## 2 Materials and Methods
### Feature Extraction  
We provide here a detailed description of the procedure of feature extraction used in this work. For each of reconstructed \(\mu_{a}\) and \(\mu_{s}^{\prime}\) images, we first selected various regions of interest and defined all data points in each region of interest as the unique data set, and then features were extracted from each unique data set [34]. In total, 11 unique data sets were generated: two 3D volume data sets and nine 2D image data sets. These 11 data sets comprise the entire unstructured or structured volume data (UV, SV), the data sets from the summation of all coronal, sagittal, transverse slices  
Figure 1: An illustration of joints and a typical finger in the human hand. (**a**) The PIP joint (marked in red circle) is located approximately in the middle of the finger, referred to as PIP II-IV (index, middle, and ring fingers, respectively). (**b**) The positions of the 11 sources are indicated on the posterior surface of the finger. (**c**) The positions of detectors (i.e., mesh nodes that map directly to a pixel on the CCD camera-based detection system) are shown on the anterior surface of the finger section. Figure 2: Transillumination captured by the ICCD camera on the posterior (palmar) surface of a PIP joint belonging to (**a**) a subject with RA and (**b**) a healthy control. respectively (SC, SS, ST), the data sets from the variance between all coronal, sagittal, transverse slices, respectively (VC, VS, VT), and also the data sets from the average of all coronal, sagittal, transverse slices in the joint region (i.e., within \(\pm 2\) mm from the center of the PIP joint), respectively (JC, JS, JT). From each of all 11 data sets (UV, SV, SC, SS, ST, VC, VS, VT, JC, JS, JT), the following 76 features were extracted and denoted by F01 to F76 consecutively: (1) five basic features, (2) seven Gaussian mixture model (GMM) parameters, and (3) 13 (in 2D) or 63 (in 3D) FFT (fast Fourier transform) coefficients. The seven basic statistical features are the maximum, minimum, mean, variance, and the ratio of maximum to minimum of the data set. Except for the unstructured volume data set, the additional seven features were extracted by fitting DOT images with a 2D or 3D multivariate GMM [42] as  
\[G(\mu)=A_{0}\exp\left[\frac{1}{2}(\mu-\mu_{0})^{T}\sum^{-1}(\mu-\mu_{0})\right] \tag{10}\]  
where amplitude \(A_{0}\), covariance matrix \(\sum\), and Gaussian mean \(\mu_{0}\) are estimated. The extracted features with Equation (10) are the absolute error (1 feature) between the mixture model and the original data and the eigenvalues of the dominant positive (3 features) and negative (3 features) Gaussians. In addition, we performed a discrete FFT of the 2D or 3D data sets, extracted the distinct FFT coefficients that best represent the original image, and used those coefficients as the features--13 FFT coefficients for the 2D data set and 63 coefficients for the 3D data set. All data sets and features used in this work are summarized in Tables A1-A4 in Appendix A. To succinctly refer to all the features, we use the notation \"Feature #:Data SetOptical Property\" throughout the rest of this work, in which Feature # represents F1 to F76, Data Set represents UV to JT, and Optical Property represents the \(\mu_{a}\) (denoted by \"a\") or \(\mu_{s}^{\prime}\) (denoted by \"s\") distribution. For example, F1:JSa indicates the maximum value (\"F1\") of the joint sagittal (\"JS\") slices in \(\mu_{a}\) image (\"a\"). The FFT feature starts with F13 and ends with F26 (in the 2D data set) and F76 (in the 3D data set). As a result, this leads to a total of 572 features (= (55 basic features + 52 GMM features + 170 FFT features) \(\times\) 2 optical property maps). This is too a large number of features to consider all the possible combinations of features (i.e., \(2^{572}-1\) possible combinations) in classification analysis.