Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications
Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation <inline-formula><math xmlns=&qu...
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2025-05-01
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| author | Hung-Tsai Huang Zi-Cai Li Yimin Wei Ching Yee Suen |
| author_facet | Hung-Tsai Huang Zi-Cai Li Yimin Wei Ching Yee Suen |
| author_sort | Hung-Tsai Huang |
| collection | DOAJ |
| description | Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation <i>T</i>. We propose improved techniques that entirely bypass nonlinear solutions for <i>T</i>, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations <i>T</i>. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under <i>T</i>. We obtain the optimal convergence rates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mi>H</mi><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for the piecewise bilinear interpolations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>) and smooth images, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mspace width="3.33333pt"></mspace><mo>(</mo><mo>≪</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> denotes the mesh resolution of an optical scanner, and <i>N</i> is the division number of a pixel split into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>N</mi><mn>2</mn></msup></semantics></math></inline-formula> sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mi>β</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mi>β</mi></msup><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For piecewise continuous images with interior and exterior greyness jumps, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msqrt><mi>H</mi></msqrt><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msqrt><mi>H</mi></msqrt><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI). |
| format | Article |
| id | doaj-art-bce3ce76411b4e009e93677466c55efc |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-bce3ce76411b4e009e93677466c55efc2025-08-20T02:32:37ZengMDPI AGMathematics2227-73902025-05-011311177310.3390/math13111773Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and ApplicationsHung-Tsai Huang0Zi-Cai Li1Yimin Wei2Ching Yee Suen3Department of Data Science and Analytics, I-Shou University, Kaohsiung 84001, TaiwanDepartment of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, TaiwanShanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, ChinaCenter for Pattern Recognition and Machine Intelligence, Concordia University, Montreal, QC H3G 1M8, CanadaGeometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>T</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation <i>T</i>. We propose improved techniques that entirely bypass nonlinear solutions for <i>T</i>, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations <i>T</i>. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under <i>T</i>. We obtain the optimal convergence rates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><mi>H</mi><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for the piecewise bilinear interpolations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>) and smooth images, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mspace width="3.33333pt"></mspace><mo>(</mo><mo>≪</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> denotes the mesh resolution of an optical scanner, and <i>N</i> is the division number of a pixel split into <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>N</mi><mn>2</mn></msup></semantics></math></inline-formula> sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mi>β</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mi>H</mi><mi>β</mi></msup><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For piecewise continuous images with interior and exterior greyness jumps, we have <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mrow><mo>(</mo><msqrt><mi>H</mi></msqrt><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msqrt><mi>H</mi></msqrt><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI).https://www.mdpi.com/2227-7390/13/11/1773image geometric transformationserror analysissplitting-integrating methodharmonic and Poisson modelsfinite difference methodpattern recognition |
| spellingShingle | Hung-Tsai Huang Zi-Cai Li Yimin Wei Ching Yee Suen Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications Mathematics image geometric transformations error analysis splitting-integrating method harmonic and Poisson models finite difference method pattern recognition |
| title | Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications |
| title_full | Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications |
| title_fullStr | Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications |
| title_full_unstemmed | Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications |
| title_short | Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications |
| title_sort | improved splitting integrating methods for image geometric transformations error analysis and applications |
| topic | image geometric transformations error analysis splitting-integrating method harmonic and Poisson models finite difference method pattern recognition |
| url | https://www.mdpi.com/2227-7390/13/11/1773 |
| work_keys_str_mv | AT hungtsaihuang improvedsplittingintegratingmethodsforimagegeometrictransformationserroranalysisandapplications AT zicaili improvedsplittingintegratingmethodsforimagegeometrictransformationserroranalysisandapplications AT yiminwei improvedsplittingintegratingmethodsforimagegeometrictransformationserroranalysisandapplications AT chingyeesuen improvedsplittingintegratingmethodsforimagegeometrictransformationserroranalysisandapplications |