Performance Comparison of Motion Encoders: Hassenstein–Reichardt and Two-Detector Models

Abstract

Several motion-detection models have been proposed based on insect visual system studies. We specifically examine two models, the Hassenstein-Reichardt (HR) model and the two-detector (2D) model, before selecting model the more efficient motion encoders. We analytically obtained the mean and variance of stationary responses of the HR and the 2D models to white noise to evaluate performances of the two models. Especially when analyzing the 2D model, we calculated higher-order cumulants of a rectified Gaussian. Results show that the 2D model gives almost equal performance to that of the HR model in a biologically reasonable case.

1 Introduction

Several motion detector models, classified as correlation-type elementary motion detectors (correlation-type EMDs), have been proposed based on insect visual system studies. In this type of model, speed and motion directions are extracted by calculating delayed cross-correlation of the two input signals. Reportedly, correlation-type EMD can be implemented in early visual systems of insects, especially in Drosophila melanogaster [1,2,3]. Additionally, the possibility has been suggested that the correlation-type EMD is a common algorithm for motion detection in many species including rabbit [4] and human models [5, 6]. Furthermore, correlation-type EMD is expected to be extremely useful for artificial visual processing for micro-unmanned aerial vehicles, for reducing computational effort [7, 8].

This paper presents specific examination of two correlation-type EMD models: the Hassenstein–Reichardt (HR) model and the two-detector (2D) model. Since the HR model was first proposed (Fig. 1A), it has been extended to several models called the 2D model (Fig. 1B) and the four detector (4D) model, which has biologically more reasonable structures than the original HR model [1, 9]. The extended models consist of a set of standard Reichardt detector (SRD) units corresponding to the original HR model. To satisfy Dale’s rule, these separated SRD units respectively process various pairs of half-wave rectified signals. The whole input–output relation of the 4D model is mathematically equivalent to that of the HR model [9, 10], whereas the whole input–output relation of the 2D model differs from that of the HR model.

Fig. 1.

Models are the following: A, the Hassenstein–Reichardt model (HR model), a motion encoder using delayed cross-correlation of the two input signals; B: the two-detector model (2D model). The 2D model has two parallel pathways, each of which functions similarly to the HR model and which processes only positive or negative components. f, M, and SUM respectively represent low-pass filter, multiplication, and summation. $w_1$ and $w_2$ are weight coefficients.

Recently, some research groups have attempted mathematical evaluation of the performance of the two models in terms of motion encoders. The HR model was evaluated analytically by calculating its response to sine waves and white noise [5, 11,12,13,14,15]. However, no analytical evaluation of the 2D model is available: only numerical evaluation by simulating its response to synthetic natural scene [16]. These earlier studies have mainly addressed amplitude response as a measure of encoding performance. However, to evaluate the robustness as an encoder, one must address the fluctuation of response of the models. It is difficult to treat the fluctuation of response of the correlation-type EMD analytically, especially the 2D model with the half-wave rectifiers.

For this study, we analytically obtained the mean and variance of responses of the HR and the 2D models to white noise; then we derived the signal-to-fluctuation-noise ratio (SFNR) to evaluate performances of the two models in terms of motion encoders. Especially, when analyzing the 2D model, we calculated the fourth-order cumulants of a rectified Gaussian distribution. Results showed that the 2D model gives almost identical performance to that of the HR model in a biologically reasonable case.

2 Methods

2.1 Formulation of Correlation-Type Elementary Motion Detectors

Hassenstein–Reichardt Model: We briefly explain the HR model (Fig. 1A).

Let I be the input signal. Here, $x_{L}$ and $x_{R}$ respectively denote a left-side signal and a right-side signal in the first processing stage, which are delayed through a low-pass filter. In the second processing stage, $x_{L}$ and $x_{R}$ are multiplied with each opposite input signal.

$$\begin{aligned} y_{L}(t)= & {} \int _0^\infty dt' f(t') I(t-t')I(t+ \varDelta t)\end{aligned}$$

(1)

$$\begin{aligned} y_{R}(t)= & {} \int _0^\infty dt' f(t') I(t+\varDelta t-t')I(t) \end{aligned}$$

(2)

Therein, f(t) is an impulse response function of the low-pass filter. For mathematical tractability, $f(t)=0$ is satisfied if $t\le 0$. $\varDelta t$ is the difference of detection time between the left-side and the right-side photoreceptors.

In the final stage, the multiplied signals $y_{L}$ and $y_{R}$ are summed up with the weight.

$$\begin{aligned} \mathcal {R}_{HR}=w_1y_{L}+w_2y_{R} \end{aligned}$$

(3)

In that equation, $w_1$ and $w_2$ are weight coefficients with mutually different signs.

