Spatial and temporal point tracking in real hyperspectral images
© Aminov et al; licensee Springer. 2011
Received: 22 March 2010
Accepted: 26 July 2011
Published: 26 July 2011
In this article, we consider the problem of tracking a point target moving against a background of sky and clouds. The proposed solution consists of three stages: the first stage transforms the hyperspectral cubes into a two-dimensional (2D) temporal sequence using known point target detection acquisition methods; the second stage involves the temporal separation of the 2D sequence into sub-sequences and the usage of a variance filter (VF) to detect the presence of targets using the temporal profile of each pixel in its group, while suppressing clutter-specific influences. This stage creates a new sequence containing a target with a seemingly faster velocity; the third stage applies the Dynamic Programming Algorithm (DPA) that tracks moving targets with low SNR at around pixel velocity. The system is tested on both synthetic and real data.
KeywordsHyperspectral Track before detect Dynamic programming algorithm Infrared tracking Variance filter
In the intervening years, interest in hyperspectral sensing has increased dramatically, as evidenced by advances in sensing technology and planning for future hyperspectral missions, increased availability of hyperspectral data from airborne and space-based platforms, and development of methods for analyzing data and new applications .
This article addresses the problem of tracking a dim moving point target from a sequence of hyperspectral cubes. The resulting tracking algorithm will be applicable to many staring technologies such as those used in space surveillance and missile tracking applications. In these applications, the images consist of targets moving at sub-pixel velocities on a background consisting of evolving clutter and noise. The demand for a low false alarm rate on the one hand, and a high probability of detection on the other makes the tracking a challenging task. We posit that the use of hyperspectral images will be superior to current technologies using broadband IR images due to the ability of the hyperspectral image technique to simultaneously exploit two target-specific properties: the spectral target characteristics and the time-dependent target behavior.
The goal of this article is to describe a unique system for tracking dim point targets moving at sub-pixel velocities in a sequence of hyperspectral cubes or, simply put, in a hyperspectral movie. Our system uses algorithms from two different areas, target detection in hyperspectral imagery [1–9] and target tracking in IR sequences [10–19]. Numerous works have addressed each of these problems separately, but to the best of our knowledge, to date no attempts have been made to combine the two fields.
We chose the most intuitive approach to tackle the problem, namely, divide and conquer; we separate the problem into three sub-problems and sequentially solve each one separately. Thus, we first transform each hyperspectral cube into a two-dimensional (2D) image using a hyperspectral target detection method. The next step involves a temporal separation of the movie (sequence of images) into sub-movies and the usage of a variance filter (VF) [10–13] algorithm. The filter detects the presence of targets from the temporal profile of each pixel, while suppressing clutter-specific influences. Finally, a track-before-detect (TBD) approach is implemented by a dynamic programming algorithm (DPA), to perform target detection in the time domain [14–17, 19]. Performance metrics are defined for each step and are used in the analysis and optimization.
To evaluate the complete system, we need to obtain a hyperspectral movie. Since data of this kind are not yet available to us, an algorithm was developed for the creation of a hyperspectral movie, based on a real-world IR sequence and real-world signatures, including an implanted synthetic moving target, given by Varsano et al. .
1 System Architecture
where MaxT is the target's maximum peak amplitude and σ is the standard deviation.
Parts of this study have been published previously by our group: we will, therefore, refer extensively to those publications. Algorithms for target detection in single hyperspectral cubes are described in Raviv and Rotman , the details of the VF and of the generation of the hyperspectral movie are presented in Varsano et al. , and the DPA is described in Nichtern and Rotman . In this article, we present an overall integration of the system; in particular, the article analyzes the integration of the VF and the DPA and provides an overall evaluation of the system.
Step 1: Transformation of the hyperspectral cube into a 2D image - the hyperspectral reduction algorithm
Three different reduction tests - spectral average, scalar product, and MF - were applied on each temporal hypercube individually. Each of these methods is characterized by a mathematical operator, which is calculated on each pixel. In every frame, a map of pixel scores is obtained (the result of the mathematical operator) and used to create the movie.
