Abstract
The solubility of the class of nonlinear optimization problems arising in image registration is discussed. The necessary optimility conditions (Euler-Lagrange equation) for such kind of problems is a nonlinear Neumann boundary value problem which is not known to have a solution in general. However, in the image registration context some assumptions can be made that let us move a little bit further in this question.
Highlights
ÊÌ ÁÆÌ ×ÓÐÙ Ð ØÝ Ó Ø Ð ×× Ó ÒÓÒÐ Ò Ö ÓÔØ Ñ Þ Ø ÓÒ ÔÖÓ Ð Ñ× Ö × Ò Ò Ñ Ö ×ØÖ Ø ÓÒ × × Ù×× o Ì Ò ×× ÖÝ ÓÔØ Ñ Ð ØÝ ÓÒ Ø ÓÒ× ́ ÙÐ Ö1Ä Ö Ò ÕÙ 1 Ø ÓÒμ ÓÖ ×Ù Ò Ó ÔÖÓ Ð Ñ× × ÒÓÒÐ Ò Ö Æ ÙÑ ÒÒ ÓÙÒ ÖÝ Ú ÐÙ ÔÖÓ Ð Ñ Û × ÒÓØ ÒÓÛÒ ØÓ Ú ×ÓÐÙØ ÓÒ Ò Ò Ö Ðo ÀÓÛ Ú Ö Ò Ø Ñ Ö ×ØÖ Ø ÓÒ ÓÒØ ÜØ ×ÓÑ ××ÙÑÔØ ÓÒ× ÒÑØØÐØÙ× ÑÓÚ Ð ØØÐ Ø ÙÖØ Ö Ò Ø × ÕÙ ×Ø ÓÒo.
Ô Ò ÒØ Ö ÙÐ Ö Þ Ò Ø ÖÑ a(·, ·) × Ð Ò Ö ÓÙÒ ÓÖÑ Ò W × W o Ì Ñ × ITØ ÑÔÐ Ø μ Ò IRÖ Ö Ò μ Ö ÒÓÒÒ Ø Ú ÙÒ Ø ÓÒ× Ò Ωo Ì ÓÑ Ò Ω Ò Rd Û Ö Ñ × Ö Ò × ××ÙÑ ØÓ ÓÙÒ Ò Ä Ô× ØÞo Ï Ð×Ó ××ÙÑ Ø Ø IT Ò ÜØ Ò Ý Þ ÖÓ ØÓ Rd ×Ó Ø Ø IT (x−u(x)) Ñ ×
ØÖ ØÑ ÒØoO 1⁄2o Ü ÑÔÐ Ó Ø ÑÔÐ ØÒÖÖÒ Ñ Ò R3 ÇÖØ Ó ÓÒ Ð ÑÙÐØ ÔÐ Ò Ö ÔÖÓ Ø ÓÒ× Ó Ì1× Ò Ó ÙÑ Ò ÑÙÖo ÄØÒÖØ × ÓÛ ×Ð × ÖÓÑ ØÛÓ ÚÓÐÙÑ × ØÒÛØØÒØØÑ ÒØ ÖÚ Ð Ó ÓÒ Ý Ö ÜØÖ Ø Ø ÔÔÖÓÜ Ñ Ø ÐÝ Ø × Ñ Ò ØÓÑ Ð ÔÓ× Ø ÓÒo.
Summary
Ì ×ÓÐÙ Ð ØÝ Ó Ø Ð ×× Ó ÒÓÒÐ Ò Ö ÓÔØ Ñ Þ Ø ÓÒ ÔÖÓ Ð Ñ× Ö × Ò Ò Ñ Ö ×ØÖ Ø ÓÒ × × Ù×× o Ì Ò ×× ÖÝ ÓÔØ Ñ Ð ØÝ ÓÒ Ø ÓÒ× ́ ÙÐ Ö1Ä Ö Ò ÕÙ 1 Ø ÓÒμ ÓÖ ×Ù Ò Ó ÔÖÓ Ð Ñ× × ÒÓÒÐ Ò Ö Æ ÙÑ ÒÒ ÓÙÒ ÖÝ Ú ÐÙ ÔÖÓ Ð Ñ Û × ÒÓØ ÒÓÛÒ ØÓ Ú ×ÓÐÙØ ÓÒ Ò Ò Ö Ðo ÀÓÛ Ú Ö Ò Ø Ñ Ö ×ØÖ Ø ÓÒ ÓÒØ ÜØ ×ÓÑ ××ÙÑÔØ ÓÒ× ÒÑØØÐØÙ× ÑÓÚ Ð ØØÐ Ø ÙÖØ Ö Ò Ø × ÕÙ ×Ø ÓÒo. Ô Ò ÒØ Ö ÙÐ Ö Þ Ò Ø ÖÑ a(·, ·) × Ð Ò Ö ÓÙÒ ÓÖÑ Ò W × W o Ì Ñ × ITØ ÑÔÐ Ø μ Ò IRÖ Ö Ò μ Ö ÒÓÒÒ Ø Ú ÙÒ Ø ÓÒ× Ò Ωo Ì ÓÑ Ò Ω Ò Rd Û Ö Ñ × Ö Ò × ××ÙÑ ØÓ ÓÙÒ Ò Ä Ô× ØÞo Ï Ð×Ó ××ÙÑ Ø Ø IT Ò ÜØ Ò Ý Þ ÖÓ ØÓ Rd ×Ó Ø Ø IT (x−u(x)) Ñ ×
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