The time-domain encoding of the intensity information automatically optimizes the exposure time separately for each pixel instead of imposing a fixed integration time for the entire array, resulting in an exceptionally high dynamic range and improved signal to noise ratio. The pixel-individual change detector allows to reduce largely temporal redundancies, resulting in a sparse encoding of the image data. Frames are absent from this acquisition process. They can however be reconstructed, when needed, at frequencies limited only by the temporal resolution of the pixel circuits up to hundreds of kiloframes per second.
Static objects and background information, if required, can be recorded as a snapshot at the start of an acquisition; henceforth, the moving objects in the visual scene describe a spatio-temporal surface at a very high temporal resolution. In the following we will present a general way to apply linear transformations on the change detector events.
The spatio-temporal space of imaging events: Static objects and scene background are acquired first. Then, dynamic objects trigger pixel-individual, asynchronous gray-level events after each change. Samples of generated images from the presented spatio-temporal space are shown in the upper part of the figure.
The AER used in the silicon retina encodes visual information as spatio-temporal events instead of a sequence of frames. This introduces a new paradigm in computer vision. Research on processing techniques suitable for AER has been prolific since these past few years, and several results have been achieved in the use of the silicon retinas. An interesting fact on most of previously published works is the exclusive use of change events to extract useful information from the scene.
One reason for this is that former silicon retinas were able to output only change events. Direct translations of state of the art computer vision algorithms are usually achieved by using the illuminance information estimated by local integration of the change events.
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This approach is adopted by several previous works, for example in using event correlation for stereomatching Kogler et al. The second reason to use only change events is that, for most of machine vision problems, time is proven to be an information medium that substitutes surprisingly well for illuminance.
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Stereovision reformulated for the asynchronous silicon retinas is an interesting example showing that classic projective geometry combined with a high temporal accuracy provide an accurate criterion for matching events and triangulating 3D structures Rogister et al. Tracking algorithms that take advantage of the accurate timing have been developed for event-based visual signals: the event-based reformulation of Hough-transform based circle tracker in Ni et al. Time as the main information medium is emphasized with HFirst Orchard et al. It demonstrates that visual learning can be achieved through temporal information.
This list of event-based signal processing algorithms, while not comprehensive, gives an overview of the state-of-the-art event-based visual signal processing methods.
As mentioned above, these algorithms process only change events. Only a handful of studies dealing directly with the event-based illuminance encoded as gray-levels can be listed so far from the literature. A compressive sensing reconstruction of the illuminance has been implemented on hardware in Orchard et al.
The idea behind it is to exploit the stochastic false change detection due to noise in the ATIS. The high temporal accuracy of the sensor is then traded off to reconstructing the missing illuminance information, and as a result, 28 Hz videos achieving state-of-the-art visual quality are obtained. In Ieng et al.
The illuminance information is a supplementary visual information for the above listed algorithms, but it is a mandatory information for displaying event-based signal in a realistic and human-friendly way.
The use of illuminance information is a step toward a unified formulation of visual signal processing that encompasses both frame-based and event-based representation. Using such an approach one can tackle spatial frequency analysis, image compression and even high dynamic range imaging that are heavily relying on the illuminance information, and explore the impact of the integration of illuminance information to the event-based signal processing.
One important observation should be emphasized about the present paper is the context of this work that focuses exclusively proposing an iterative, event-by-event adaption of the classical basis transformations. A complexity analysis is provided to show the inherent possibility to save computation power thanks to the low redundancy of the event-based signal to process. The problem of sparse representation has been widely tackled by the communities of adaptative and compressive sensing, the main concern of these domains is the initial signal reconstruction from one sparse basis to another one Candes et al.
This is however a totally different problem that we are not aiming to step in as signal reconstruction is an extremly costly offline processing.
Rather, in this work we are aiming to provide an easy to implement and computationaly cheap event-based algorithm that can process events provided by an event-based sensor on the fly. The event-based representation assumes that only a few pixels change at a given time, implying only local updates of the signal content. Let us first consider a one-dimensional sensor whose output x is a column vector of length m. Each linear transform f can be represented by a matrix M for which. On the whole then. Event update step.
Since there are n elements in M i , this event update rule for y takes n multiply-and-accumulate MAC operations. This shows that the event update rule 3 does not introduce overhead in the computations for a general linear transform. This mechanism can be generalized to more complex and non-linear transforms if the assumption of infinitesimal changes of x i holds i. In such a case, we can use a first order approximation to update y :. Let us consider transformations of the form. Many practically important 2D transformations — such as the Fourier transform, the discrete cosine transform DCT , and DWT — can be written in this form.
