map-based exploration of intrinsic shape differences and variability
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这篇主要用矩阵来测量两个shape之间的area distortion和comformal distortion
2) 图二中
3)Theorem 2中,
4)公式(3)的推导
根据Theorem 2,
得证
5) By applying a matrix conjugation by
乘以
乘以
乘以
6) Option 1
7) Option 2 Laplace-Beltrai basis is orthnormal
则
关于
8)functoriality
9)localization and relation to existing measures
注意
10) First, we vectorize the area … commensurable both within and across plots.
这句话得意思是先把shape difference matrices向量化, 找出最大的两个principal component directions, 然后单位化, 这样不仅在于一副图内可以比较, 在图于图之间也可以比较.
11) second, we obtain … where
这里函数
其中有句话 Note that we are not discussing the average amount of distortion, but rather the deviation of the distortion from the average.
是说下式求出来的是不是平均形变量, 而是某一具体形变到平均形变的差距, 因为后面说了Since PCA centers data around the average, 所以中心点是平均形变量(从base shape变过来的), 而式子结果就是某一具体形变到这个平均形变的形变量.
12) 图3, 说明shape collection中 不同形变在PCA坐标系下的表示, 最后一列说明, 在base shape上面, 形变值的分布
13) 图4, 说明area distortion and conformal distortion之间还是差距的
14)图5, 是讲fuzzy map, 是说不确定的函数映射关系, 原本函数映射
15) 图6, 是说根据ROI然后在collection找出ROI相似的shape
16) 图7, 是说根据从Rest到ROI的形变矩阵, 然后多乘几次, 就可以这种形变的加强版magnification
17) 图8, 是说从Initial 到 Final的形变向量之间均匀取几个值, 然后从collection里面跳出跟这几个值最接近的模型
18) 图9, 从A到B的差异找从C到D的差异
19) 图10, 从A到B的差异, 找出
20) 图11是two shape collections之间的匹配, without a cross map.
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