {\partial \lambda =0}\) readily yields the estimator for λ.

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BIRADAR, R.L., KOHIR, V.V. A novel image inpainting technique based on median diffusion. Sadhana 38, 621–644 (2013). https://doi.org/10.1007/s12046-013-0152-2

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