A Fast 3D Poisson Solver of Arbitrary Order Accuracy

Braverman, Elena; Israeli, M.; Averbuch, Amir; Vozovoi, L.

doi:10.1006/jcph.1998.6001

Cited by 44 publications

(36 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, our image approximation experiments using four standard images demonstrated the superiority of HWT over LLST, PWT, FWT, and WOI. We also note that the extension of HWT to a higher dimension is straightforward thanks to the efficient Laplace solver for higher dimension [3]. …”

Section: Resultsmentioning

confidence: 99%

Harmonic Wavelet Transform and Image Approximation

Zhang

Saito

2010

J Math Imaging Vis

View full text Add to dashboard Cite

In 2006, Saito and Remy proposed a new transform called the Laplace Local Sine Transform (LLST) in image processing as follows. Let f be a twice continuously differentiable function on a domain . First we approximate f by a harmonic function u such that the residual component v = f − u vanishes on the boundary of . Next, we do the odd extension for v, and then do the periodic extension, i.e. we obtain a periodic odd function v * . Finally, we expand v * into Fourier sine series. In this paper, we propose to expand v * into a periodic wavelet series with respect to biorthonormal periodic wavelet bases with the symmetric filter banks. We call this the Harmonic Wavelet Transform (HWT). HWT has an advantage over both the LLST and the conventional wavelet transforms. On the one hand, it removes the boundary mismatches as LLST does. On the other hand, the HWT coefficients reflect the local smoothness of f in the interior of . So the HWT algorithm approximates data more efficiently than LLST, periodic wavelet transform, folded wavelet transform, and wavelets on interval. We demonstrate the superiority of HWT over the other transforms using several standard images.

show abstract

Section: Resultsmentioning

confidence: 99%

Harmonic Wavelet Transform and Image Approximation

Zhang

Saito

2010

J Math Imaging Vis

View full text Add to dashboard Cite

show abstract

“…For more complicated situations such as end value jumps at four corners or solutions with higher order accuracy, we refer the readers to their original papers [1,4].…”

Section: {B (I)mentioning

confidence: 99%

“…It also enables us to interpret the frequency contents of each block without being influenced by the surrounding blocks and without the edge effect such as the Gibbs phenomenon. Moreover, as long as the boundary data are stored and the normal derivatives at the boundary are available, the polyharmonic components can be computed quickly by utilizing the FFT-based Laplace solver developed by Averbuch, Braverman, Israeli, and Vozovoi [1,4], which we shall call the ABIV method. Combining the fast solver ABIV with the quickly decaying Fourier sine coefficients of the residuals, the usefulness of PHLST to image approximation when the degree of polyharmonicity m = 1 was demonstrated in the papers [18,19].…”

Section: Introductionmentioning

confidence: 99%

PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform

2007

View full text Add to dashboard Cite

show abstract

“…The proof of this theorem and its generalization can be found in [17]. The polyharmonic components can be computed quickly by utilizing the computationally-fast and numerically-accurate Laplace/Poisson solver developed by Averbuch, Braverman, Israeli, and Vozovoi [23], [24] as long as the boundary data are available. Combining this feature with the quickly decaying expansion coefficients of the residuals, the usefulness of PHLST to image approximation was demonstrated [16], [17].…”

Section: Theoremmentioning

confidence: 99%

Improvement of DCT-Based Compression Algorithms Using Poisson's Equation

Yamatani

Saito

2006

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

We propose two new image compression-decompression methods that reproduce images with better visual fidelity, less blocking artifacts, and better PSNR, particularly in low bit rates, than those processed by the JPEG Baseline method at the same bit rates. The additional computational cost is small, i.e., linearly proportional to the number of pixels in an input image. The first method, the "full mode" polyharmonic local cosine transform (PHLCT), modifies the encoder and decoder parts of the JPEG Baseline method. The goal of the full mode PHLCT is to reduce the code size in the encoding part and reduce the blocking artifacts in the decoder part. The second one, the "partial mode" PHLCT (or PPHLCT for short), modifies only the decoder part, and consequently, accepts the JPEG files, yet decompresses them with higher quality with less blocking artifacts. The key idea behind these algorithms is a decomposition of each image block into a polyharmonic component and a residual. The polyharmonic component in this paper is an approximate solution to Poisson's equation with the Neumann boundary condition, which means that it is a smooth predictor of the original image block only using the image gradient information across the block boundary. Thus the residual-obtained by removing the polyharmonic component from the original image block-has approximately zero gradient across the block boundary, which gives rise to the fast-decaying DCT coefficients, which in turn lead to more efficient compression-decompression algorithms for the same bit rates. We show that the polyharmonic component of each block can be estimated solely by the first column and row of the DCT coefficient matrix of that block and those of its adjacent blocks and can predict an original image data better than some of the other AC prediction methods previously proposed. Our numerical experiments objectively and subjectively demonstrate the superiority of PHLCT over the JPEG Baseline method and the improvement of the JPEG-compressed images when decompressed by PPHLCT. K. Yamatani is with the

show abstract

A Fast 3D Poisson Solver of Arbitrary Order Accuracy

Cited by 44 publications

References 18 publications

Harmonic Wavelet Transform and Image Approximation

Harmonic Wavelet Transform and Image Approximation

PHLST5: A Practical and Improved Version of Polyharmonic Local Sine Transform

Improvement of DCT-Based Compression Algorithms Using Poisson's Equation

Contact Info

Product

Resources

About