[PPT] - Polyharmonic Local Cosine Transforms for Improving JPEG-Compressed PowerPoint Presentation

SLIDE 1

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

SLIDE 2

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 2 / 51

SLIDE 3

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

SLIDE 4

Motivations

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 4 / 51

SLIDE 5

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

SLIDE 6

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

20 40 60 80 100 120 20 40 60 80 100 120

(b) JPEG: 0.162 bpp

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 5 / 51

SLIDE 7

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

SLIDE 8

Review of Fourier Cosine Series & PHLST

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 7 / 51

SLIDE 9

Review of Fourier Cosine Series & PHLST

Review of Fourier Cosine Series

Let Ω = (0,1)2 ⊂ R2 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 ck of f decay as

O(k−1), where k = (k1,k2) ∈ Z2.

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

SLIDE 10

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

n ∂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

SLIDE 11

Review of Fourier Cosine Series & PHLST

Review of Polyharmonic Local Sine Transform . . .

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

Original Signal Supported on [0,1]

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

After Periodization

frequency |fy| −1.0 −0.5 0.0 0.5 1.0 10^−15 10^−11 10^−710^−410^−1

DFT Coefficients

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

Original Signal Supported on [0,1]

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

After Even Reflection

frequency |fy| −1.0 −0.5 0.0 0.5 1.0 10^−15 10^−11 10^−710^−410^−1

DCT Coefficients

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

Original Signal Supported on [0,1]

x y −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

After Lin Removal+Odd Reflect

frequency |fy| −1.0 −0.5 0.0 0.5 1.0 10^−15 10^−11 10^−710^−410^−1

LLST Coefficients saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 10 / 51

SLIDE 12

Polyharmonic Local Cosine Transform

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 11 / 51

SLIDE 13

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:

∆u = K

in Ω;

∂νu = ∂ν f

n ∂Ω,

where the constant source term K :=

1 |Ω|

∂Ω

∂ν f (x)dσ(x) is necessary

for the solvability of the Neumann problem. Then, the Fourier cosine coefficients of the residual decay as O(k−4) because ˜

v, the even extension of v to ˜ Ω becomes a periodic C 2( ˜ Ω)

function thanks to ∂νv = 0 on ∂Ω.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 12 / 51

SLIDE 14

Polyharmonic Local Cosine Transform

Why Poisson instead of Laplace?

Green’s second identity claims that for any u,v ∈ C 1(Ω),

Ω

(u∆v − v∆u) dx =

∂Ω

(u ∂νv − v ∂νu) dσ(x),

where dσ(x) is a surface (or boundary) measure. Setting v = 1 with the Neumann boundary condition, we have

Ω

∆u dx =

∂Ω

∂νu dσ(x) =

∂Ω

∂ν f dσ(x).

This is a necessary condition that u must satisfy. Now, the source term of Poisson’s equation is K :=

1 |Ω|

∂Ω ∂ν f dσ(x), where |Ω| is the volume of the

block Ω.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 13 / 51

SLIDE 15

Computational Aspects of PHLCT

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 14 / 51

SLIDE 16

Computational Aspects of PHLCT PHLCT from DCT coefficients

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 15 / 51

SLIDE 17

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 fi,j be a midpoint sample f (xi, y j) with

xi = (i +0.5)/N, y j = (j +0.5)/N, i, j = 0,1,...,N −1.

Let F ∈ RN×N be a DCT coefficient matrix of {fi,j}:

Fk1,k2 := λk2

2

N

N−1

j=0
λk1
2

N

N−1

i=0

f (xi, y j)cosπk1xi

cosπk2y j

where λ0 = 1/

2, λk = 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

SLIDE 18

Computational Aspects of PHLCT PHLCT from DCT coefficients

PHLCT from DCT coefficients . . .

Assume for the moment that the discretized Neumann boundary data at each edge of Ω = [0,1]2 are available:

g (1)

i

:= −fy(xi,0), g (2)

i

:= fy(xi,1), g (3)

j

:= −fx(0, y j), g (4)

j

:= fx(1, y j). Ω Ω(0,1) Ω(−1,0) Ω(0,−1) Ω(1,0) ✲ ✞ Γ(1) ✛✆ Γ(2) ✻ ✝ Γ(4) ❄ ☎ Γ(3) ❄ x ✲ y

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 17 / 51

SLIDE 19

Computational Aspects of PHLCT PHLCT from DCT coefficients

PHLCT from DCT coefficients . . .

Let {G(ℓ)

k } be the 1D-DCT coefficients of {g (ℓ) i

}.

