A Probabilistic Framework for Point-Based Shape Modeling in by Heike Hufnagel

By Heike Hufnagel

Heike Hufnagel develops a mathematically sound statistical form version. end result of the specific attributes of the version, the difficult integration of specific and implicit representations might be played in a sublime mathematical formula, hence combining the benefits of either particular version and implicit segmentation approach.

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Additional resources for A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis (Aktuelle Forschung Medizintechnik – Latest Research in Medical Engineering)

Example text

I=1 {si , CP (T • ÓÖ Ö ÓÑÔÙØ T k Ø Ò Cº Ö ×ØÖ Ø ÓÒ¸ Ø Ø ÑÒÑÞ × Ø ÖÖÓÖ Ø ÓÒ Ó T ØÓ S ÐÓÓ × Ð ÔÔÐ T Ñ Ò ×ÕÙ Ö si = Rsi + t ØÛ Ò ÐÐ ÔÓ ÒØ Ô Ö× Ø × ∀i Û Ø Ø ÖÓØ Ø ÓÒ Ñ ØÖ Ü R ∈ R3x3 Ò Ø ØÖ Ò×Ð Ø ÓÒ Ú ØÓÖ t ∈ R3 º Ì Ñ Ò Ñ Þ Ø ÓÒ Ó Ø ÖÖÓÖ ØÛ Ò ÐÐ ÔÓ ÒØ Ô Ö× Ò C × ÓÑÔÙØ Ù× Ò Ø Ä ×Ø ËÕÙ Ö × Ö Ø Ö ÓÒ T = argmin T = argmin R,t 1 Ns 1 Ns Ns mij − T i=1 Ns si 2 mij − Rsi − t 2 . i=1 Ì Á È Ð ÓÖ Ø Ñ ÓÒÚ Ö × ÐÛ Ý× ÑÓÒÓØÓÒ ÐÐÝ ØÓ Ø Ò Ö ×Ø ÐÓ Ð Ñ Ò ÑÙÑ Û Ö Ò Ö ×Ø × Ñ ÒØ Ò Ø × Ò× Ó Ñ Ò¹×ÕÙ Ö ×Ø Ò Ñ ØÖ º × Ñ Ò × Ú ÒØ Ø ÑÙ×Ø ÒÓØ Ø Ø Ø Á È × ×Ù× ÔØ Ð ØÓ ÖÓ×× ×Ø Ø ×Ø Ð ÓÙØÐ Ö׺ Ë Ú Ö Ð ÔÔÖÓ × Ð Û Ø Ø × ÔÖÓ Ð Ñ Ý º º ÔÖÓÔÓ× Ò ÖÓ Ù×Ø ×Ø Ñ ØÓÖ× Ò ½ ¸ Å ×Ù ½ º ÅÓÖ ÓÚ Ö¸ × ÒÝ Ñ Ø Ó Ñ Ò Ñ Þ Ò ÒÓÒ¹ ÓÒÚ Ü Ó×Ø ÙÒ Ø ÓÒ¸ Ø Á È Ð × ÖÓ Ù×ØÒ ×× Û Ø Ö ×Ô Ø ØÓ Ø Ò Ø Ð ØÖ Ò× ÓÖÑ Ø ÓÒ Ù× Ó ¾º¾ Ì ÓÖÖ ×ÔÓÒ Ò ÈÖÓ Ð Ñ ½¿ s2 s1 ?

25 ? 20 mj ? 35 mj ?

S4 mj ? , π]º ÌÛÓ Ú ÖØ × Ú ØÓ × Ð Ø × Ø ÔÓÐ × ÓÖ Ø × ÔÖÓ ×׺ Ì Ð Ø ØÙ × ÓÙÐ ÖÓÛ ×ÑÓÓØ ÐÝ ÖÓÑ 0 Ø Ø ÒÓÖØ ÔÓÐ ØÓ π Ø Ø ×ÓÙØ ÔÓÐ º Ì ÐÓÒ ØÙ ÓÒ Ø ÓØ Ö Ò × Ý Ð Ô Ö Ñ Ø Öº Ä Ø Ü¸ Ý Ò Þ ÒÓØ ÖØ × Ò Ó Ø ×Ô ÓÓÖ Ò Ø ×º Ì ÙÒ Ø ÓÒ Û ×Ô × Ø Ñ ÔÔ Ò Ó Ø ÓÓÖ Ò Ø × ÖÓÑ Ø ÙÒ Ø ×Ô Ö ÓÒ Ø ×ÙÖ × ½ ÔØ Ö ¾º ×Ô ÙÖÖ ÒØ Å Ø Ó × Ò ËØ Ø ×Ø ÐË Ô Ò ÐÝ× × ÛØ ⎛ ⎞ x(θ, φ) v(θ, φ) = ⎝ y(θ, φ) ⎠ . z(θ, φ) Û Ö v(φ, θ) Ì × ÖÙÒ× ÓÚ Ö Ø ÓÓÖ Ò Ø Û ÓÐ ×ÙÖ ¹×ÔÐ Ò × ÓÖ Û Ú Ð Ø׺ Ì Ø ÝÓ ÖØ Ú ÒØ ÓÖÖ ×ÔÓÒ Ò Ó ÙÒ Ø ÓÒ× ÓÙÐ ËÈÀ ÊÅ Ó Ö Ö Ý Ú Ö ÓÙ× Ð ÓÖ Ø Ñ Ñ Ð× Ø ÖÑ Ò Ø ÓÒ Øº Ô Ö Ñ Ø ÖÞ Ö × Ù× × × ÙÒ Ø ÓÒ× × º º Ó ×Ô Ö Ô Ö ÔÖ × ÒØ Ø ÓÒ Û Ð Ö ½ º ÌÝÔ ÐÐݸ Ø Ð ÖÑÓÒ × Ò ÐÐÝ ÐØ Ø ×Ø ÓÐÐÓÛ Ò ØÖÙÒ × Ø × Ö × ÜÔ Ò× ÓÒ × Ù× R r v(θ, φ) = m cm r Yr (θ, φ) r=0 −r Û Yrm Ö ÒÓØ × Ø ÓÑÔÐ Ø Ó ÙÒ Ø ÓÒ Ó Ò Ø ÓÒ cm r ÒØ× ÓÑÔÙØ Ö Ý Ø Ò ¿ Ú ÓÙÒ ØÓÖ× Û Ø Ú ÒØÙ ÐÐݸ × 0 0 Ô ×ÙÖ Ù ØÓ Ø Ö Ö Ö ÑÓ Ð Ð × Ô Ö 1 × Ò × Ô ÔÓ ÒØ ×ÙÖ Ò ÓÖÖ ×ÔÓÒ ×Ô º º Ö ÖÓÒ×Ø Ò ¾¼¼¼ º (x, y, z)º Ò v Ø Ì × ÓÖÑ ÐÐݸ Ø Ô × Ö ÔØÓÖ Ó ÒØ× × Ö Ñ ÜÑ Ð ×ÙÖ Ý × Ø R Ò Ø ØÓ Ò ÐÐ Ô×Ó ×Ô × Ö ´¾º½µ × ØÓ Ö Ý Ô Ö Ñ Ø Ö Þ Ø ÓÒ ØÓ ØÚ Û Ö × × ÙÒ Ø ÓÒ v(θ, φ)Yrm (θ, φ)dφ sin θdθ.

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