Probabilistic analysis on the splitting-shooting method for image transformations

Li, Zi‐Cai; Bai, Zhidong

doi:10.1016/s0377-0427(98)00062-4

Cited by 4 publications

(10 citation statements)

References 40 publications

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“…The computed sequential errors are O(N −(3/2) ), where N 2 is the division number of a pixel into subpixels. These computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in Li and Bai [12]. In computation, the average absolute errors of restored pixel grayness can be smaller than 2 out of 256 grayness levels.…”

supporting

confidence: 77%

“…More importantly, the FVM may be suited for large face images with a huge number of pixels under transformations, such as O (10 6 )−O(10 8 ). 4) The sequential errors during transformations are computed for the samples and are observed as O(N −1.5 ) to coincide with the error analysis in [12] for the CSIM for μ = 0, 1, where N is the division number of a 2-D pixel into subpixels along one direction. The absolute average errors of the restored face images are about 2 grayness levels, which are small compared with the 256 grayness levels used.…”

Section: E Comparisons With the Fdm And The Fem For Harmonic Modelsmentioning

confidence: 65%

“…In [9] and [12], error bounds are derived for image transformations under the splitting algorithms and for those under harmonic transformations. In this section, we only provide error bounds without proof.…”

Section: B Error Boundsmentioning

confidence: 99%

“…The transformation T is said to be regular if x(ξ, η) ∈ C 2 (Ω) and y(ξ, η) ∈ C 2 (Ω), and the Jacobian determinant satisfies 0 < J 0 ≤ J ≤ J M (see [12]). Assume that…”

Section: B Error Boundsmentioning

confidence: 99%

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Face Transformation With Harmonic Models by the Finite-Volume Method With Delaunay Triangulation

Chiang

Suen

2010

IEEE Trans. Syst., Man, Cybern. B

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Abstract-To carry out face transformation, this paper presents new numerical algorithms, which consist of two parts, namely, the harmonic models for changes of face characteristics and the splitting techniques for grayness transition. The main method in this paper is a combination of the finite-volume method (FVM) with Delaunay triangulation to solve the Laplace equations in the harmonic transformation of face images. The advantages of the FVM with Delaunay triangulation are given as follows: 1) easy to formulate the linear algebraic equations; 2) good in retaining the pertinent geometric and physical need; and 3) less central processing unit time needed. Numerical and graphical experiments have been conducted for the face transformation from a female (woman) to a male (man), and vice versa. The computed sequential errors are O(N −(3/2) ), where N 2 is the division number of a pixel into subpixels. These computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in the previous paper of Li and Bai. In computation, the average absolute errors of restored pixel grayness can be smaller than 2 out of 256 grayness levels. The FVM is as simple as the finite-difference method (FDM) and as flexible as the finite-element method (FEM). Hence, the FVM is particularly useful when dealing with large face images with a huge number of pixels in shape distortion. The numerical transformation of face images in this paper can be used not only in pattern recognition but also in resampling, image morphing, and computer animation.

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supporting

confidence: 77%

Section: E Comparisons With the Fdm And The Fem For Harmonic Modelsmentioning

confidence: 65%

Section: B Error Boundsmentioning

confidence: 99%

“…The transformation T is said to be regular if x(ξ, η) ∈ C 2 (Ω) and y(ξ, η) ∈ C 2 (Ω), and the Jacobian determinant satisfies 0 < J 0 ≤ J ≤ J M (see [12]). Assume that…”

Section: B Error Boundsmentioning

confidence: 99%

See 2 more Smart Citations

Face Transformation With Harmonic Models by the Finite-Volume Method With Delaunay Triangulation

Chiang

Suen

2010

IEEE Trans. Syst., Man, Cybern. B

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“…The remarkable advantage of the original combination CSIM of splitting-shooting-integration methods is the omission of nonlinear solutions, if the explicit functions of the forward transformations T are known. The sequential errors are proven to be of O(1/ N) by Li (1996) and O p (1/N 1.5 ) in probability of Li and Bai (1998). As the initial sequential errors of CSIM are large, the number N is also large for a system with multiple (e.g., 256) levels of intensity.…”

mentioning

confidence: 99%

New splitting algorithms for geometric transformations of digital images and their error analysis

Chiang

Suen

2011

Int J Imaging Syst Tech

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For digital images and patterns under the nonlinear geometric transformation, T: (n, h) ? (x, y), this study develops the splitting algorithms (i.e., the pixel-division algorithms) that divide a 2D pixel into N 3 N subpixels, where N is a positive integer chosen as N 5 2 k (k ! 0) in practical computations. When the true intensity values of pixels are known, this method makes it easy to compute the true intensity errors. As true intensity values are often unknown, the proposed approaches can compute the sequential intensity errors based on the differences between the two approximate intensity values at N and N/2. This article proposes the new splitting-shooting method, new splitting integrating method, and their combination. These methods approximate results show that the true errors of pixel intensity are O(H), where H is the pixel size. Note that the algorithms in this article do not produce any sequential errors as N ! N 0 , where N 0 (!2) is an integer independent of N and H. This is a distinctive feature compared to our previous papers on this subject. The other distinct feature of this article is that the true error bound O(H) is well suited to images with all kinds of discontinuous intensity, including scattered pixels.

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Advanced combinations of splitting–shooting–integrating methods for digital image transformations

1999

Journal of Computational and Applied Mathematics

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Probabilistic analysis on the splitting-shooting method for image transformations

Cited by 4 publications

References 40 publications

Face Transformation With Harmonic Models by the Finite-Volume Method With Delaunay Triangulation

Face Transformation With Harmonic Models by the Finite-Volume Method With Delaunay Triangulation

New splitting algorithms for geometric transformations of digital images and their error analysis

Advanced combinations of splitting–shooting–integrating methods for digital image transformations

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