Reza Rezaie

Reza Rezaie, X. Rong Li

Conditionally Markov (CM) sequences are powerful mathematical tools for modeling random phenomena. There are several classes of CM sequences one of which is the reciprocal sequence. Reciprocal sequences have been widely used in many areas including image processing, intelligent systems, and acausal systems. To use them in application, we need not only their applicable dynamic models, but also some general approaches to designing parameters of dynamic models. Dynamic models governing two important classes of nonsingular Gaussian (NG) CM sequences (called $CM_L$ and $CM_F$ models), and a dynamic model governing the NG reciprocal sequence (called reciprocal $CM_L$ model) were presented in our previous work. In this paper, these models are studied in more detail and general approaches are presented for their parameter design. It is shown that every reciprocal $CM_L$ model can be induced by a Markov model and parameters of the reciprocal $CM_L$ model can be obtained from those of the Markov model. Also, it is shown how NG CM sequences can be represented in terms of an NG Markov sequence and an independent NG vector. This representation provides a general approach for parameter design of $CM_L$ and $CM_F$ models. In addition, it leads to a better understanding of CM sequences, including the reciprocal sequence.

Reza Rezaie, X. Rong Li

The conditionally Markov (CM) sequence contains different classes including Markov, reciprocal, and so-called $CM_L$ and $CM_F$ (two special classes of CM sequences). Each class has its own forward and backward dynamic models. The evolution of a CM sequence can be described by different models. For example, a Markov sequence can be described by a Markov model, as well as by reciprocal, $CM_L$, and $CM_F$ models. Also, sometimes a forward model is available, but it is desirable to have a backward model for the same sequence (e.g., in smoothing). Therefore, it is important to study relationships between different dynamic models of a CM sequence. This paper discusses such relationships between models of nonsingular Gaussian (NG) $CM_L$, $CM_F$, reciprocal, and Markov sequences. Two models are said to be explicitly sample-equivalent if not only they govern the same sequence, but also a one-one correspondence between their sample paths is made explicitly. A unified approach is presented, such that given a forward/backward $CM_L$/$CM_F$/reciprocal/Markov model, any explicitly equivalent model can be obtained. As a special case, a backward Markov model explicitly equivalent to a given forward Markov model can be obtained regardless of the singularity/nonsingularity of the state transition matrix of the model.

Reza Rezaie, X. Rong Li

The conditionally Markov (CM) sequence contains several classes, including the reciprocal sequence. Reciprocal sequences have been widely used in many areas of engineering, including image processing, acausal systems, intelligent systems, and intent inference. In this paper, the reciprocal sequence is studied from the CM sequence point of view, which is different from the viewpoint of the literature and leads to more insight into the reciprocal sequence. Based on this viewpoint, new results, properties, and easily applicable tools are obtained for the reciprocal sequence. The nonsingular Gaussian (NG) reciprocal sequence is modeled and characterized from the CM viewpoint. It is shown that an NG sequence is reciprocal if and only if it is both $CM_L$ and $CM_F$ (two special classes of CM sequences). New dynamic models are presented for the NG reciprocal sequence. These models (unlike the existing one, which is driven by colored noise) are driven by white noise and are easily applicable. As a special reciprocal sequence, the Markov sequence is also discussed. Finally, it can be seen how all CM sequences, including Markov and reciprocal, are unified.

Reza Rezaie, X. Rong Li

In some problems there is information about the destination of a moving object. An example is an airliner flying from an origin to a destination. Such problems have three main components: an origin, a destination, and motion in between. To emphasize that the motion trajectories end up at the destination, we call them \textit{destination-directed trajectories}. The Markov sequence is not flexible enough to model such trajectories. Given an initial density and an evolution law, the future of a Markov sequence is determined probabilistically. One class of conditionally Markov (CM) sequences, called the $CM_L$ sequence (including the Markov sequence as a special case), has the following main components: a joint endpoint density (i.e., an initial density and a final density conditioned on the initial) and a Markov-like evolution law. This paper proposes using the $CM_L$ sequence for modeling destination-directed trajectories. It is demonstrated how the $CM_L$ sequence enjoys several desirable properties for destination-directed trajectory modeling. Some simulations of trajectory modeling and prediction are presented for illustration.

Reza Rezaie, X. Rong Li

Markov processes are widely used in modeling random phenomena/problems. However, they may not be adequate in some cases where more general processes are needed. The conditionally Markov (CM) process is a generalization of the Markov process based on conditioning. There are several classes of CM processes (one of them is the class of reciprocal processes), which provide more capability (than Markov) for modeling random phenomena. Reciprocal processes have been used in many different applications (e.g., image processing, intent inference, intelligent systems). In this paper, nonsingular Gaussian (NG) CM sequences are studied, characterized, and their dynamic models are presented. The presented results provide effective tools for studying reciprocal sequences from the CM viewpoint, which is different from that of the literature. Also, the presented models and characterizations serve as a basis for application of CM sequences, e.g., in motion trajectory modeling with destination information.

Reza Rezaie

A multiple model track-before-detect (TBD) particle filter-based approach for detection and tracking of low signal to noise ratio (SNR) objects based on a sequence of image frames in the presence of noise and clutter is briefly studied in this letter. At each time instance after receiving a frame of image, first, some preprocessing approaches are applied to the image. Then, it is sent to the multiple model TBD particle filter for detection and tracking of an object. Performance of the approach is evaluated for detection and tracking of an object in different scenarios including noise and clutter.

Reza Rezaie

A sequential detection and tracking (SDT) approach is proposed for detection and tracking of very low signal-to-noise (SNR) objects. The proposed approach is compared with two existing particle filter track-before-track (TBD) methods. It is shown that the former outperforms the latter. A conventional detection and tracking (CDT) approach, based on one-data-frame thresholding, is considered as a benchmark for comparison. Simulations demonstrate the performance.

Reza Rezaie, X. Rong Li

Conditionally Markov (CM) sequences are powerful mathematical tools for modeling problems. One class of CM sequences is the reciprocal sequence. In application, we need not only CM dynamic models, but also know how to design model parameters. Models of two important classes of nonsingular Gaussian (NG) CM sequences, called $CM_L$ and $CM_F$ models, and a model of the NG reciprocal sequence, called reciprocal $CM_L$ model, were presented in our previous works and their applications were discussed. In this paper, these models are studied in more detail, in particular their parameter design. It is shown that every reciprocal $CM_L$ model can be induced by a Markov model. Then, parameters of each reciprocal $CM_L$ model can be obtained from those of the Markov model. Also, it is shown that a NG $CM_L$ ($CM_F$) sequence can be represented by a sum of a NG Markov sequence and an uncorrelated NG vector. This (necessary and sufficient) representation provides a basis for designing parameters of a $CM_L$ ($CM_F$) model. From the CM viewpoint, a representation is also obtained for NG reciprocal sequences. This representation is simple and reveals an important property of reciprocal sequences. As a result, the significance of studying reciprocal sequences from the CM viewpoint is demonstrated. A full spectrum of dynamic models from a $CM_L$ model to a reciprocal $CM_L$ model is also presented. Some examples are presented for illustration.