## 3 Methods
### MRI Data Processing and Analysis  
The main goal was to explore the methods proposed by [PERSON] et al. [5] and their applicability to study EP nasal vowels from articulatory data extracted from RT-MRI sequences. For this purpose, we have expanded the number of grids in order to include the labial and velar signals, which are particularly relevant to study the nasal vowel dynamics. To find the vocal tract apertures, we have adapted the function to fit and process our acquired data. The proposed method consists of a pipeline with five main steps:  
1. Image registration--each frame of the dataset is aligned to compensate any movements of the speaker during the recordings.
2. Semi-polar grid--after the alignment, a 28-line grid is positioned throughout the vocal tract, starting at the glottis and going to the alveolar ridge.
3. Air-tissue boundary detection--as in the other steps, this semi-automatically processes each image to locate the outer boundary of each grid line.
4. Aperture estimation --the aperture is calculated from the number of pixels in the gridline that are below a certain threshold of intensity. The gridlines are also grouped in regions (alveolar, palatal, velar, and hyper-pharyngeal region), and this will later be useful for interpreting the results.
5. Velum and lip aperture--these are obtained separately: for the velum, principal component analysis (PCA) is used, and lip aperture is estimated based on a region of interest (ROI) and by counting the amount of the dark pixels.  
Figure 1 illustrates, in the top row, the steps entailed in positioning the gridlines over the tract and finding the corresponding tract aperture values, along with the manual selection of the velum and lips region, in the second row. To create the new dataset with the data to explore the generalized additive mixed model (GAMM) visualizations, we have followed the approach taken by [PERSON] et al. [5]. However, as we have interest in the movement of the velum and lips, the data of the semi-polar grid was complemented with the data extracted for velar and lips aperture. To the grid lines obtained in the previous steps, two more data points were added, representing the computed values obtained for the lip aperture obtained in step 5. Additionally, the tract aperture values associated with the grid lines corresponding to the velar region were modified by combining their values with those obtained from the velar aperture PCA computed in step 5. Specifically, a factor of this value was subtracted from the tract aperture in that region, in an attempt to make velar aperture have an impact on the differences between oral and nasal sounds when computing the GAMMs. The bottom row of Figure 1 shows illustrative examples of the results obtained from the dataset.  
The GAMMs were computed with **R** using the _mgcv_ package. The generalized additive model (GAM) function and parameters used were the same as those considered in [PERSON] et al. [5]. Figure 2 shows the code to compute the GAMMs and FLMMs.  
The data considered to compute the GAMMs includes the preceding and following consonants as a context to what is happening in the vocal tract before and after the vowel. Each GAMM is represented considering a normalized time axis with a duration of 1.0: the vowel is at the center one-third, with the preceding consonant--a /p/ or an /m/-- and the subsequent consonant--a /t/ or a /d/--shown to the left and right, respectively. In addition to the GAMMs visualizations, two different functional linear mixed models (FLMMs) were computed, in order to have a more detailed analysis: the first FLMM was computed using the vocal tract configuration (apertures) at 20%, 50%, and 80% of the vowel interval, to have a more detailed comparison at the beginning, middle, and end of the vowels; the second was obtained based on the data extracted from the velum and lips, to model their behavior over time, supporting an overall analysis of their coordination.