Test 1: spectral average
where x denotes the pixel's spectrum, x n the spectral value of the nth band, and N the number of spectral bands.
Test 2: scalar product
where x is the pixel being examined, t is the known target signature, and m is the background estimation based on neighboring pixels.
Test 3: MF
where x is the pixel being examined, t is the known target signature, and m and Φ are the background and covariance matrix estimations, respectively.
The MF test was run with different target factors (intensities). The target factor (intensity) can be controlled manually by the hyperspectral data creation algorithm, as an external parameter to the three tests mentioned previously. A higher target factor value, i.e., stronger intensity of the implanted target, poses less difficulty to the detection and tracking algorithm. Since the target implantation model is linear, it is directly proportional to the target factor (intensity of implantation).
Overall, the input to the first stage is a hyperspectral cube; the output of the first stage is a 2D image obtained from the hypercube. Details of the signal processing algorithm using a hyperspectral MF can be found in Raviv et al. .
Step 2. Temporal separation of the 2D sequence: the temporal processing algorithm
where N is the number of profile frames, G0 is the overlap between each of the sub-profiles, and G is the length of the sub-profile.
Following temporal separation, the temporal processing algorithm is applied. The temporal processing algorithm is based on a comparison of the sub-profile overall linear background estimation (defined as DC) to the single highest fluctuation within the sub-profile. The overall linear background estimation (DC) fit, is done using a wider temporal window of samples to achieve best background estimation. The background estimation is performed by calculating a linear fit by means of least squares estimation (LSE) . The fluctuation or short-term variance estimation is performed on a short temporal window of samples (susceptible to temporal variations, i.e., the target entering/leaving pixel), after removing the estimated baseline background. The algorithm is presented in the following two steps, although, in practice, the processing can be performed in real time using a finite size buffer.
Background estimation using a linear fit model
where n is the noise, a and b are the coefficients that must be estimated, M is the number of samples for each long-term window, and y is the DC signal. The goal of this step is to estimate the long-term DC baseline using a least-squares fit to the linear model represented by a coefficients vector .
Short-term variance estimation
where σ i 2 is the estimated variance of the ith window.
where is the zero-mean temporal profile, n is noise, and t is the target signal. is the estimated variance when assuming a target is present; is the variance estimated assuming the absence of a target.
where for 1 ≤ i ≤ K denotes the K minimal variance values of each temporal profile. The value of K is chosen to be smaller than W, so as not to include values that might be caused by the presence of a target.
The performance of the algorithm depends on a wise choice of parameters, i.e., the sizes of the short-term and long-term windows and the length of the sub-profile. The long-term window size serves as the baseline for DC estimation. Since the pixel might be affected by clutter, the baseline DC is not constant. It is assumed that the presence of clutter will cause a monotonic rise or fall pattern in the values of the pixel's temporal profile at least during the duration of the long-term window. Thus, the long-term window should be long enough to facilitate accurate estimation of background, on the one hand, and short enough to enable the influence of clutter to be tracked, on the other hand. Thus, the long-term window should be minimally longer than the target base width to avoid suppressing it . The short-term window is used for variance estimation. It should be matched (or reduced) to the target width (which depends on the target velocity). If the short-term window is significantly longer than the target width, the change in variance caused by the target will be reduced. The sub-profile length matches a pixel target velocity; it should be matched to the target temporal width. The importance of these two window sizes and the overall window parameters will be discussed in the experimental section of the article. We note that the temporal algorithm presented here does not assume a particular target shape and width. It does, however, assume a maximum temporal size of the target, (affecting the target temporal profile), and a positive adding of the target intensity to the background.
To determine the optimal set of window sizes on a real data sequence, the algorithm was run with various sets of parameters.
Dynamic Programming Algorithm
The target size is one pixel or less.
