Dagstuhl Seminar Inpainting-Based Image Compression Optimising Data for PDE-Based Inpainting and Compression Laurent Hoeltgen hoeltgen@b-tu.de Chair for Mathematics of Engineering & Numerical Optimisation Brandenburg University of Technology, Cottbus - Senftenberg November 17th, 2016 This work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license.
Motivation (1) PDE-Based Image Compression � alternative to transform based approaches � has three important pillars � each requires advanced optimisation Image Compression Data Data Data Storage Selection Reconstruction
Motivation (2) Challenges in Mathematical Modelling We have to take a few tough decisions: � How to reconstruct the image? � PDE has significant influence on our design principles � What are our optimisation criteria? � Which data should we optimise? � Should we maximise the quality or minimise the file size?
Outlook � Data Reconstruction Data Selection � Data Selection in the Domain � Data Selection in the Co-Domain � � Summary and Conclusion
Data Reconstruction (1) PDE-Based Image Inpainting We could use any diffusion type PDE. However, we want it to be: � simple to analyse � linear, parameter free, ... � applicable to any domain and co-domain � no restriction on image type, size, quantisation, ... � fast to carry out Laplace interpolation fulfils all these requirements.
Data Reconstruction (2) Laplace Interpolation ( Noma, Misuglia, 1959) � consider Laplace equation with mixed boundary conditions: ∂ Ω Ω 8 − ∆ u = 0, on Ω Ω K > < u = g , on Ω K Ω K > ∂ n u = 0, on ∂ Ω : � Ω K : represents known data � Ω \ Ω K : region to be inpainted (i.e. missing data) � image reconstructions given as solutions u
Data Selection (1) Highest Accuracy or Smallest Memory Footprint? Optimisation strategies can be split into 2 groups: 1. maximise accuracy at the expense of file size � data locations have no apparent structure � data values are real valued � large amounts of data are preferred 2. minimise memory footprint at the expense of accuracy � data positions stored in memory efficient structures � data values are quantised � sparse sets of data are preferred
Data Selection (2) Finding Optimal Interpolation Data � We can’t optimise accuracy and file size at the same time. � Using all image data yields perfect accuracy. � Using no image data at all yields perfect file size. Our Strategy: � We target a fixed amount of image data and optimise accuracy. � In case of identical accuracy, we chose the one with smaller size.
Data Selection (3) � brute force search of optimal pixel locations is impossible � 10 5600 ways to choose 5% of data from 0.07 megapixel image � most smart-phones yield 12 megapixel photographs � for comparison: 10 23 × =
Data Selection in the Domain (1) Laplace Interpolation (Reminder) ∂ Ω Ω 8 − ∆ u = 0, on Ω \ Ω K Ω K > < u = g , on Ω K Ω K > ∂ n u = 0, on ∂ Ω : can be rewritten as (Mainberger et al. 2011): ( c ( u − g ) + (1 − c ) ( − ∆ ) u = 0, on Ω on ∂ Ω ∂ n u = 0, with c ≡ 1 on Ω K and c ≡ 0 on Ω \ Ω K
Data Selection in the Domain (2) Regularised Laplace Interpolation ( c ( u − g ) + (1 − c ) ( − ∆ ) u = 0, on Ω ∂ n u = 0, on ∂ Ω � PDE makes also sense if c : Ω → R � can be seen as regularisation � Regions with c > 1 violate Max-Min principle � PDE resembles backward diffusion process � may also be interpreted as a Helmholtz equation � contrast enhancement � unconditional existence of a solution difficult to show
Data Selection in the Domain (3) An Optimal Control Model for Sparse Masks (H., Setzer, Weickert 2013) � optimal non-binary and sparse masks c given by Z 1 ff 2 ( u − g ) 2 + λ | c | + ε 2 | c | 2 d x arg min u , c Ω ( c ( u − g ) + (1 − c )( − ∆ ) u = 0, on Ω subject to: ∂ n u = 0, on ∂ Ω � model has strong similarities to Belhachmi et al. 2009 2 ( u − g ) 2 favours accurate reconstructions 1 � � λ | c | favours sparse sets of data 2 | c | 2 required for technical reasons ε � � PDE enforces feasible solutions
Data Selection in the Domain (4) Interpretation Z 1 ff 2 ( u − g ) 2 + λ | c | + ε 2 | c | 2 d x arg min u , c Ω ( c ( u − g ) + (1 − c )( − ∆ ) u = 0, on Ω subject to: ∂ n u = 0, on ∂ Ω � energy reflects trade-off between accuracy and sparsity � objectives cannot be fulfilled simultaneously � λ steers sparsity of the interpolation data � small, positive λ : many data points, but good reconstruction � large, positive λ : few data points, but bad reconstruction � non-convex, non-smooth, and large scale optimisation
