Invariant chromatic descriptor for LADAR data processing

Abstract

A new LADAR data descriptor is proposed. This descriptor is produced from the application of the chromatic methodology to extract features from the LADAR data by applying invariant spatial chromatic processors. The descriptor developed has a high discrimination capability, robust to the effects that disturb LADAR data, and requires less storage space and computational time for recognition. The performance of the proposed LADAR descriptor is evaluated using simulated LADAR data, which are generated from special software called LADAR simulator. The simulation results show high discrimination capability for the new descriptor over the traditional techniques such as Moments descriptor. This Moments descriptor is used to benchmark the results. The results also show the robustness of the proposed descriptor in the presence of noise, low resolution, view change, rotation, translation, and scaling effects.

Appendices

Appendix A: Procedure of extracting invariant chromatic features

The spatial distributions of the chromatic processors make them variant to the shift and scale effects. If the signal is moved from its original location, or enlarged the processed chromatic values will change. To make these processors invariant, a special approach was proposed, which makes the centres of the processors and their widths adaptable to the input signal type [5]. Referring to Fig. 11, the approach starts by calculating the centroid $c_{m}$ of the input signal $P_{R}(l_\mathrm{o})$ of $\ell$ length by

$$\begin{aligned} c_{m}=\sum _{l_\mathrm{o}=1}^{\ell } l_\mathrm{o} P_{R}(l_\mathrm{o})/\sum _{l_\mathrm{o}=1}^{\ell }P_{R}(l_\mathrm{o}) \end{aligned}$$

(3)

This centroid is then used as a boundary condition for calculating two processors’ centres $C_{R}$ and $C_{B}$ as follows:

$$\begin{aligned}&\!\!\! C_{R}=\sum _{l_\mathrm{o}=1}^{c_{m}} l_\mathrm{o} P_{R}(l_\mathrm{o})/\sum _{l_\mathrm{o}=1}^{c_{m}}P_{R}(l_\mathrm{o})\end{aligned}$$

(4)

$$\begin{aligned}&\!\!\! C_{B}=\sum _{l_\mathrm{o}=c_{m}}^{\ell } l_\mathrm{o} P_{R}(l_\mathrm{o})/\sum _{l_\mathrm{o}=c_{m}}^{\ell }P_{R}(l_\mathrm{o}) \end{aligned}$$

(5)

The third processor centre $G_{G}$ can be then determined from the following equation:

$$\begin{aligned} C_{G}=(C_{R}+C_{B})/2 \end{aligned}$$

(6)

The widths for these three processors $W_{RGB}$ are all equal and they define as

$$\begin{aligned} W_{RGB}=(C_{B}-C_{R})/2 \end{aligned}$$

(7)

Fig. 11

Calculated centres and widths for Gaussian processors using the proposed approach

The calculated centres and widths are then used to determine the response profiles ($R (l_\mathrm{o}), G (l_\mathrm{o}), B (l_\mathrm{o})$) for these Gaussian processors using the following equations, respectively:

$$\begin{aligned} R (l_\mathrm{o})=\exp \left( \frac{-2.8(l_\mathrm{o}-C_{R})^{2}}{W_{RGB}^{2}}\right) \end{aligned}$$

(8)

$$\begin{aligned} G (l_\mathrm{o})=\exp \left( \frac{-2.8(l_\mathrm{o}-C_{G})^{2}}{W_{RGB}^{2}}\right) \end{aligned}$$

(9)

$$\begin{aligned} B (l_\mathrm{o})=\exp \left( \frac{-2.8(l_\mathrm{o}-C_{B})^{2}}{W_{RGB}^{2}}\right) \end{aligned}$$

(10)

The processors (of the adapted centres and widths), are then applied on the discrete input signal $P_{R}(l_\mathrm{o})$ to extract its features, where their outputs ($R _\mathrm{o},G _\mathrm{o},B _\mathrm{o}$) are calculated by applying the following equations, respectively: [20]:

$$\begin{aligned} R _\mathrm{o}=\sum _{l_\mathrm{o}=1}^{\ell }R (l_\mathrm{o})\,P_{R}(l_\mathrm{o})\end{aligned}$$

(11)

$$\begin{aligned} G _\mathrm{o}=\sum _{l_\mathrm{o}=1}^{\ell }G (l_\mathrm{o})\,P_{R}(l_\mathrm{o})\end{aligned}$$

(12)

$$\begin{aligned} B _\mathrm{o}=\sum _{l_\mathrm{o}=1}^{\ell }B (l_\mathrm{o})\,P_{R}(l_\mathrm{o}) \end{aligned}$$

(13)

The approach then is to evaluate combinations of these cross correlations to yield coordinates which define the signal in one of several chromatic modes (e.g. x:y, Lab, $HLS$, etc.) depending on the nature of the information sought [20].