Only one target exists in each spatial block.
The target may move in any possible direction.
Target velocity is within 0-2 pixels per frame (ppf).
Images are too noisy to allow detection of a threshold on a single frame.
Jitter of up to 0.5 ppf is allowed only in the horizontal and vertical directions and is uniformly distributed.
Since the target velocity is within the range of 0-2 ppf with a possible jitter of 0.5 ppf, the pixel can move up to 2.5 ppf in the horizontal and vertical directions; hence, a valid area from which a pixel might origin from in the previous frame is a 7 × 7 pixel area (matrix). Such a search area can be resized according to different velocity ranges and jitter values. The search area will define the probability matrices that contain the probabilities of pixels in the previous frames being the origin of the pixel in the current frame. To take into account unreasonable changes of direction, penalty matrices are introduced with the aim of building probability matrices for the different possible directions of movement. These matrices give high probabilities to pixels in the estimated direction and decreasing probabilities (punishment) as the direction varies from the estimated direction.
2 System Evaluation
Evaluation of the temporal algorithm on synthetic data
Creation of synthetic IR frames
To evaluate the performance of the spatial and temporal tracking algorithms, synthetic temporal profiles that simulate different types of clutter and background behavior were created. A target signal was implanted into these background signals to simulate a target traversing both clutter and noise-dominated scenes. On the basis of the study of Silverman et al.  showing that the temporal noise is closely matched to white Gaussian noise, we used white Gaussian noise at various SNRs to test the temporal algorithm.
The base width of the target corresponds to the target velocity. The simulations showed that there were no significant performance differences between the sine and the triangular shaped targets.
Examination of the temporal algorithm on synthetic data
The background type.
The SNR, which is a function of the noise variance and the target's amplitude (factor). SNR is a function of MaxT - the target's peak amplitude.
Parameters of the windowing procedure:
the window size to estimate the background baseline DC: the grouping spatial window size to convert sub-pixel target velocity to the pixel target velocity in the frame (as an input to the DPA)
the size of the short-term variance windows for each sample and for each grouping
the step size of each window (overlapping).
The dependence of the performance of the algorithm on these factors is described below.
The factor most influenced by the background type is the DC estimation capability of the algorithm. It is expected that DC estimation will be easiest for signals having a constant DC level (signals of types 4 and 5) and for signals having a slowly changing DC (signals of types 2 and 3), since the linear regression is capable of estimating parameters of the linear model. Type 1 signals are the most problematic, since the DC of such signals does not have an overall fit with a linear model, but depends on piecewise matching of the DC to windows sizes, as explained below.
As can been seen in Figure 11, the increase in the variance of sub-profiles 2, 8, and 9 may be attributed to the imprecise DC estimation of the background. This case simulates a cloud entering and exiting. Nevertheless, the variance score of the target sub-profiles 5 and 6 is still much higher than that of sub-profiles 2, 8, and 9. The variance of the other sub-profiles 1, 3, 4, and 7 is close to zero.
The DC estimation for signals of types 2 and 3 is precise, since the signal fits a linear model. The variance increases significantly when the target passes through the pixel and is close to zero at other times.
Figures 16, 17, 18, 19, and 20 show the results of temporal processing (variance estimation) for the different signal types, for the cases where no target is present. Not all the sub-profiles are shown, since the values of the exhibited variance values are around zero.
From Figure 16 it can be seen that, by analogy with Figure 11, the variance increases for sub-profiles where the cloud enters and exits into/from frame. Such cases, i.e., with no target, may cause false alarms. Figures 17, 18, 19, and 20 show that the temporal processing score is close to zero for such signals in the absence of a target.
Target/noise (T/N) ratio for various signal types
T/N ratio [Varsano et al. ]
As expected, the performance of the type 1 signal is the worst among the different signals. Signals of types 2-5 all have similar good performance. If we compare our sub-temporal processing algorithm with a temporal processing algorithm as described in , it can be clearly seen (Table 1, 2nd column) that the performance improves by a factor of at least 2 for signals of types 2-5, but not for the type 1 signal, for which the performance improvement is insignificant.