Then, we have a solution to Poisson’s equation as (see [Yamatani-Saito 2006] for details):

u(x, y) =

2

N

N−1

k=0

λk

G(1)

k ψk(y −1)+G(2) k ψk(y)

cosπkx

+

G(3)

k ψk(x −1)+G(4) k ψk(x)

cosπky
+c ,

where c is a constant to be determined and

ψk(t) :=

t2/2

if k = 0;

(coshπkt)/(πk sinhπk)

therwise.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 18 / 51

SLIDE 20

Computational Aspects of PHLCT PHLCT from DCT coefficients

PHLCT from DCT coefficients . . .

Applying 2D DCT to u above, we obtain U = (Uk1,k2) as

Uk1,k2 = G(1)

k1 ηk1,k2 +G(2) k1 η∗ k1,k2 +G(3) k2 ηk2,k1 +G(4) k2 η∗ k2,k1,

where ηk1,k2, η∗

k1,k2 are independent from the input image:

ηk,m := λm

2

N

N−1

i=0

ψk(xi −1)cosπmxi, η∗

k,m

:= λm

2

N

N−1

i=0

ψk(xi)cosπmxi,

Can set the DC component U0,0 ≡ 0 because the solution to the Poisson-Neumann problem is unique modulo an additive constant. In fact this is achieved by c = − 4N 2−1

24N 2.5

G(1)

0 +G(2) 0 +G(3) 0 +G(4)

. This will

become important in our algorithms.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 19 / 51

SLIDE 21

Computational Aspects of PHLCT Approximation of the Neumann Boundary Data

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 20 / 51

SLIDE 22

Computational Aspects of PHLCT Approximation of the Neumann Boundary Data

Approximation of the Neumann Boundary Data

In practice, we need to estimate the Neumann boundary data {g (ℓ)

i

}

from the image samples of the current and adjacent blocks. Let

f (s,t)

i,j

= f (xi + s, y j + t) and Ω(s,t) be: Ω Ω(0,1) Ω(−1,0) Ω(0,−1) Ω(1,0) ✲ ✞ Γ(1) ✛✆ Γ(2) ✻ ✝ Γ(4) ❄ ☎ Γ(3) ❄ x ✲ y

Let I5 := {(0,−1),(−1,0),(0,0),(1,0),(0,1)} be the indices of the current and adjacent blocks. Note: Ω(0,0) = Ω

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 21 / 51

SLIDE 23

Computational Aspects of PHLCT Approximation of the Neumann Boundary Data

Approximation of the Neumann Boundary Data . . .

Approximate {g (ℓ)

i

} using column & row averages: g (1)

i

≃ X (−1)

i

− X (0)

i

; g (2)

i

≃ X (1)

i

− X (0)

i

; g (3)

j

≃ Y (−1)

j

−Y (0)

j

; g (4)

j

≃ Y (1)

j

−Y (0)

j

X (t)

i

:= 1 N

N−1

j=0

f (0,t)

i,j

, Y (s)

j

:= 1 N

N−1

i=0

f (s,0)

i,j

,

Then, {G(ℓ)

k } can be expressed using the first row & column of F (s,t).

Consequently, for (k1,k2) = (0,0), we have

Uk1,k2 = 1

N
F (0,−1)

k1,0

−Fk1,0

ηk1,k2 +
F (0,1)

k1,0 −Fk1,0

η∗

k1,k2

+

F (−1,0)

0,k2

−F0,k2

ηk2,k1 +
F (1,0)

0,k2 −F0,k2

η∗

k2,k1

.

(1)

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 22 / 51

SLIDE 24

Computational Aspects of PHLCT Approximation of the Neumann Boundary Data

Approximation of the Neumann Boundary Data . . .

2 4 6 8 10 12 14 16 170 175 180 185 190 195 200

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 23 / 51

SLIDE 25

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT PHLCT from DCT coefficients Approximation of the Neumann Boundary Data Modifying PHLCT for Practice / Inverse PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 24 / 51

SLIDE 26

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Modifying PHLCT for Practice

Approximating Eq.(1) only using the DC components F0,0 and F (s,t)

0,0

allows us to simplify our algorithms:

Uk1,k2 =             

if k1 = k2 = 0;

1

N
F (−1,0)

0,0

−F0,0

η0,k1 +
F (1,0)

0,0

−F0,0

η∗

0,k1

if k1 = 0 = k2;

1

N
F (0,−1)

0,0

−F0,0

η0,k2 +
F (0,1)

0,0

−F0,0

η∗

0,k2

if k1 = 0 = k2;

Uk1,k2 as Eq.(1)

therwise.

