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By Heike Hufnagel

Heike Hufnagel develops a mathematically sound statistical form version. as a result of the specific attributes of the version, the tough integration of particular and implicit representations should be played in a sublime mathematical formula, hence combining some great benefits of either particular version and implicit segmentation technique.

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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.

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.

S3 ? 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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