Outlier removal in biomaterial image segmentations using a non-stationary Bayesian learning

Abstract

Segmentation of dried amnion biomaterial tends to produce invalid (outlier) contour point detections due to texture and colour inhomogeneity of the biomaterial. In this paper, a novel implementation of a non-stationary Bayesian learning process for outlier contour point removal of amnion segmentations is presented. This outlier removal method is independent to algorithms used for the contour detection. The Bayesian process uses a non-stationary kernel to learn a function with complex shape that maps image features in a region-of-interest around each contour point to a discrete output. Based on this output, a contour point can be determined as valid or invalid (outlier). The hyper-parameters of the non-stationary kernel are learned by maximising the marginal likelihood of the combined likelihood of data and the prior of the kernel parameters. Moreover, a novel combination of gradient-ascend and harmonic heuristic search methods is presented to find the optimal hyper-parameters. To validate the method, experiments are conducted to detect and ignore invalid contour points on amnion biomaterial images. A comparison of the proposed method with a logistic regression classification as the baseline is performed. The results show that the proposed method can significantly improve the contour detection by removing outliers and, hence, can reduce waste of uncut biomaterials.

Appendices

Appendix A: Derivation of the posterior distribution estimation for learning

By re-writing Eq. 4, we have:

$$P\left({\varvec{\uptheta}}|D;\,M\right)\propto P\left(D|{\varvec{\uptheta}};\,M\right) P\left({\varvec{\uptheta}};\,M\right)\leftrightarrow P\left({\varvec{\uptheta}}|\mathbf{X},\mathbf{y};\,M\right)\propto P\left(\mathbf{y}|\mathbf{X},{\varvec{\uptheta}};\,M\right) P\left({\varvec{\uptheta}};\,M\right)$$

(21)

Both the likelihood $P\left(\mathbf{y}|\mathbf{X},\theta ;\,M\right)$ and the prior $P\left(\theta ;\,M\right)$ are assumed to be Gaussian. The model $M$ is assumed to be a linear model $f({\mathbf{X}}^{T}{\varvec{\uptheta}})$. Hence:

$$P\left({\varvec{\uptheta}}|\mathbf{X},\mathbf{y};\,M\right)\propto \mathrm{exp}\left(-\frac{1}{2{\sigma }^{2}}{\left(\mathbf{y}-{\mathbf{X}}^{T}{\varvec{\uptheta}}\right)}^{T}\left(\mathbf{y}-{\mathbf{X}}^{T}{\varvec{\uptheta}}\right)\right)\mathrm{exp}\left(-\frac{1}{2}{{\varvec{\uptheta}}}^{T}{\sum }^{-1}{\varvec{\uptheta}}\right)$$

(22)

where $\sum$ is a positive semi-definite matrix with size that follows the size of ${\varvec{\uptheta}}$. Hence:

$$\propto \mathrm{exp}(-\frac{1}{2{\sigma }^{2}}{\left(\mathbf{y}-{\mathbf{X}}^{T}{\varvec{\uptheta}}\right)}^{T}\left(\mathbf{y}-{\mathbf{X}}^{T}{\varvec{\uptheta}}\right)-\frac{1}{2}{{\varvec{\uptheta}}}^{T}{\sum }^{-1}{\varvec{\uptheta}})$$

(23)

$$\propto \mathrm{exp}\left(-\frac{1}{2{\sigma }^{2}}{\mathbf{y}}^{T}\mathbf{y}+\frac{1}{2{\sigma }^{2}}{\mathbf{y}}^{T}\left({\mathbf{X}}^{T}{\varvec{\uptheta}}\right)+\frac{1}{2{\sigma }^{2}}{{\varvec{\uptheta}}}^{T}\mathbf{X}\mathbf{y}-\frac{1}{2{\sigma }^{2}}\mathbf{X}{{\varvec{\uptheta}}}^{T}\left({\mathbf{X}}^{T}{\varvec{\uptheta}}\right)-\frac{1}{2}{{\varvec{\uptheta}}}^{T}{\sum }^{-1}{\varvec{\uptheta}}\right)$$

(24)

By completing the square with respect to ${\varvec{\uptheta}}$, hence:

$$\propto \mathrm{exp}\left(-\frac{1}{2}{{\varvec{\uptheta}}}^{T}\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right){\varvec{\uptheta}}+\frac{1}{{\sigma }^{2}}{{\varvec{\uptheta}}}^{T}\mathbf{X}\mathbf{y}-\frac{1}{2{\sigma }^{2}}{\mathbf{y}}^{T}\mathbf{y}\right).$$

(25)

From Eq. (25), one can obtain:

$$P\left({\varvec{\uptheta}}|\mathbf{X},\mathbf{y};\,M\right)\sim N\left( mean \overline{{\varvec{\uptheta}} }={\sigma }^{2}{({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1})}^{-1}\mathbf{X}\mathbf{y},{cov=\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\right).$$

