A new Truncated Fourier Transform algorithm

Truncated Fourier Transforms (TFTs), first introduced by Van der Hoeven, refer to a family of algorithms that attempt to smooth "jumps" in complexity exhibited by FFT algorithms. We present an in-place TFT whose time complexity, measured in terms of ring operations, is comparable to existing not-in-place TFT methods. We also describe a transformation that maps between two families of TFT algorithms that use different sets of evaluation points.Comment: 8 pages, submitted to the 38th International Symposium on Symbolic and Algebraic Computation (ISSAC 2013

Transformation Based Ensembles for Time Series Classification

Until recently, the vast majority of data mining time series classification (TSC) research has focused on alternative distance measures for 1-Nearest Neighbour (1-NN) classifiers based on either the raw data, or on compressions or smoothing of the raw data. Despite the extensive evidence in favour of 1-NN classifiers with Euclidean or Dynamic Time Warping distance, there has also been a flurry of recent research publications proposing classification algorithms for TSC. Generally, these classifiers describe different ways of incorporating summary measures in the time domain into more complex classifiers. Our hypothesis is that the easiest way to gain improvement on TSC problems is simply to transform into an alternative data space where the discriminatory features are more easily detected. To test our hypothesis, we perform a range of benchmarking experiments in the time domain, before evaluating nearest neighbour classifiers on data transformed into the power spectrum, the autocorrelation function, and the principal component space. We demonstrate that on some problems there is dramatic improvement in the accuracy of classifiers built on the transformed data over classifiers built in the time domain, but that there is also a wide variance in accuracy for a particular classifier built on different data transforms. To overcome this variability, we propose a simple transformation based ensemble, then demonstrate that it improves performance and reduces the variability of classifiers built in the time domain only. Our advice to a practitioner with a real world TSC problem is to try transforms before developing a complex classifier; it is the easiest way to get a potentially large increase in accuracy, and may provide further insights into the underlying relationships that characterise the problem

Time-causal and time-recursive spatio-temporal receptive fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and Vision, published online Dec 201

Cortical Specification of a Fast Fourier Transform Supports a Convolution Model of Visual Perception

Currently, the full extent of the role Fourier analysis plays in biological vision is unclear. Although we have examples of sensory organs that perform Fourier transforms, e.g. the lens of the eye and the cochlear, to date there is no direct empirical evidence for its implementation in cortical architecture. However, there does exist intriguing theoretical evidence that suggests a role for the Fourier transform in a primate’s primary visual cortex (area V1) which emerges from recent developments in our knowledge of contextual modulation. This paper proposes a new Fourier transform and a specification of how this transform has a natural implementation in cortical architecture. The significance of this new Fourier transform and its specification in neural circuitry is that it provides a plausible explanation for previously unexplained observable properties of the primate vision system.Full Tex

Idealized computational models for auditory receptive fields

This paper presents a theory by which idealized models of auditory receptive fields can be derived in a principled axiomatic manner, from a set of structural properties to enable invariance of receptive field responses under natural sound transformations and ensure internal consistency between spectro-temporal receptive fields at different temporal and spectral scales. For defining a time-frequency transformation of a purely temporal sound signal, it is shown that the framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters, with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal temporal window functions. When applied to the definition of a second-layer of receptive fields from a spectrogram, it is shown that the framework leads to two canonical families of spectro-temporal receptive fields, in terms of spectro-temporal derivatives of either spectro-temporal Gaussian kernels for non-causal time or the combination of a time-causal generalized Gammatone filter over the temporal domain and a Gaussian filter over the logspectral domain. For each filter family, the spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Within each domain of either non-causal or time-causal time, these receptive field families are derived by uniqueness from the assumptions. It is demonstrated how the presented framework allows for computation of basic auditory features for audio processing and that it leads to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields measured in the inferior colliculus (ICC) and primary auditory cortex (A1) of mammals.Comment: 55 pages, 22 figures, 3 table