(2) Now set Vk1,k2 = Fk1,k2 −Uk1,k2, ∀k1,k2. Note V0,0 = F0,0!! Note also that if we know V of the current and adjacent blocks, we can reconstruct F. No need to store U! See next page. Strictly speaking, this new version of u does not satisfy the original Poisson equation (due to the uniqueness theorem modulo an additive constant) but still satisfies the Neumann condition.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 25 / 51

SLIDE 27

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT

1 Assuming V (s,t), (s,t) ∈ I5, are available, recover the first column and

row of U using the DC components, F (s,t)

0,0

= V (s,t)

0,0

, (s,t) ∈ I5 via

Eq.(2);

2 Recover the first column and row of F (s,t), (s,t) ∈ I5 by summing

those of U and V (see Eq.(2));

3 Recover other entries of U via Eq.(1) and the results of Step 2; 4 Set F =U +V ; 5 Apply Inverse 2D DCT to F to recover f . saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 26 / 51

SLIDE 28

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT: Step 0

(a) F (b) U (c) V

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 27 / 51

SLIDE 29

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT: Step 1

(a) F (b) U (c) V

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 28 / 51

SLIDE 30

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT: Step 2

(a) F (b) U (c) V

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 29 / 51

SLIDE 31

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT: Step 3

(a) F (b) U (c) V

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 30 / 51

SLIDE 32

Computational Aspects of PHLCT Modifying PHLCT for Practice / Inverse PHLCT

Inverse PHLCT: Step 4

(a) F (b) U (c) V

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 31 / 51

SLIDE 33

Full Mode PHLCT

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 32 / 51

SLIDE 34

Full Mode PHLCT

Full Mode PHLCT (FPHLCT)

FPHLCT adds simple procedures in both the encoder and the decoder parts of the JPEG Baseline method. In the encoder part, the only difference from JPEG is to: 1) compute

U from F; and 2) compute the residual V = F −U and store the

quantized version V Q instead of FQ. In the decoder part, the only difference from JPEG is to: 1) compute

UQ, the estimate of U from V Q; and 2) compute UQ +V Q as an

improved estimate of F over FQ. Because V decays faster than F, the decompressed image quality gets better than JPEG if it is compressed at the same bit rate.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 33 / 51

SLIDE 35

Partial Mode PHLCT

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 34 / 51

SLIDE 36

Partial Mode PHLCT

Partial Mode PHLCT (PPHLCT)

Only the decoder part of the JPEG Baseline method is modified: PPHLCT accepts the JPEG-compressed files. The JPEG encoder kills small DCT coefficients Fk′, i.e., FQ

k′ = 0.

PPHLCT replaces those FQ

k′ by UQ k′ if UQ k′ are also small.

This is possible because UQ can be computed solely from the first column & row of FQ and those of the adjacent blocks F (s,t)Q; see Eqs.(1), (2). Our reasoning to do this is Fk ≈Uk for large k because Vk decays quickly. We also add some quadratic polynomial to reduce the blocking artifacts further. This can be done also in the DCT domain. (See [Yamatani-Saito 2006] for the details.)

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 35 / 51

SLIDE 37

Numerical Experiments

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 36 / 51

SLIDE 38

Numerical Experiments

(a) Barbara (b) Gabor Figure: Two test images

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 37 / 51

SLIDE 39

Numerical Experiments

Numerical Experiments . . .

(a) JPEG, 23.61dB (b) FPHLCT, 24.19dB (c) PPHLCT, 23.97dB Figure: Compressed at 0.15 bits/pixel. Numerical values indicate the Peak Signal-to-Noise Ratio (PSNR).

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 38 / 51

SLIDE 40

Numerical Experiments

Numerical Experiments . . .

(a) JPEG, 25.67dB (b) FPHLCT, 26.05dB (c) PPHLCT, 25.73dB Figure: Compressed at 0.30 bits/pixel. Numerical values indicate the Peak Signal-to-Noise Ratio (PSNR).

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 39 / 51

SLIDE 41

Numerical Experiments

Numerical Experiments . . .

0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

0.

5

0.

4

0.

3

0.

2

0.

1 0. 1 0. 2 0. 3 0. 4 0. 5 Bi t Rat e ( bi t s / pi xel ) PSNR gai n ( dB ) PH LCT( Q M ) Q SFI T( Q M ) LAKH ANI ( Q M ) PPH LCT( Q M ) PQ SFI T( Q M ) PLAKH ANI ( Q M )

Figure: Comparison of PSNR gain by various methods for the Barbara image over the JPEG Baseline method

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 40 / 51

SLIDE 42

Numerical Experiments

Numerical Experiments . . .

(a) JPEG, 31.41dB (b) FPHLCT, 39.21dB (c) PPHLCT, 35.69dB Figure: Compressed at 0.15 bits/pixel. Numerical values indicate the Peak Signal-to-Noise Ratio (PSNR).

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 41 / 51

SLIDE 43

Numerical Experiments

Numerical Experiments . . .

(a) JPEG, 38.12dB (b) FPHLCT, 47.02dB (c) PPHLCT, 40.89dB Figure: Compressed at 0.30 bits/pixel. Numerical values indicate the Peak Signal-to-Noise Ratio (PSNR).

