1786 IEICE TRANS. FUNDAMENTALS, VOL.E94–A, NO.9 SEPTEMBER 2011 PAPER Asymptotically Optimum Quadratic Detection in the Case of Subpixel Targets Victor GOLIKOV †a) , Member, Olga LEBEDEVA † , Andres CASTILLEJOS MORENO †† , Nonmembers, and Volodymyr PONOMARYOV †† , Member SUMMARY This work extends the optimum Neymann-Pearson methodology to detection of a subspace signal in the correlated additive Gaussian noise when the noise power may be different under the null (H 0 ) and alternative (H 1 ) hypotheses. Moreover, it is assumed that the noise co- variance structure and power under the null hypothesis are known but under the alternative hypothesis the noise power can be unknown. This situation occurs when the presence of a small point (subpixel) target decreases the noise power. The conventional matched subspace detector (MSD) neglects this phenomenon and causes a consistent loss in the detection performance. We derive the generalized likelihood ratio test (GLRT) for such a detection problem comparing it against the conventional MSD. The designed detec- tor is theoretically justified and numerically evaluated. Both the theoreti- cal and computer simulation results have shown that the proposed detector outperforms the conventional MSD. As to the detection performance, it has been shown that the detectivity of the proposed detector depends on the additional adaptive corrective term in the threshold. This corrective term decreases the value of presumed threshold automatically and, therefore, in- creases the probability of detection. The influence of this corrective term on the detector performance has been evaluated for an example scenario. key words: subpixel targets, subpixel matched subspace detector 1. Introduction Target detection in the remotely sensed image sequences can be conducted spatially, temporally or spectrally. The need for subpixel temporally (or spectrally) detection in remotely sensed image sequences arises from the fact that the ground sampling distance is generally larger than the size of targets of interest. In this case, the targets are embedded in a single pixel sequence and cannot be detected spatially. As a result, traditional spatial-temporal analysis-based image sequence processing techniques are not applicable. We tackle the problem of subpixel for the objects de- tection in the image sequences, which arises for instance, in the electro-optical or in the infrared search and track appli- cations [1]–[13]. In many instances, the objects are small and partially obscured in a severe image-to-image fluctuat- ing background environment. For example, in the airborne visible/infrared imaging spectrometer, the targets of interest are generally smaller than the spatial resolution of hyper- spectral images (e.g., pixel resolution 20 m) [10]. When the object size is smaller than the pixel resolution, some exam- Manuscript received November 30, 2010. Manuscript revised April 13, 2011. † The authors are with UNACAR, Ciudad del Carmen, Camp., Mexico. †† The authors are with National Polytechnic Institute, ESIME- Culhuacan, Mexico, D.F. a) E-mail: vgolikov@pampano.unacar.mx DOI: 10.1587/transfun.E94.A.1786 Fig. 1 Typical target detection scenario. (a) Resolved target occupying a whole pixel, (b) subpixel target in pure background with background fill factor of b ≈ 0.5. ple are: small boats on a marine surface, people or animals on a ground (or sea) surface, small vehicles in battlefields, etc., the data analysis must rely solely on temporal or spec- tral information that can only be obtained and provided by a single pixel sequence [5], [7], [9], [12] and [13]. The ship- based video automatic detection of small floating objects (swimmers, small boats, and semi-submersible snorkels) on an agitated sea surface is other example of the subpixel tar- get detection, too. The pixel resolution can be approxi- mately 5 m at the distance of 1000 m, that why, these targets are smaller than the spatial resolution [13]. According to common sensor design, the small object is located in a single pixel where the background power de- pends on the object size and its position within a pixel area (see Fig. 1). Therefore, a non-complete knowledge of a sig- nal to be detected can lead to background power variation under hypothesis H 1 . Therefore, the background power in the target presence can be unknown and can produce neg- ative impact on a detection performance as it was shown in [9], [12]. To our knowledge, this pitfall has not been investigated yet completely in the literature. A prevailing opinion stands that the background powers or their relation is known. A lot of investigations exist dealing with small objects detection that are concentrated on clutter removing [1]–[3], multi- or hyper-spectral fusion [4], [5] and multi- frame tracking frameworks [5]–[7]. The common drawback of these works is the assumption that the background power under hypothesis H 0 remains the same value as under hy- pothesis H 1 . In optical systems, it is typically that the back- ground has the same covariance structure under hypotheses H 0 and H 1 , but different variances where the subpixel target occupies only a part of the pixel area [9], which is directly related to the fill factor of a background (the percentage of the pixel area occupied by a background). Copyright c  2011 The Institute of Electronics, Information and Communication Engineers