The algorithm responds similarly for signals of types 2-5, in agreement with expectations, i.e., the performance improves as the SNR increases. The performance of the algorithm for the type 1 signal behaves differently; first it increases with the SNR, but at a slower rate than for signals of types 2-5, and it then decreases as a result of an inaccurate DC estimation, as will be detailed later. Since the DC of this signal does not fit a linear model and the estimation must therefore be performed in piecewise fashion, the size and the position of the windows used to perform the estimation act as a limiting factor to the performance.
Both the window size for the baseline DC estimation and the window size for the short-term variance for the sub-profile have a marked impact on the performance of the algorithm. It is expected that large window sizes for baseline DC estimation would improve the DC estimation in cases where the background changes monotonically (as for signals of types 2-4). Too large a DC estimation window size might, in some cases, lead to inaccurate tracking of the clutter form and cause high false alarm rates (e.g., as for type 1 signals). Thus, for background profiles, the optimal window size is determined by the background type, i.e., for a noise-dominated background or backgrounds containing monotonically changing clutter, larger window sizes are preferred; for backgrounds characterized by rapidly changing clutter, shorter windows are preferred. For target temporal profiles, the larger the DC window, the higher the profile score, since the presence of the target peak will have a smaller influence on the DC estimation. Obviously, an estimated DC that tracks the target form is highly undesirable since it leads to target suppression. Thus, in terms of the overall algorithm performance, the optimal DC window is the one that is small enough to closely track background changes but is large enough not to track the target peak.
The sub-temporal profile length should be matched to the target sub-pixel velocity, which is expressed as the base width of the peak of the target profile and the sub-pixel velocity, although there is no acute need for an exact match. The short-term variance window size for the sub-profile should be matched to the target rise time, although once again there is no acute need for an exact match. A sub-profile length that is larger than the target width will disable the ability to track/detect a target with a sub-pixel velocity. Alternatively, a sub-profile length that is smaller than the target width will allow too few samples for the sub-profile.
A short time variance window that is larger than the target rise time will result in a lower score for the target profile, since the variance calculated on each window is normalized by the window's length. Thus, for the target profile, the optimal variance window size is expected to be less than or equal to the sub-profile length. In fact, the shorter the window, the higher the score of the target profile. On the other hand, a short variance window is more sensitive to random noise spikes in a temporal profile dominated by noise. Therefore, the optimal variance window size for noise-dominated profiles should be as large as possible so as to diminish the effect of the noise spikes. The optimal variance window for the overall algorithm's performance is the one offering the best compromise between the need to enhance the target profile score (i.e., as short as possible) and the need to suppress the short-term noise fluctuations (i.e., as long as possible).
Another factor which will impact performance is the overlap window between the sub-profiles. The overlap window should allow for the compensation of low sub-pixel velocity that derives from a small sub-profile length. The overlap window results in the creation of more sub-profiles, as defined in Equation 5, since the greater number of sub-profiles aids to achieve a more accurate tracking estimation.
Evaluation of the temporal processing algorithm on real data
Real IR sequences
Real-world IR image sequences taken from Silverman et al.  were used for evaluating the temporal algorithm. The movies comprised 95 or 100 12-bit IR frames. The sequences contain raw data of unresolved targets flying from Boston Logan Airport in Massachusetts, USA. In the available dataset, there are five scenes containing various types of clutter and sky as well as various targets moving at different velocities.
