# What kind of information can we obtain from curvature for exemplar-based inpainting?

I just want to know what kind of information does Curvature suggest to us? I've seen an article that used from the curvature of an Image. I can't understand what's the purpose of the author or it's better to say that what kind of information Curvature has!!!

Can anyone help me to handle this problem?

The article defines isophote curvature as: $$K(p) = \bigtriangledown \left[ \frac{ \bigtriangledown I_p}{ \left| \bigtriangledown I_p \right|} \right]$$

• Welcome to DSP.SE! It would be good to include more detail from the paper in your question, as the paper is behind a paywall for me. How do they define curvature? – Peter K. Mar 12 '15 at 17:58
• @PeterK, Thanks for your response: K(p) = gradient((gradient(I(p))/|(gradient(I(p))|), which I is a RGB Value ,p refers to a pixel and k is Curvature ... – mohammad mehdi derakhshani Mar 12 '15 at 18:30
• @PeterK. link of article – mohammad mehdi derakhshani Mar 12 '15 at 18:36

I haven't read the paper before, but the rationale is state in III.A:

The filling order is very crucial which directly affects the quality of image inpainting. However, during filling procedure we find the data terms may be zeros for many pixels, which make the priorities being close to zeros, although the confidence terms may be very big. This would result in incorrect filling order.[...] by introducing the curvature term, we effectively avoid the priority being close to zero while the data term is zero.

The foundation for the method used in this paper is in Object removal by exemplar-based inpainting, so be sure to read that first.

• thanks, I've implemented this article before, I mean the article which you posted for me, but the curvature is a new experience ... I could not find any useful information about that... – mohammad mehdi derakhshani Mar 12 '15 at 19:27
• The new priority function is given in (6), where the curvature is also defined. Are you familiar with the concept of curvature from computer vision or differential geometry? It defines the rate of change in edge direction, as the name suggests. You might want to consult your textbook. – Emre Mar 12 '15 at 19:42
• yeah, it's okay how we can calculate the curvature, but I can not understand what kind of interpretation it has in this formula! why the author's used from this concept in equation (6)!! – mohammad mehdi derakhshani Mar 12 '15 at 19:47
• To rephrase the rationale, it avoids assigning a low priority to regions of high confidence when the change in intensity is linear (the curvature is small). The result is a reduction of the "staircase effect", depicted in Figure 2. – Emre Mar 12 '15 at 19:55

Standard inpainting procedures first compute a distance transform and then sort the pixel values according to distances (to the closest edge of the binary mask). This way, the center of the region is inpainted at last. Curvature term basically contributes to this sorting as a second means of priority setting.

However, I don't like Eq. 7 in this paper. It is not clear to me how they weight the importance of curvature vs. color.