Temporal Continuity

In order to perform motion compensation and vasodilation assessment, it is convenient to introduce prior knowledge about the smooth nature of the arterial vasodilation process. This a priori information could be used to filter out sharp transitions in the vasodilation parameter, which arise as a consequence of registration errors and which are not physiologically plausible. Moreover, these registration errors could easily propagate to the following frames thus invalidating all subsequent measurements.

To avoid error propagation and impose constraints on the vasodilation dynamics, a recursive filter is employed in order to improve the estimation of the initial registration parameters. In the next two subsections, the elaboration of the starting estimates in the motion compensation and in the vasodilation stages are presented.

5.3.3.1 Starting Estimate in Motion Compensation

The motion compensation phase involves three parameters: two translations, tx and ty, and a rotation angle, 0. Aparametervectoristhusdefinedby x = {tx, ty, 0}. Because of the strong nonstationary behavior of motion artifacts, it is not possible to derive an elaborated linear model of the dynamics the parameter vector. Therefore, a simple first-order auto-regressive model, AR(1), was assumed to predict a suitable initialization for the registration algorithm in the nth frame, x(n). This was done according to x(n) = x(n - 1) + y (x(n - 1) - x(n - 1)) (5.7)

where x(n) denotes the parameter vector output after registering the nth frame. Note that there is some implicit delay between x(n) and x(n) since they refer to parameter vectors before and after the registration process. Equation (5.7) introduces a systematic inertia to changes in the parameter values through the constant y. This filtering tries to avoid falling into local minima in the parameter space during registration, which would not be temporally consistent with previous history of arterial motion. On the other hand, it might also slow down the ability to track sudden transitions coming from true motion artifacts. A value of y = 0.1 has been empirically shown to be a good compromise between these two competing goals and it was used throughout our experiments.

5.3.3.2 Starting Estimate During Vasodilation Assessment

In this stage we assume that the translation and rotation parameters were correctly recovered at the motion compensation stage. Therefore, only the scale factor, sy, will be tracked over time using a simplified Kalman filtering scheme [23].

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Figure 5.5: Kalman gain. (a) Estimated aw (n) used for the computation of K(n) during vasodilation assessment. It contains the expected vasodilation dynamics along the sequence. Instants with higher value of aw (n) correspond to higher uncertainty about the chosen dynamic model and, consequently, where it has to be relaxed to accommodate for possibly sudden transitions. (b) Corresponding Kalman gain for three different measurement noise, av (n), values corresponding to the minimum, median, and maximum noise levels, respectively, in our sequence database.

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Figure 5.5: Kalman gain. (a) Estimated aw (n) used for the computation of K(n) during vasodilation assessment. It contains the expected vasodilation dynamics along the sequence. Instants with higher value of aw (n) correspond to higher uncertainty about the chosen dynamic model and, consequently, where it has to be relaxed to accommodate for possibly sudden transitions. (b) Corresponding Kalman gain for three different measurement noise, av (n), values corresponding to the minimum, median, and maximum noise levels, respectively, in our sequence database.

Let us assume that sy(n) can be also modeled as an AR(1) process sy(n) = a sy(n — 1) + w(n) (5.8)

where w(n) is white noise with variance aw, and 0 < a < 1 is the coefficient of the AR(1) model, which was chosen as a = 0.95. The scaling factor has a non-stationary behavior and, therefore, aw (n) actually changes widely over time (cf. Fig. 5.5).

Let the measurement model be

where sy(n) are noisy measurements of vasodilation at frame n (obtained via image registration), and v(n) is white noise, uncorrelated with w(n). Under the assumptions of this model, it can be shown [24] that the Kalman filter state estimation equation to predict the vasodilation initialization for the registration of the nth frame, sy(n), is sy(n) = a sy(n — 1) + K(n) [sy(n — 1) — a sy(n — 1)] (5.10)

where K(n) is the Kalman filter gain. Owing to the standardized acquisition protocol, the vasodilation time series has a characteristic temporal evolution, which can be exploited to give an a priori estimation of aw (n). The value of aw (n) is high when vasodilation is expected and, it is low when no variations in the artery diameter should be found (e.g. at baselines). On the other hand, the observation noise power av (n) will be considered constant, and it is estimated from the first 60 sec when vasodilation is known to be zero. The measurement noise is assumed to be stationary as it mainly depends on the image quality, which can be considered uniform over time for a given sequence.

The temporal evolution of aw (n) is shown in Fig. 5.5. It has been estimated from the analysis of 50 vasodilation curves (from the dataset described in section 5.2) that were free from artifacts and obtained with the computerized method but using a fixed K(n) = 0.1 to assess the vasodilation. The plot indicates the average instantaneous power over the 50 realizations. The 60 initial frames are processed with K(n) = 0 to calculate av in each sequence, because no vasodi-lation is expected during this interval and any variation here can be regarded as measurement noise.

The values of aw (n) and av determine the Kalman gain, K(n), using the following equation [24]:

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