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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/11/1773 |
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| Summary: | 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). |
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| ISSN: | 2227-7390 |