Article Contents
Article Contents

# Inpainting via sparse recovery with directional constraints

• * Corresponding author: Xuemei Chen
The first author is supported by NSF DMS-2050028
• Image inpainting is a particular case of image completion problem. We describe a novel method allowing to amend the general scenario of using sparse or TV-based recovery for inpainting purposes by an efficient use of adaptive one-dimensional directional "sensing" into the unknown domain. We analyze the smoothness of the image near each pixel on the boundary of the unknown domain and formulate linear constraints designed to promote smooth transitions from the known domain in the directions where smooth behavior have been detected. We include a theoretical result relaxing the widely known sufficient condition of sparse recovery based on coherence, as well as observations on how adding the directional constraints can improve the well-posedness of sparse inpainting.

The numerical implementation of our method is based on ADMM. Examples of inpainting of natural images and binary images with edges crossing the unknown domain demonstrate significant improvement of recovery quality in the presence of adaptive directional constraints. We conclude that the introduced framework is general enough to offer a lot of flexibility and be successfully utilized in a multitude of image recovery scenarios.

Mathematics Subject Classification: Primary: 65F45, 65K10; Secondary: 94A16.

 Citation:

• Figure 1.  An image to be inpainted

Figure 2.  Forming boundary equations

Figure 3.  Forming boundary equations: pick the best direction

Figure 4.  Recovery via directional sensing only. (a) Image with a thin missing domain. (b) 'DB-2', all smooth directions per pixel included, both centered and shifted equations used. (c) 'DB-2', all smooth directions per pixel included, only centered equations used. (d) 'DB-2', one direction per pixel included, only centered equations used could be formed as the filter has length 2. (e) 'DB-4', all smooth directions included, both centered and shifted equations used

Figure 5.  Examples of directional constraints adaptively formulated for the inpainting examples discussed further in the text. In both cases the 'DB2' filter of length 4 was used to form centered equations. Green dots indicate the unknown pixels around which the constraints were formed. Red dots indicate other pixels present in those constraints. The constraints are shown only for some pixels to avoid overcrowding the images. The actual results of inpainting appear later in Figures 6 and 9

Figure 9.  Text removal of Pepper, $128\times128$

Figure 6.  Inpaint a missing block of a vertical stripe, $64\times64$

Figure 7.  Inpainting a thin missing block in an image with repetitive pattern of slanted stripes, $64\times64$

Figure 8.  Inpaint a missing annulus of a sectional image, $128\times128$

Figure 10.  Smooth stripes inpainting, $64\times64$

Figure 11.  Text removal of a colored image, $128\times128$

Table 1.  Here $\sqrt{\beta} = .75$, $F$ is the reshaped 2D DB-4 wavelet basis matrix, the first filter $h$ is a high pass DB-2 filter, all smooth directions used and only centered equations included, the second filter added is a high pass DB-4 filter

 Cosine Coherence $\mu_1$ Min singular value Max singular value $P_{\Lambda} F$ 0.8541 3.8339 0 1 $SF$ (1 filter) 0.4799 0.5888 0.0455 1.5255 $S_2F$ (2 filters) 0.3913 0.4480 0.0937 1.9837

Table 2.  Here $\sqrt{\beta} = .75$, $F$ is the reshaped 2D binary Weyl basis matrix, the filter $h$ is a high pass DB-2 filter, all smooth directions used and only centered equations included

 Cosine Coherence $\mu_1$ Min singular value Max singular value $P_{\Lambda} F$ 0.0667 0.0667 0 1 $SF$ (1 filter) 0.0452 0.0457 0.0455 1.5255
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