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Closed and Open Maps for Partial Frames
This paper concerns the notions of closed and open maps in the setting of partial frames, which, in contrast to full frames, do not necessarily have all joins. Examples of these include bounded distributive lattices,
σ
- and
κ
-frames and full frames. We define closed and open maps using geometrically intuitively appealing conditions involving preservation of closed, respectively open, congruences under certain maps. We then characterize them in terms of algebraic identities involving adjoints. We note that partial frame maps need have neither right nor left adjoints whereas frame maps of course always have right adjoints. The embedding of a partial frame in either its free frame or its congruence frame has proved illuminating and useful. We consider the conditions under which these embeddings are closed, open or skeletal. We then look at preservation and reflection of closed or open maps under the functors providing the free frame or the congruence frame. Points arise naturally in the construction of the spectrum functor for partial frames to partial spaces. They may be viewed as maps from the given partial frame to the 2-chain or as certain kinds of filters; using the former description we consider closed and open points. Any point of a partial frame extends naturally to a point on its free frame and a point on its congruence frame; we consider the closedness or openness of these.
Journal Article
Reproducing formulas for generalized translation invariant systems on locally compact abelian groups
In this paper we connect the well-established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we let Γj\\Gamma _{\\!j}, j∈Jj \\in J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian group GG and study systems of the form ⋃j∈J{gj,p(⋅−γ)}γ∈Γj,p∈Pj\\bigcup _{j \\in J}\\{ g_{j,p}(\\cdot - \\gamma )\\}_{\\gamma \\in \\Gamma _{\\!j}, p \\in P_j} with generators gj,pg_{j,p} in L2(G)L^2(G) and with each PjP_j being a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technical α\\alpha local integrability condition (α\\alpha-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas for L2(G)L^2(G). This generalizes results on generalized shift invariant systems, where each PjP_j is assumed to be countable and each Γj\\Gamma _{\\!j} is a uniform lattice in GG, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform lattices Γj\\Gamma _{\\! j}, our characterizations improve known results since the class of GTI systems satisfying the α\\alpha-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames for L2(G)L^2(G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform in L2(Rn)L^2(\\mathbb {R}^n) are special cases of the same general characterizing equations.
Journal Article
Constructing relay fusion frames in Hilbert spaces
In this work, we start by introducing a general methodology to generate new relay fusion frames from given ones, namely the Spatial Complement Method, and analyze the relationships between the parameters of the original and the new relay fusion frame. We then present another simple approach to obtain relay fusion frames by considering fusion frames for its components. An explicit characterization concerning the existence of Parseval relay fusion frame consisting of two initial subspaces is given. Moreover, we obtain a necessary and sufficient condition under which the spatial complements of alternate dual relay fusion frames remain to be alternate dual relay fusion frames. Some results about Bessel relay fusion sequences are included. Finally, several examples are also given.
Journal Article
Towards Frame Rate Agnostic Multi-object Tracking
Multi-object Tracking (MOT) is one of the most fundamental computer vision tasks that contributes to various video analysis applications. Despite the recent promising progress, current MOT research is still limited to a fixed sampling frame rate of the input stream. They are neither as flexible as humans nor well-matched to industrial scenarios which require the trackers to be frame rate insensitive in complicated conditions. In fact, we empirically found that the accuracy of all recent state-of-the-art trackers drops dramatically when the input frame rate changes. For a more intelligent tracking solution, we shift the attention of our research work to the problem of Frame Rate Agnostic MOT (FraMOT), which takes frame rate insensitivity into consideration. In this paper, we propose a Frame Rate Agnostic MOT framework with a Periodic training Scheme (FAPS) to tackle the FraMOT problem for the first time. Specifically, we propose a Frame Rate Agnostic Association Module (FAAM) that infers and encodes the frame rate information to aid identity matching across multi-frame-rate inputs, improving the capability of the learned model in handling complex motion-appearance relations in FraMOT. Moreover, the association gap between training and inference is enlarged in FraMOT because those post-processing steps not included in training make a larger difference in lower frame rate scenarios. To address it, we propose Periodic Training Scheme to reflect all post-processing steps in training via tracking pattern matching and fusion. Along with the proposed approaches, we make the first attempt to establish an evaluation method for this new task of FraMOT. Besides providing simulations and evaluation metrics, we try to solve new challenges in two different modes, i.e., known frame rate and unknown frame rate, aiming to handle a more complex situation. The quantitative experiments on the challenging MOT17/20 dataset (FraMOT version) have clearly demonstrated that the proposed approaches can handle different frame rates better and thus improve the robustness against complicated scenarios.
Journal Article