(26)

Appendix B: Derivation of the distribution estimation for prediction

By re-writing Eq. 6, we have:

$$P\left({\mathbf{y}}^{*}|{\mathbf{X}}^{\boldsymbol{*}},\mathbf{X},\mathbf{y};\,M\right)=\int P\left({\mathbf{y}}^{*}|{\mathbf{X}}^{\boldsymbol{*}},{\varvec{\uptheta}};\,M\right) P\left({\varvec{\uptheta}}|\mathbf{X},\mathbf{y};\,M\right) d{\varvec{\uptheta}}$$

(27)

The marginalisation is carried out as follows (by neglecting the constant term of the Gaussian probability):

$$P\left({\mathbf{y}}^{*}|{\mathbf{X}}^{\boldsymbol{*}},\mathbf{X},\mathbf{y};\,M\right)=\int \mathrm{exp}({\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left(cov {\mathbf{y}}^{*}\right)}^{-1}\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)) \mathrm{exp}\left(-\frac{1}{2}{{\varvec{\uptheta}}}^{T}\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right){\varvec{\uptheta}}+\frac{1}{{\sigma }^{2}}{{\varvec{\uptheta}}}^{T}\mathbf{X}\mathbf{y}-\frac{1}{2{\sigma }^{2}}{\mathbf{y}}^{T}\mathbf{y}\right) d{\varvec{\uptheta}}$$

(28)

where $cov {\mathbf{y}}^{*}={\mathbf{X}}^{*T}{{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}}^{*}$ due to the relation to the covariance of the posterior of ${\varvec{\uptheta}}$. By only considering the exponent terms, hence:

$$\leftrightarrow {\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left({\mathbf{X}}^{*T}{{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}}^{*}\right)}^{-1}\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)-\frac{1}{2}{{\varvec{\uptheta}}}^{T}\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right){\varvec{\uptheta}}+\frac{1}{{\sigma }^{2}}{{\varvec{\uptheta}}}^{T}\mathbf{X}\mathbf{y}-\frac{1}{2{\sigma }^{2}}{\mathbf{y}}^{T}\mathbf{y}$$

(29)

To simplify the form, let ${\mathbf{A}=\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}$, $\mathbf{B}=\frac{1}{{\sigma }^{2}}\mathbf{X}\mathbf{y}$, $\mathbf{C}={\mathbf{y}}^{T}\mathbf{y}$. Hence,

$$\leftrightarrow {\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left({\mathbf{X}}^{*T}{{\left(\mathbf{A}\right)}^{-1}\mathbf{X}}^{*}\right)}^{-1}\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)-\frac{1}{2}{{\varvec{\uptheta}}}^{T}\left(\mathbf{A}\right){\varvec{\uptheta}}+{{\varvec{\uptheta}}}^{T}\mathbf{B}-\frac{1}{2{\sigma }^{2}}\mathbf{C}$$

(30)

Hence, by completing the square with respect to ${\varvec{\uptheta}}$ (marginalisation over ${\varvec{\uptheta}}$), Eq. (30) becomes:

$$\leftrightarrow -\frac{1}{2}{\left({\varvec{\uptheta}}-{\mathbf{A}}^{-1}\mathbf{B}\right)}^{T}\mathbf{A}\left({\varvec{\uptheta}}-{\mathbf{A}}^{-1}\mathbf{B}\right)+\frac{1}{2}{\mathbf{B}}^{T}{\mathbf{A}}^{-1}\mathbf{B}+{\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left({\mathbf{X}}^{*T}{{\left(A\right)}^{-1}\mathbf{X}}^{*}\right)}^{-1}\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)-\frac{1}{2{\sigma }^{2}}C$$

(31)

By inserting Eq. (31) into Eq. (28), after some algebraic calculations, it is obtained:

$$=\sqrt{\left|{\left(2\pi \right)}^{n}{\mathbf{A}}^{-1}\right|}\mathrm{exp}\left({\left({{\varvec{y}}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left({\mathbf{X}}^{*T}{{\left(A\right)}^{-1}\mathbf{X}}^{*}\right)}^{-1}\left({{\varvec{y}}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)\right)\mathrm{exp}(\frac{1}{2}{\mathbf{B}}^{T}{\mathbf{A}}^{-1}\mathbf{B}-\frac{1}{2{\sigma }^{2}}C)\int N\left({\varvec{\uptheta}}|{\mathbf{A}}^{-1}\mathbf{B},{\mathbf{A}}^{-1}\right) d{\varvec{\uptheta}}$$

(32)

With respect to ${\mathbf{y}}^{*}$ and by considering other terms without ${\mathbf{y}}^{*}$ as constant, hence:

$$P\left({\mathbf{y}}^{*}|{\mathbf{X}}^{\boldsymbol{*}},\mathbf{X},\mathbf{y};\,M\right)\propto \mathrm{exp}\left({\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)}^{T}{\left({\mathbf{X}}^{*T}{{\left(A\right)}^{-1}\mathbf{X}}^{*}\right)}^{-1}\left({\mathbf{y}}^{*}-{\mathbf{X}}^{*T}\overline{{\varvec{\uptheta}} }\right)\right)$$