2D Model: We briefly explain the 2D model, which model consists of two SRD units equivalent to the HR model as shown in Fig. 1B. In the first processing stage of the left side and the right side of the 2D model, the input signal I(t) is split into $I_+(t)$ and $I_-(t)$ through two half-wave rectifiers. The left SRD unit processes $I_+(t)$ and $I_+(t+\varDelta t)$; the right SRD unit processes $I_-(t)$ and $I_-(t+\varDelta t)$ (Fig. 1B). Additionally, $x_{+L}$ and $x_{+R}$ respectively denote a left-side signal and a right-side signal in the left SRD, which are delayed through the low-pass filter f(t). Then, $y_{+L}$ and $y_{+R}$ are multiplied with each opposite rectified input signal as follows.

$$\begin{aligned} y_{+L}(t)= & {} \int _0^\infty dt' f(t') I_+ (t-t')I_+(t+\varDelta t)\end{aligned}$$

(4)

$$\begin{aligned} y_{+R}(t)= & {} \int _0^\infty dt' f(t') I_+ (t+\varDelta t-t')I_+(t) \end{aligned}$$

(5)

It is noteworthy that the right SRD unit is formulated similarly. Therefore, we omit the explanation of the right SRD unit.

In the final stage, the multiplied signals $y_{+L}$ and $y_{+R}$ in the left SRD unit and $y_{-L}$ and $y_{-R}$ in the right SRD unit are summed up with the weight.

$$\begin{aligned} \mathcal {R}_{2D}=w_1y_{+L}+w_1y_{-L}+w_2y_{+R}+w_2y_{-R} \end{aligned}$$

(6)

In that equation, $w_1$ and $w_2$ are weight coefficients with mutually different signs.

2.2 Numerical Evaluation

We evaluate the analytical solutions by comparison with numerical simulation results. The conditions of numerical evaluations in Sects. 3.2 and 3.3 are given as shown below.

In the HR model and the 2D model SRD unit, the low-pass filter of the first processing stage was implemented with the first-order low-pass filter, as in earlier studies [9, 15].

$$\begin{aligned} f(t) = \left\{ \begin{array}{ll} \frac{1}{\tau }\mathrm {exp}(-\frac{1}{\tau }t) &{} (0 < t) \\ 0 &{} (t\le 0) \end{array} \right. \end{aligned}$$

(7)

For mathematical tractability, $f(t)=0$ is satisfied if $t\le 0$.

In both models, the weight coefficients $w_1$ and $w_2$ of the final stage were given as the same as those of earlier studies [9, 11].

$$\begin{aligned} w_1=1,\ \ w_2=-\alpha \end{aligned}$$

(8)

where $0<\alpha \le 1$.

3 Results

3.1 Analytical Solutions of HR and 2D Models

Analysis of the 2D Model: We obtained the mean value of stationary response of the 2D model to the white Gaussian noise to rewrite the second joint moment derived from the above formulation as the sum of the products of the cumulants (we call this expression cumulant expansion).

$$\begin{aligned} <\mathcal {R}_{2D}>= & {} (\mathcal {A}_1+\mathcal {A}_2)\sigma ^2\\ \mathcal {A}_1= & {} \frac{\pi -1}{\pi }(w_1 f(-\varDelta t)+w_2 f(\varDelta t))\nonumber \\ \mathcal {A}_2= & {} \frac{1}{\pi }(w_1+w_2)\int _0^\infty dt'f(t')\nonumber \end{aligned}$$

(9)

Therein, brackets $<X>$ denote the mean of X.

Next, we obtained the variance of the stationary response of the 2D model to the white Gaussian noise to rewrite the fourth-order joint moment derived from the formulation as the cumulant expansion.

$$\begin{aligned} V[\mathcal {R}_{2D}]= & {} (\mathcal {B}_1+\mathcal {B}_2+\mathcal {B}_3+\mathcal {B}_4+\mathcal {B}_5)\sigma ^4\\ \mathcal {B}_1= & {} \frac{3\pi ^2\!-\!2\pi \!-\!2}{2\pi ^2}(w_1^2(f(-\varDelta t))^2+w_2^2(f(\varDelta t))^2)\nonumber \\ \mathcal {B}_2= & {} \frac{\pi \!+\!2}{\pi ^2}(w_1^2f(-\varDelta t)+w_2^2f(\varDelta t))\int _0^\infty dt'f(t')\nonumber \\ \mathcal {B}_3= & {} \frac{\pi \!-\!2}{\pi ^2}w_1w_2\int _0^\infty dt'f(t'+\varDelta t)f(t')\nonumber \\ \mathcal {B}_4= & {} \frac{\pi \!-\!1}{2\pi }(w_1^2+w_2^2)\int _0^\infty dt'(f(t'))^2\nonumber \\ \mathcal {B}_5= & {} \frac{\pi \!-\!2}{2\pi ^2}(w_1^2+w_2^2)(\int _0^\infty dt'f(t'))^2\nonumber \end{aligned}$$

(10)

In those expressions, V[X] represents the variance of X.

