Polyharmonic Local Cosine Transforms for Improving JPEG-Compressed Images Naoki Saito Department of Mathematics University of California, Davis Dagstuhl Seminar #16462: Inpainting-Based Image Compression November 15, 2016 saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 1 / 51
Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 2 / 51
Acknowledgment Acknowledgment ONR Grants: N00014-00-1-0469; N00014-16-1-2255 NSF Grants: DMS-0410406; DMS-1418779 Jean François Remy (DriveScale, Inc.) Katsu Yamatani (Meijo Univ., Japan) saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 3 / 51
Motivations Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 4 / 51
Motivations Motivations Want to improve the quality of images (e.g., less blocking artifacts/visible discontinuities between blocks) reconstructed from the low bit rate JPEG files. 20 40 60 80 100 120 20 40 60 80 100 120 (a) Original: 8 bpp saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 5 / 51
Motivations Motivations Want to improve the quality of images (e.g., less blocking artifacts/visible discontinuities between blocks) reconstructed from the low bit rate JPEG files. 20 20 40 40 60 60 80 80 100 100 120 120 20 40 60 80 100 120 20 40 60 80 100 120 (a) Original: 8 bpp (b) JPEG: 0.162 bpp saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 5 / 51
Motivations Motivations . . . Want to develop a local image transform that generates faster decaying expansion coefficients than block DCT used in JPEG and our Polyharmonic Local Sine Transform (PHLST) because the faster decay of coefficients = ⇒ the more efficient compression Want to fully incorporate the infrastructure provided by the JPEG standard, e.g., the block DCT algorithm, the quantization method, the file format, etc. saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 6 / 51
Review of Fourier Cosine Series & PHLST Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 7 / 51
Review of Fourier Cosine Series & PHLST Review of Fourier Cosine Series Let Ω = (0,1) 2 ⊂ R 2 and f ∈ C 2 ( Ω ) but not periodic: the periodically extended version of f is discontinuous at ∂ Ω . Then the size of the complex Fourier coefficients c k of f decay as O ( � k � − 1 ) , where k = ( k 1 , k 2 ) ∈ Z 2 . Instead, expanding f into the Fourier cosine series gives rise to the decay rate O ( � k � − 2 ) because it is equivalent to the complex Fourier series expansion of the extended version of f via even reflection that is continuous at ∂ Ω . This is one of the main reasons why the JPEG Baseline method adopts Discrete Cosine Transform (DCT) instead of Discrete Fourier Transform (DFT) or Discrete Sine Transform (DST) saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 8 / 51
Review of Fourier Cosine Series & PHLST Review of Polyharmonic Local Sine Transform We now consider a decomposition f = u + v ∈ C 2 ( Ω ) . The u (or polyharmonic) component satisfies Laplace’s equation with the Dirichlet boundary condition : ∆ u = 0 in Ω ; u = f on ∂ Ω . The u component is solely represented by the boundary values of f via the fast and highly accurate Dirichlet problem solver of Averbuch, Israeli, & Vozovoi (1998). The residual v = f − u vanishes on ∂ Ω = ⇒ The Fourier sine coefficients of v decay as O ( � k � − 3 ) because ˜ v , the odd extension of v to ˜ Ω : = [ − 1,1] 2 , becomes a periodic C 1 ( ˜ Ω ) function. This is a multidimensional extension of the idea of Lanczos (1938). See [Saito-Remy 2006] for the details. saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 9 / 51
Review of Fourier Cosine Series & PHLST Review of Polyharmonic Local Sine Transform . . . Original Signal Supported on [0,1] Original Signal Supported on [0,1] Original Signal Supported on [0,1] 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 y y y −0.5 −0.5 −0.5 −1.0 −1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 x x x After Periodization After Even Reflection After Lin Removal+Odd Reflect 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 y y y −0.5 −0.5 −0.5 −1.0 −1.0 −1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 x x x DFT Coefficients DCT Coefficients LLST Coefficients 10^−15 10^−11 10^−710^−410^−1 10^−15 10^−11 10^−710^−410^−1 10^−15 10^−11 10^−710^−410^−1 |fy| |fy| |fy| −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 frequency frequency frequency saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 10 / 51
Polyharmonic Local Cosine Transform Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 11 / 51
Polyharmonic Local Cosine Transform Polyharmonic Local Cosine Transform Want to use DCT for fully utilizing the JPEG infrastructure. Want coefficients decaying faster than O ( � k � − 3 ) . To do so, we need to solve Poisson’s equation with the Neumann boundary condition : in Ω ; on ∂ Ω , ∆ u = K ∂ ν u = ∂ ν f 1 � where the constant source term K : = ∂ ν f ( x )d σ ( x ) is necessary | Ω | ∂ Ω for the solvability of the Neumann problem. Then, the Fourier cosine coefficients of the residual decay as O ( � k � − 4 ) v , the even extension of v to ˜ Ω becomes a periodic C 2 ( ˜ because ˜ Ω ) function thanks to ∂ ν v = 0 on ∂ Ω . saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 12 / 51
Polyharmonic Local Cosine Transform Why Poisson instead of Laplace? Green’s second identity claims that for any u , v ∈ C 1 ( Ω ) , � � ( u ∆ v − v ∆ u ) d x = ( u ∂ ν v − v ∂ ν u ) d σ ( x ), Ω ∂ Ω where d σ ( x ) is a surface (or boundary) measure. Setting v = 1 with the Neumann boundary condition, we have � � � ∆ u d x = ∂ ν u d σ ( x ) = ∂ ν f d σ ( x ). ∂ Ω ∂ Ω Ω This is a necessary condition that u must satisfy. Now, the source term of 1 � Poisson’s equation is K : = ∂ Ω ∂ ν f d σ ( x ) , where | Ω | is the volume of the | Ω | block Ω . saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 13 / 51
Computational Aspects of PHLCT Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 14 / 51
Computational Aspects of PHLCT PHLCT from DCT coefficients Outline Motivations 1 Review of Fourier Cosine Series & PHLST 2 Polyharmonic Local Cosine Transform 3 Computational Aspects of PHLCT 4 PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT Full Mode PHLCT 5 Partial Mode PHLCT 6 Numerical Experiments 7 Speculation: Use of the Helmholtz Equation 8 Conclusion 9 10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 15 / 51
Computational Aspects of PHLCT PHLCT from DCT coefficients PHLCT from DCT coefficients Want to achieve the PHLCT representation of f = u + v entirely in the DCT domain, F = U + V . Let f ( x , y ) ∈ C 2 ( Ω ) , and f i , j be a midpoint sample f ( x i , y j ) with x i = ( i + 0.5)/ N , y j = ( j + 0.5)/ N , i , j = 0,1,..., N − 1 . Let F ∈ R N × N be a DCT coefficient matrix of { f i , j } : � � � N − 1 � N − 1 2 2 � � F k 1 , k 2 : = λ k 2 λ k 1 f ( x i , y j )cos π k 1 x i cos π k 2 y j N N j = 0 i = 0 � where λ 0 = 1/ 2 , λ k = 1 for all k ≥ 1 . Now let’s compute the DCT coefficient matrix U of the polyharmonic component u using F . saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 16 / 51
Recommend
More recommend