(33)

Finally, the prediction distribution is (by writing again ${\mathbf{A}=\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}$ and $\overline{{\varvec{\uptheta}} }={\sigma }^{2}{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}\mathbf{y}$):

$$P\left({\mathbf{y}}^{*}|{\mathbf{X}}^{\boldsymbol{*}},\mathbf{X},\mathbf{y};\,M\right)\sim N({\overline{{\mathbf{y} }^{*}}=\mathbf{X}}^{*T}{\sigma }^{2}{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}\mathbf{y},{cov\left(\overline{{\mathbf{y} }^{*}}\right)=\mathbf{X}}^{*T}{{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}}^{*})$$

(34)

Appendix C: Derivation of the distribution estimation for prediction with Kernel

As being previously defined, let ${\mathbf{A}=\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}$. Subsequently, with matrix algebra, one can obtain:

$${\sigma }^{-2}\mathbf{X}\left({\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right)=\mathbf{A}\sum \mathbf{X}$$

(35)

$${\mathbf{A}}^{-1}\left[{\sigma }^{-2}\mathbf{X}\left({\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right)\right]{\left({\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right)}^{-1}={\mathbf{A}}^{-1}\left[\mathbf{A}\sum \mathbf{X}\right]{\left({\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right)}^{-1}$$

(36)

$${\mathbf{A}}^{-1}{\sigma }^{-2}\mathbf{X}=\sum \mathbf{X}{\left({\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right)}^{-1}$$

(37)

By inserting Eq. (37) into the mean in Eq. (34), the mean becomes:

$${{{\varvec{X}}}^{*T}{\sigma }^{2}{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}\mathbf{y}=\mathbf{X}}^{*T}\sum \mathbf{X}\left[{\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right]\mathbf{y}$$

(38)

For the covariance in Eq. (34), with matrix inversion lemma [48], the term ${\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}$ can be re-written as (by setting $\mathbf{Z}={\sum }^{-1}, \mathbf{U}=\mathbf{V}=\mathbf{X}$ and $W={\sigma }^{-2}$):

$${\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}=\sum -\sum \mathbf{X}{\left[{\sigma }^{2}+{\mathbf{X}}^{T}\sum \mathbf{X}\right]}^{-1}\sum \mathbf{X}$$

(39)

Hence, the covariance in Eq. (34) becomes:

$${\mathbf{X}}^{*T}{{\left({\sigma }^{-2}\mathbf{X}{\mathbf{X}}^{T}+{\sum }^{-1}\right)}^{-1}\mathbf{X}}^{*}={\mathbf{X}}^{*T}\sum {\mathbf{X}}^{\boldsymbol{*}}-{\mathbf{X}}^{*T}\sum \mathbf{X}\left[{\mathbf{X}}^{T}\sum \mathbf{X}+{\sigma }^{2}\mathbf{I}\right]{\mathbf{X}}^{*T}\sum \mathbf{X}$$

(40)

From Eq. (38) and Eq. (40), all multiplications involving input vector ${\mathbf{X}}^{\boldsymbol{*}}\sum \mathbf{X}$ become $\langle {\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}\rangle =\mathbf{K}\left({\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}\right)$ and $\mathbf{X}\sum \mathbf{X}$ become $\langle \mathbf{X},\mathbf{X}\rangle =\mathbf{K}\left(\mathbf{X},\mathbf{X}\right)$. By this substitution, the input multiplication is carried out in kernel space having a non-linear function with variable ${\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}$ or $\mathbf{X},\mathbf{X}$ or ${\mathbf{X}}^{\boldsymbol{*}},{\mathbf{X}}^{\boldsymbol{*}}$. Hence, the mean $\overline{{\mathbf{y} }^{*}}$ and variance of the prediction $cov\left(\overline{{\mathbf{y} }^{*}}\right)$ become:

$$\overline{{\mathbf{y} }^{*}}=\mathbf{K}\left({\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}\right){[\mathbf{K}(\mathbf{X},\mathbf{X})+{\sigma }^{2}\mathbf{I}]}^{-1}\mathbf{y}$$

(41)

$$cov\left(\overline{{\mathbf{y} }^{*}}\right)=\mathbf{K}\left({\mathbf{X}}^{\boldsymbol{*}},{\mathbf{X}}^{\boldsymbol{*}}\right)-\left(\mathbf{K}\left({\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}\right){\left[\mathbf{K}\left(\mathbf{X},\mathbf{X}\right)+{\sigma }^{2}\mathbf{I}\right]}^{-1}\mathbf{K}\left({\mathbf{X}}^{\boldsymbol{*}},\mathbf{X}\right)\right)$$

(42)