The Hue–Lightness–Saturation ($HLS$) scheme is used in this work. The choice of this scheme enables the intuitive methods of colour science to be related to signal defining factors, where $L$ represents the strength of the signal, $S$ its spread in the measured domain and $H$ is the dominant measured value [51].

The transformation of the processors’ outputs ($R _\mathrm{o}$, $G _\mathrm{o}$, $B _\mathrm{o}$) to $HLS$ is performed using the following relationships [20, 51]:

$$\begin{aligned} H= & {} 0.667-0.333\left( \frac{g_\mathrm{o}}{g_\mathrm{o}+b_\mathrm{o}}\right) ,\quad r_\mathrm{o}=0 \\= & {} 1.000-0.333\left( \frac{b_\mathrm{o}}{b_\mathrm{o}+r_\mathrm{o}}\right) ,\quad g_\mathrm{o}=0 \nonumber \\= & {} 0.333-0.333\left( \frac{r_\mathrm{o}}{r_\mathrm{o}+g_\mathrm{o}}\right) ,\quad b_\mathrm{o}=0 \nonumber \end{aligned}$$

(14)

$$\begin{aligned} L= & {} \frac{R _\mathrm{o}+G _\mathrm{o}+B _\mathrm{o}}{3}\end{aligned}$$

(15)

$$\begin{aligned} S= & {} \frac{\mathrm{max}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})-\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})}{\mathrm{max}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})+\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})} \end{aligned}$$

(16)

where

$$\begin{aligned} r_\mathrm{o}= & {} R _\mathrm{o}-\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})\end{aligned}$$

(17)

$$\begin{aligned} g_\mathrm{o}= & {} G _\mathrm{o}-\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})\end{aligned}$$

(18)

$$\begin{aligned} b_\mathrm{o}= & {} B _\mathrm{o}-\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o}) \end{aligned}$$

(19)

$\mathrm{\mathrm{max}}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})$ and $\mathrm{min}(R _\mathrm{o},G _\mathrm{o},B _\mathrm{o})$ represents the parameter ($R _\mathrm{o},G _\mathrm{o},B _\mathrm{o}$) having the highest and lowest values, respectively. If $R _\mathrm{o} \, \& \, G _\mathrm{o} \, \& \, B _\mathrm{o} =0$ then $S =0$ and $H$ is undefined.

The $H$ and $S$ processors’ outputs only are used without $L$, because of their robustness to the scale effect and their built-in ability to have a fixed range of values ([0 1]), while $L$ requires normalisation and it is sensitive to scale effect.

Appendix B: Object rotation angle

The object rotation angles $\Phi _\mathrm{o}$ can be calculated using different methods such as moments [6] and principle component analysis PCA [16, 25]. It is the angle between the natural axis $x_{\mathfrak {R}}^{'}$ for that object and the $x_{\mathfrak {R}}$-axis as shown in Fig. 12.

Fig. 12

Car image rotation angle $\Phi _\mathrm{o}$ and its natural axis ($x_{\mathfrak {R}}^{'}$, $y_{\mathfrak {R}}^{'}$)

The object natural axis ($x_{\mathfrak {R}}^{'}$, $y_{\mathfrak {R}}^{'}$) can be simply calculated by determining the eigenvalues of the image covariance matrix. This matrix is defined by [16]:

$$\begin{aligned} C_\mathbf{sv }= & {} \frac{1}{K_{s}}\sum _{k_{s}=1}^{K_{s}}\mathbf{sv }_{k_{s}}\mathbf{sv }^{T}_{k_{s}}- \mathbf m _\mathbf{sv }\mathbf m _\mathbf{sv }^{T}\end{aligned}$$

(20)

$$\begin{aligned} \mathbf m _\mathbf{sv }= & {} \frac{1}{K_{s}}\sum _{k_{s}=1}^{Ks}\mathbf{sv }_{k_{s}} \end{aligned}$$

(21)

where $\mathbf sv$ is the pixels’ distribution vector for the image and $K_{s}$ is the number of the vector samples $\mathbf sv$.

The matrix $C_\mathbf{sv }$ is real and symmetric; therefore, its eigenvalues are nonnegative real numbers [25]. These values are sorted in a non-increasing order to find the corresponding eigenvectors, which represent the natural axis of the object. The eigenvectors have mirror symmetry which can be handled by making the positive axis direction towards the highest standard deviation (STD) of the vector’s lengths (lengths between the pixel locations and the origin). More information about this procedure can be found in [36, 37]. If the STD are equal in both directions (positive and negative) then the natural axis of the second highest eigenvalues is used instead.