Analysis of the HR Model: We obtained the mean value of stationary response of the HR model to the white Gaussian noise as the following equation.

$$\begin{aligned} <\mathcal {R}_{HR}>=(w_1f(-\varDelta t)+w_2f(\varDelta t))\sigma ^2 \end{aligned}$$

(11)

This result is the same as the result obtained from an earlier study [5] for $w_1=1$ and $w_2=-1$.

We obtained the variance of stationary response of the HR model as shown below.

$$\begin{aligned} V[\mathcal {R}_{HR}]= & {} (\mathcal {C}_1+\mathcal {C}_2)\sigma ^4\\ \mathcal {C}_1= & {} w_1^2(f(-\varDelta t))^2+w_2^2(f(\varDelta t))^2\nonumber \\ \mathcal {C}_2= & {} (w_1^2+w_2^2)\int _0^\infty dt'(f(t'))^2\nonumber \end{aligned}$$

(12)

3.2 Numerical Evaluation of Analytical Solutions

We compared the analytical solutions with those obtained from numerical simulations. Figure 2 shows the mean and variance of responses of the two models as a function of $\tau$ and $\alpha$. The analytical solutions matched those obtained by numerical simulations. When $\tau$ was small, however, a mismatch resulted from discretized approximation.

Fig. 2.

Comparison between the analytical solutions of the stationary responses of the HR and the 2D models to white Gaussian noise and those obtained by numerical simulations. A and B respectively show the mean and variance as a function of $\tau$, when $\alpha$ fixed to 0.75. C and D show them as a function of $\alpha$, when $\tau$ fixed to 100 [ms]. Solid and dashed lines respectively show numerical solutions and analytical solutions. Here, the Gaussian noise variance is set to $1^2$. (Color figure online)

3.3 Signal-to-Fluctuation-Noise Ratios of the HR and 2D Models

SFNR of the HR and 2D Models: To evaluate the performance of the two models as an encoder, we calculated the signal-to-fluctuation-noise ratio (SFNR)[17]. The SFNR is defined as

$$\begin{aligned} \mathrm {SFNR}_{x}={\sqrt{\frac{ <\mathcal {R}_{x}>^2 }{V[\mathcal {R}_{x} ] } } }, \end{aligned}$$

(13)

where $x\in \{\mathrm {HR},\,\mathrm {2D}\}$ that indicates either the HR or 2D model. The larger SFNR stands for more precise motion detection. We analytically obtained the mean and variance of responses of the HR and the 2D models. Therefore, we can obtain the SFNR analytically.

Fig. 3.

Gray-scale plots of the SFNRs of the HR and the 2D models and their differences: A, SFNRs of the 2D model; B, SFNRs of the HR model; C, differences obtained when SFNRs of the HR model are subtracted from the corresponding SFNRs of the 2D model. The white dashed line represents the difference of SFNRs becoming zero. The markers denote the values of parameters estimated in the earlier works (diamond [9], circle [18], and cross [11]). $\varDelta t$ is set to 1 [ms]. (Color figure online)

Figures 3A and 3B respectively show SFNR as a function of $\alpha$ and $\tau$ in the 2D and the HR models. In the case of the 2D model, $\mathrm {SFNR}_{\mathrm {2D}}$ became independent of $\tau$ if $\tau$ is larger than about 50 [ms]. In this region, $\mathrm {SFNR}_{\mathrm {2D}}$ was larger as $\alpha$ approached zero. However, in the case of the HR model, $\mathrm {SFNR}_{\mathrm {HR}}$ changed dependent on both of $\tau$ and $\alpha$. The changes were smaller than those of the 2D model.

Figure 3C presents the difference of SFNRs between the two models. The white dashed lines represent differences of SFNRs of zero. Markers on the panel denote the values of parameters estimated in earlier studies [9, 11, 18]. They were near the dashed line (Fig. 3C).

4 Discussion

We analytically evaluated responses of two typical models of the correlation-type EMD to the white Gaussian noise. We analytically obtained the mean and variance of responses of the HR and the 2D models to the white noise. The HR model is fundamentally equivalent to the four detector models as described above. Therefore, we must deal with almost all correlation-type EMD models by analyzing the HR model. Here we have overcome the difficulty of analyzing the 2D model with half-wave rectifiers. We were able to obtain higher-order joint moments in the 2D model strictly by calculating higher-order cumulants of the rectified Gaussian.

Furthermore, to evaluate the performance of the two models as motion encoders, we calculated SFNR based on the analytical solutions. As shown in Fig. 3C, $\mathrm {SFNR}_{\mathrm {2D}}$ and $\mathrm {SFNR}_{\mathrm {HR}}$ were comparable at biologically reasonable parameter values. This result suggests that the 2D model can provide almost equivalent performance to the HR model in a biologically reasonable case.

The HR model must be implemented biologically with forms of the 4D model because of the restriction of the Dale’s rule. The 2D model is structurally simpler than the 4D model. Therefore, we conclude that the 2D model has almost equivalent performance and lower implementation costs compared to those of the HR model.