Description of the IR sequences
Two targets in wispy clouds
One slow target in clear of cloudy scene
One fast target in bright clouds
Two targets in fluffy clouds
One weak target in hot hazy night sky
Silverman et al.  suggested several performance metrics for the evaluation of temporal algorithms. A derived version of the metric was defined. Each frame in the sequence was divided into H × N blocks (30 × 30 were used), and the algorithm was run over nine blocks, i.e., the target block (TB) and its eight adjacent non-target blocks (NTB). The SNRs of the TB and its eight adjacent NTBs were calculated. Thereafter, the algorithm score was calculated on the basis of the resulting S NRs.
where v i, j is the set of pixels belonging to the (i, j)th block, M is a set containing the five pixels with the highest gray level in that block, σ is the standard deviation of the block pixels. E[v i, j ∈ M] is the expected value of the highest pixels (target) and the E[v i, j ∉ M] is the expected value of the rest of the pixels (background).
The block formula performs a subtraction between the expectation value of the highest pixels (target) and the expectation value of the rest of the pixels (background), divided by the standard deviation of block pixels. Since the probability matrices of the DPA introduce the influence of target pixels on adjacent pixels, these influenced pixels might accumulate higher values than unaffected pixels (background), and can be regarded as target pixels. This might lower the expectation value of the target, but will also lower the standard deviation of the background, since these high pixels are higher than the statistics of the background.
The final grade of the algorithm serves as a tool for comparison between the suggested temporal processing algorithm and other temporal processing algorithms that deal with the same problems. The grade is a reflection of the difference between the score of the block containing the target and the expected values of the rest of the blocks in the image, normalized by their standard deviations.
Real data results
Algorithm performance grade for each sequence
Varsano et al. 
The variety of the target scores for the different sequences can be understood by examining the amplitude of the maximum target peak relative to the profiles average values.
Target maximal peak for the different IR sequences
Optimal window size
Choosing the appropriate set of parameters for the temporal algorithm is crucial for the detection capabilities of the system. The dependence of the algorithm on the window size was evaluated on real data, and the optimal set of parameters was obtained for each IR sequence.
The expected optimal window sizes depend on both the shape of the target's temporal profile, mainly on the target's peak width, which is inversely proportional to the target's velocity, and on the background scene, i.e., on the presence of clouds and their size and velocity, as stated in section "Window size".
To determine the optimal set of window sizes on a real data sequence, the algorithm was run on the sequence with various sets of parameters. The set that yielded the highest algorithm grade, defined in Equation 18, was chosen for the evaluation.
Optimal window sets for three IR sequences
Evaluation of the complete system on real data
The hyperspectral movie is created as described in Varsano et al. . The movie consists of a sequence of 30 × 30 × 96 cubes (width × height × bands). A synthetic target is implanted into the sequence. The target is sine-shaped, 2 × 2 pixels wide, and has a horizontal and vertical velocity of 0.1 ppf. White Gaussian noise is added to each spectral signature and the noise variance is set to be [0.1 × Max(signature)]2.
The movie then constitutes the input into our system. The MF with the estimated covariance matrix is applied for the first-stage processing of each hypercube, and the temporal processing algorithm described in section "System architecture" is used for the target detection in the second stage. The output of the second stage constitutes the input into the DPA. The DPA allows us to track the target from pixel to pixel; an updated summation score for each pixel is kept for each pixel, based on its similarity to expected target behavior. Penalties are introduced to lower the score of pixels apparently acting in a non-physical manner. In the final stage, the last processed frame is taken and the highest pixel is declared as a "target" and its track is found. In this section, we will define three tests to evaluate the effect of each stage of the algorithm.
The performance metric, Equations 17 and 18, was used to evaluate various stages of the analysis. The three tests based on this metric were applied. Test 3 uses a MF for the cube collapsing. The resulting cubes are then evaluated as described in Equations 17 and 18; the cube with the highest score will be evaluated as representative of the efficacy of using the MF alone.
Test 4 (full system test) uses a MF detector as the input of the temporal processing velocity filter. Test 5 (full system test) adds to Test 4 the DPA. These tests were created to evaluate the effect of the IR tracking algorithms on the overall score of the hyperspectral tracking system.