Data Selection in the Domain (5) A Solution Strategy � linearise constraint to handle non-convexity: T ( u , c ) := c ( u − g ) + (1 − c )( − ∆ ) u T ( u , c ) ≈ T ( u k , c k ) + D u T ( u k , c k )( u − u k ) + D c T ( u k , c k )( c − c k ) � add proximal term and solve iteratively Z 1 2 ( u − g ) 2 + λ | c | + ε 2 | c | 2 arg min u , c Ω ff + µ u − u k ” 2 + µ c − c k ” 2 “ “ d x 2 2 T ( u k , c k ) + D u T ( u k , c k )( u − u k ) + D c T ( u k , c k )( c − c k ) = 0 until fixed point is reached
Data Selection in the Domain (6) Theoretical Findings algorithm yields several interesting results: 1. energy decreasing as long as: 1 “ � u k +1 − g � 2 2 − � u k − g � 2 ” 6 2 2 + ε “ ” “ ” � c k +1 � 1 − � c k � 1 � c k +1 � 2 2 − � c k � 2 λ 2 2 gain in sparsity must outweigh loss in accuracy 2. fixed-points fulfil necessary optimality conditions: u − g − D u T ( u , c ) ⊤ p = 0 λ ∂ ( �·� 1 ) ( c ) + ε c + D c T ( u , c ) p ∋ 0 T ( u , c ) = 0
Data Selection in the Domain (7) Example (Input)
Data Selection in the Domain (8) Evolution of the Iterates mean squared error energy 0 100 200 300 400 500 600 700 iteration
Data Selection in the Domain (8) Evolution of the Iterates mean squared error energy 0 100 200 300 400 500 600 700 iteration
Data Selection in the Domain (8) Evolution of the Iterates mean squared error energy 0 100 200 300 400 500 600 700 iteration
Data Selection in the Domain (8) Evolution of the Iterates mean squared error energy 0 100 200 300 400 500 600 700 iteration
Data Selection in the Domain (9) Example (5% of Mask Points and Reconstruction)
Data Selection in the Co-Domain (1) Tonal Optimisation � optimal pixel values necessary to maximise quality � quantisation to n ≪ 256 colours essential for compression � best number of colours hard to determine � often 30% less memory needed � optimisation criteria should take file size into account � hard to predict
Data Selection in the Co-Domain (2) The Inpainting Operator � given mask c , solution u = u ( c ) of c ( u − g ) + (1 − c ) ( − ∆ ) u = 0, on Ω ∂ n u = 0, on ∂ Ω can be expressed in terms of a linear inpainting operator M c : u = M c ( cg ) � optimal data values can be found by solving: Z ff | M c ( cx ) − f | 2 d x arg min x Ω
Data Selection in the Co-Domain (3) Continuous Tonal Optimisation � least squares model suggested by Mainberger et al. 2011 � yields arbitrary values in R � equivalence to mask optimisation optimal non-binary masks c equivalent to optimal colours in R with binary masks (H., Weickert 2015) � maximises reconstruction quality � storage of colour values in R is very expensive
Data Selection in the Co-Domain (4) Remarks � The presented model yields high quality data. � The optimisation is time consuming. � Storing this data as-is is too expensive for compression tasks. We need algorithms to simplify the data: � memory efficient representation of arbitrary binary masks � good colour quantisation strategies
Data Selection in the Co-Domain (5) Quantisation of Optimal Data Values � Optimal colour values gather around few dominant colours. � replacing all colours by their closest dominant colour yields: � reconstruction error becomes slightly larger � data becomes much easier to compress � achievable by using clustering strategies Important questions: � Which clustering method is the best? � How to chose our feature values?
Data Selection in the Co-Domain (6) Distribution of Optimal Colour Values optimal colour values of the trui test image on mask positions 60 40 20 0 64 128 192
Data Selection in the Co-Domain (7) Evaluated Strategies we tested with various feature choices: � k-means++ � hierarchical clustering � Gaussian mixture models Findings: k-means++ on colour values at mask positions works quite well
Data Selection in the Co-Domain (8) Discrete/Continuous Tonal Optimisation (H., Breuß 2016) � apply k-means++ on colour values from mask locations hierarchical mean squared error k-means 55 probabilistic no optimisation 50 45 10 15 20 25 30 35 40 45 50 55 60 65 70 75 number of clusters � k-means++ 30 colours outperforms original data (170 colours) � optimal quantisation found with silhouette statistics
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