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Grouping variables in an underdetermined system for invariant object recognition


We study the problem of object recognition invariant to transformations, such as translation, rotation and scale. A system is underdetermined if its degrees of freedom (number of possible transformations and potential objects) exceed the available information (image size). The regularization theory solves this problem by adding constraints [1]. It is unclear what constraints biological systems use. We suggest that rather than seeking constraints, an underdetermined system can make decisions based on available information by grouping its variables. We propose a dynamical system as a minimum system for invariant recognition to demonstrate this strategy.

A dynamical system for invariant recognition

Assume there are q objects in the gallery, and p possible transformations. An input image I is generated by one of the objects through a transformation. The task is to recover the object and the transformation that generate I. The system variables are C = (c1,..., c p )Tfor transformation and D = (d1,..., d q )Tfor object selection. When p + q > n, where n is the size of the image, the system is underdetermined, having many solutions.

Our system structure is shown in Figure 1. The state variables C and D follow the dynamics described by a system of linear differential equations. Figure 2 top row shows a solution of a toy system (n = 8*8, p = 72, q = 2), with I generated by c1 = 1, d1 = 1. In this underdetermined system, the solution depends on the initial condition.

Figure 1
figure 1

System architecture.

Figure 2
figure 2

Solutions ( C and D ) in a toy system.

The system can be made overdetermined by grouping variables such that all variables within a group share the same dynamics. When the total activity of the system is below a predefined level, we then let the variables in the top group resume their individual dynamics. Under this dynamics with grouping, the solution to the same toy system is shown in Figure 2 bottom row. It is close to the true value.


Our example shows that, in an underdetermined system for invariant recognition, it is plausible to recover a sparse solution by grouping variables and then fine-tune the winning group. The applicability of this strategy depends on the structure of transformations and of objects. Our system could provide a model system to study the coarse-to-fine processing which is evident in biological systems [2].


  1. Poggio T, Koch C: Ill-posed problems in early vision: From computational theory to analog networks. Proceedings of the Royal Society London B. 1985, 226: 303-323. 10.1098/rspb.1985.0097.

    Article  Google Scholar 

  2. Hegdé J: Time course of visual perception: Coarse-to-fine processing and beyond. Progress in Neurobiology. 2008, 84: 405-439. 10.1016/j.pneurobio.2007.09.001.

    Article  PubMed  Google Scholar 

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Supported by EU project "SECO" and the Hertie Foundation.

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Correspondence to Junmei Zhu.

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Open Access This article is published under license to BioMed Central Ltd. This is an Open Access article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Zhu, J., Malsburg, C.v.d. Grouping variables in an underdetermined system for invariant object recognition. BMC Neurosci 10 (Suppl 1), P308 (2009).

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