Maciej Liśkiewicz

On the complexity of affine image matching

The problem of image matching is to find for two given digital images A and B an admissible transformation that converts image A as close as possible to B. This problem becomes hard if the space of admissible transformations is too complex. Consequently, in many real applications, like the ones allowing nonlinear elastic transformations, the known algorithms solving the problem either work in exponential worst-case time or can only guarantee to find a local optimum. Recently Keysers and Unger have proved that the image matching problem for this class of transformations is NP-complete, thus giving evidence that the known exponential time algorithms are justified. On the other hand, allowing only such transformations as translations, rotations, or scalings the problem becomes tractable. In this paper we analyse the computational complexity of image matching for a larger space of admissible transformations, namely for all affine transformations. In signal processing there are no efficient algorithms known for this class. Similarly, the research in combinatorial pattern matching does not cover this set of transformations neither providing efficient algorithms nor proving intractability of the problem, although it is a basic one and of high practical importance. The main result of this paper is that the image matching problem can be solved in polynomial time even allowing all affine transformations.

Space efficient algorithms for directed series-parallel graphs

The subclass of directed series-parallel graphs plays an important role in computer science. Whether a given graph is series-parallel is a well studied problem in algorithmic graph theory, for which fast sequential and parallel algorithms have been developed in a sequence of papers. Also methods are known to solve the reachability and the decomposition problem for series-parallel graphs time efficiently. However, no dedicated results have been obtained for the space complexity of these problems when restricted to series-parallel graphs - the topic of this paper.Deterministic algorithms are presented for the recognition, reachability, decomposition and the path counting problem for series-parallel graphs that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in Logspace, the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterization of series-parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problem is shown to be L-complete with respect to AC0-reductions. They have also implications for the parallel time complexity of these problems when restricted to series-parallel graphs.Finally we sketch how these results can be generalized to extension of the series-parallel graph family: to graph with multiple sources or multiple sinks and to the minimal vertex series-parallel graphs.

Negative selection algorithms without generating detectors

Negative selection algorithms are immune-inspired classifiers that are trained on negative examples only. Classification is performed by generating detectors that match none of the negative examples, and these detectors are then matched against the elements to be classified. This can be a performance bottleneck: A large number of detectors may be required for acceptable sensitivity, or finding detectors that match none of the negative examples may be difficult. In this paper, we show how negative selection can be implemented without generating detectors explicitly, which for many detector types leads to polynomial time algorithms whereas the common approach to sample detectors randomly takes exponential time in the worst case. In particular, we show that negative selection on strings with generating all detectors can be efficiently simulated without detectors if, and only if, an associated decision problem can be answered efficiently, regardless the detector type. We also show how to efficiently simulate the more general case in which only a limited number of detectors is generated. For many detector types this non-exhaustive negative selection is more meaningful but it can be computationally more difficult, which we illustrate using Boolean monomials.

Combinatorial Bounds and Algorithmic Aspects of Image Matching under Projective Transformations

Image matching is an important problem in image processing and arises in such diverse fields as video compression, optical character recognition, medical imaging, watermarking etc. Given two images, image matching determines a transformation that changes the first image such that it most closely resembles the second. Common approaches require either exponential time, or find only an approximate solution, even when only rotations and scalings are allowed. This paper provides the first general polynomial time algorithm to find the exact solution to the image matching problem under projective, affine or linear transformations. Subsequently, nontrivial lower bounds on the number of different transformed images are given which roughly induce the complexity of image matching under the three classes of transformations.

Two-Dimensional Pattern Matching with Combined Scaling and Rotation

The problem of two-dimensional pattern matching invariant under a given class of admissible transformations

Provably secure steganography and the complexity of sampling

\mathcal{F}

Privacy in Non-private Environments

is to find in text Tmatches of transformed versions f(P) of the pattern P, for all fin

Reversal complexity classes for alternating turing machines

\mathcal{F}

On reversal bounded alternating turing machines

. In this paper, pattern matching invariant under compositions of real scaling and rotation are investigated. We give a new discretization technique for this class of transformations and prove sharp lower and upper bounds on the number of different possibilities to transform a pattern in this way. Subsequently, we present the first efficient pattern matching algorithm invariant under compositions of scaling and rotation. The algorithm works in time O(m2n6) for patterns of size m2and texts of size n2. Our method can also be applied to the image matching problem, the well known issue in the image processing research.

Fast simulations of time-bounded one-tape Turing machines by space-bounded ones

Recent work on theoretical aspects of steganography resulted in the construction of oracle-based stegosystems. It has been shown that these can be made secure against the steganography equivalents of common cryptographic attacks. In this paper we use methods from complexity theory to investigate the efficiency of sampling from practically relevant types of channels. We show that there are channels that cannot be efficiently used in oracle-based stegosystems. By classifying channels based on their usability for stegosystems, we provide a means to select suitable channels for their practical implementation.