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 42 / 51

SLIDE 44

Numerical Experiments

Numerical Experiments . . .

0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1

2

2 4 6 8 10 Bi t Rat e ( bi t s / pi xel ) PSNR gai n ( dB ) PH LCT( Q M ) Q SFI T( Q M ) LAKH ANI ( Q M ) PPH LCT( Q M ) PQ SFI T( Q M ) PLAKH ANI ( Q M )

Figure: Comparison of PSNR gain by various methods for the Gabor image over the JPEG Baseline method

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 43 / 51

SLIDE 45

Speculation: Use of the Helmholtz Equation

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 44 / 51

SLIDE 46

Speculation: Use of the Helmholtz Equation

Problems with Oscillatory Textures

Solutions of the Laplace/Poisson equations quickly attenuate

scillatory patterns at the boundary when evaluated at the inside of

the domain. This leads to inefficient u + v decomposition for oscillatory patterns.

50 100 150 200 50 100 150 200

(a) Enemy f

50 100 150 200 50 100 150 200

(b) Harmonic u

50 100 150 200 50 100 150 200

(c) Residual v Figure: The solution of Laplace’s equation may lead to an inefficient u + v decomposition.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 45 / 51

SLIDE 47

Speculation: Use of the Helmholtz Equation

Using the Helmholtz Equation for Oscillatory Textures

The Helmholtz equation may rescue us:

∆u +k2u = 0

in Ω;

∂νu = ∂ν f

n ∂Ω.

50 100 150 200 50 100 150 200

(a) Enemy f

50 100 150 200 50 100 150 200

(b) Helmholtz u

50 100 150 200 50 100 150 200

(c) Residual v Figure: The Helmholtz equation may lead to an efficient u + v decomposition.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 46 / 51

SLIDE 48

Speculation: Use of the Helmholtz Equation

Using the Helmholtz Equation for Oscillatory Textures . . .

Important to use k (the wavenumber parameter) that should be estimated from the oscillatory patterns on ∂Ω.

50 100 150 200 50 100 150 200

(a) Enemy f

50 100 150 200 50 100 150 200

(b) Helmholtz u

50 100 150 200 50 100 150 200

(c) Residual v Figure: A wrong k may harm you.

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 47 / 51

SLIDE 49

Conclusion

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 48 / 51

SLIDE 50

Conclusion

Both the FPHLCT and PPHLCT algorithms were patented as “Data Compression/Decompression Method, Program, and Device,” Japan Patent # 4352110, Granted 8/7/09; US Patent # 8,059,903, Granted 11/15/11. More extensive numerical experiments (see [Yamatani-Saito 2006]) indicate that FPHLCT reduces the bit rates about 15% over JPEG whereas PPHLCT does about 7% to achieve the same PSNR in the relatively low bit rate range. If one can afford to use the higher bit rates, then our methods naturally approach to the performance of JPEG. PPHLCT is particularly useful because it accepts the files already compressed by the JPEG standard. On the other hand, FPHLCT is better than PPHLCT if one can afford to modify the encoder part of the JPEG standard. Additional computational cost of both methods over JPEG is small: linearly proportional to the number of pixels of an input image. Should be useful for zooming, interpolation, feature extraction. Using the Helmholtz equation with the Neumann boundary condition for u should be investigated! Any collaborations?

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 49 / 51

SLIDE 51

References

Outline

1

Motivations

2

Review of Fourier Cosine Series & PHLST

3

Polyharmonic Local Cosine Transform

4

Computational Aspects of PHLCT

5

Full Mode PHLCT

6

Partial Mode PHLCT

7

Numerical Experiments

8

Speculation: Use of the Helmholtz Equation

9

Conclusion

10 References saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 50 / 51

SLIDE 52

References

K. Yamatani & N. Saito: “Improvement of DCT-based compression

algorithms using Poisson’s equation,” IEEE Trans. Image Proc., vol.15, no.12, pp.3672–3689, 2006.

N. Saito & J.-F. Remy: “The polyharmonic local sine transform: A

new tool for local image analysis and synthesis without edge effect,”

Appl. Comp. Harm. Anal., vol.20, no.1, pp.41–73, 2006.
A. Averbuch, M. Israeli, & L. Vozovoi: “A fast Poisson solver of

arbitrary order accuracy in rectangular regions,” SIAM J. Sci. Comp., vol.19, no.3, pp.933–952, 1998.

G. K. Wallace: “The JPEG still picture compression standard,” IEEE
Trans. Consumer Electronics, vol.38, no.1., pp.xviii–xxxiv, 1992.

The ApproxFun.jl Package in Julia: https://github.com/ApproxFun/ApproxFun.jl Thank you very much for your attention!

saito@math.ucdavis.edu (UC Davis) PHLCT Compression Dagstuhl Seminar #16462 51 / 51