The MF and the temporal processing algorithm create images with pixel scores according to their likelihood of being a target, whereas the DPA accumulates the scores of pixels according to the probability of the path going through them to be the target's path.
Discussion of the results obtained on real data
Complete system simulation parameters
Hyperspectral movie parameters
Identical to the ones presented in Varsano et al. , Figure 13
30 × 30 pixels
Number of spectral bands
IR source sequence
NA23, taken from Silverman et al. 
White Gaussian noise factor of 0.05 (std = noise factor * 0.05)
Synthetic target properties
Half sine, 2 × 2 pixels, integral of the spatial distribution normalized to 0.5
Noise factor of 0.25, as described in Varsano et al. , section 6.3
Hyperspectral cube reduction
Match filter, selected according to best performance
37 (weak clutter), 38 (sky only), 39 (strong clutter)
10, 20, 40, 60, 80, 100, 500, 1000
Temporal processing parameters
Sub profile length
DC step size
[0...6] for g = 0, [0...1] for g = 1
Evaluation results of Test 1-spectral average
Partial weak clutter
Mainly strong clutter
Evaluation results of Test 2-scalar product
Partial weak clutter
Mainly strong clutter
Evaluation results of Test 3 - Match filter
Partial weak clutter
Mainly strong clutter
Evaluation results of Test 4-match filter and temporal processing
Partial weak clutter
Mainly strong clutter
Evaluation results of Test 5-match filter, temporal processing and DPA
Partial weak clutter
Mainly strong clutter
Comparison of Tables 9 and 10 shows that scores for Test 4 were lower than those for Test 3. This difference may be attributed to differences in the calculation of the metrics and the temporal processing method, i.e., the difference in the metric calculation due to the fact that we have more score frames and hence calculate their avarage (Figure 25); and as described in Table 1, the temporal processing method used gives us a better target to noise ratio.
Previous research  has shown that Test 1 allows a rough assessment of the relative amplitude of the target pixels vis-à-vis their background. The low values of the results indicate that the implantation of the target and taking the maximal score without any processing is not sufficient for target detection; in other words, the implantation method does not allow an "easy" detection. The highest values of Test 1 were obtained for the weak clutter scenes, which is reasonable since the implantation method is additive and in weak clutter surroundings, the amplitudes are obviously higher than those in clear sky scenes. A comparison of Tests 1 and 2 allowed us to estimate the improvement conferred by using primitive hyperspectral processing, i.e., simply taking the average of all the bands. Although this method led to an improvement in the sky and weak clutter scenes, it had negative impact on strong clutter scenes, a finding that indicates that simply averaging the bands is disastrous for certain sets of spectral signatures and cannot itself be used as a detection method. Thus, the discussion will focus on the use of "smart" hyperspectral processing (Test 3), hyperspectral processing and temporal processing (Test 4), and hyperspectral processing, temporal processing and the DPA (Test 5). The results of Varsano et al.  have shown that there is an obvious advantage of using both hyperspectral detection (MF) and temporal processing (Test 4 vs. Tests 1-3).
When the target is implanted in clear sky scenes, the use of temporal processing significantly improves the performance vis-à-vis hyperspectral detection alone. In most cases, the use of the MF compared to simple averaging was clearly advantageous, with the exception of block 31, for which the performance was similar for the two techniques. This similarity may be attributed to the relative "easiness" of detection in this kind of scene and the fact that the high level of noise might confer a disadvantage on the MF but an advantage on the averaging filter. When weak clutter was present, temporal processing combined with the MF detector was always better than hyperspectral detection alone.
In this study, a complete system for the tracking of dim point targets moving at sub-pixel velocities in a sequence of hyperspectral cubes or, simply put, a hyperspectral movie was presented. Our research incorporates algorithms from two different areas, target detection in hyperspectral imagery and target tracking in IR sequences.
Performance metrics are defined for each step and are used in the analysis and optimization; a comparison is made to previous work in this area.
dynamic programming algorithm
least squares estimation
linear mixture model
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