In other words, the event update changes only the j th column of W , and thus this step requires k MACs. This observation will be useful as we consider wavelet transforms in section 2. The event-based formulation assumes the processing of the data on arrival of each individual event in a sequential manner, however Equation 9 is extendable to events that occur at the same time in an almost straightforward manner. The update equation is the finite sum of the N events contributions:.
As such, the number of MACs is still increasing linearly with the number of events in the set, hence the global complexity is unchanged. However, by extending to a set of simultaneous events, we are actually getting away from the event-based hypothesis and get closer to frame representation.
A strategy to switch to fast and optimized classic transformations FFT,… when they perform better is necessary. In the following we apply the results presented in section 2.
For the discrete cosine transform the transform matrix U equals. In conventional signal processing this form is not computationally the most efficient, and the wavelet transform is preferably implemented with the filter bank approach as introduced in Mallat While the matrix multiplication scheme requires more MACs in the frame-based approach than the filter-bank approach does, we will show that it is an efficient way to perform the wavelet transform using the event-based update Equations 3 and 8.
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Computation of the discrete wavelet transform using consecutive filtering and down-sampling by two. Here h n denotes the high-pass filter and g n denotes the low-pass filter of the wavelet transform. Let us first consider the Haar wavelet transform, which is one of the most important wavelet transforms due to its simplicity. As explained in Appendix 0. The number of MACs required by the 2D transform update step 9 is. The number of bits needed to store H is O n log n , since H is sparse, and the number of MACs required by the event update step is O log n 2 , which compares favorably to the O n 2 MACs required by the update step of a general dense 2D linear transformation.
This complexity reduction is not only due to the event-by-event processing of the data but also because of the sparse structure of H. This is also benefiting frame-based calculation of the wavelet transform. For a general DWT, there is no obvious iterative way based on the Kronecker product to build the matrix H. However, H has a general structure that can be used to determine an upper bound to the number of non-zero elements per column. In this subsection, we analyze the structure of the matrix H and derive an a upper bound to the number of non-zero element in each of its columns.
Let us denote by h and g the finite impulse response filters of the considered DWT. Let us assume that h and g contain only non-zero coefficients, and let p the length of the longer of these two filters. The transform matrix H has then the following structure:. Each of the rows has at most p non-zero elements corresponding to the high-pass filter coefficients.
Wavelet transform matrix at the first level. The left structure is a schematic of the transform matrix split into submatrix A i. The first step is represented by the bottom half of the matrix, A 1 and corresponds to a high-pass filter followed by the downsampling by two elements. These coefficients are obtained by applying the high-pass filter h onto a low-pass filtered and downsampled vector.
Due to the convolution of the high-pass and low-pass filters, each row contains at most 2 p non-zero coefficients, and due to the two separate downsamplings by two, each row is circularly shifted by four steps. Wavelet transform matrix at the second level. Notice that the topmost submatrix is defined from row 1 to row 2 l.
Clarity is King
In the bottom inset the number of MACs per an event update, normalized by the total number of pixels n 2 , is compared to a dense basis transform. As can be seen, the normalized number of MACs decreases with n for the wavelet transform, while this ratio remains constant for the dense transform. Bottom inset: Number of MACs required by a single event update per the total number of pixels n 2. As can be seen, the discrete wavelet transform is much more efficient in terms of numbers of MACs than a general dense transform. The MACs estimation for the standard transforms are established assuming dense and non-symmetric transform matrices in general.
For specific transforms such as Fourier, symmetry can be exploited to produce fast and efficient algorithms such as the FFT.
Theses and dissertations
For transformations like the wavelets' ones, the sparsity is an additional property that should be taken into account. While we only compare the event-based approach with the classic filterbank architecture, it would have been fairer to compare with the improvement introduced in Daubechies and Sweldens by the lifting scheme. However this is not changing fundamentally the results shown in the next section since as reported in Daubechies and Sweldens , the complexity of the lifting scheme is still linear and the number of operations can be reduced up to half of what is needed for the classic filterbank technique.
More complex optimization techniques can help in reducing frame-based wavelet computation: in Andreopoulos and van der Schaar , an incremental wavelet computation is introduced to exploit the idea that a non-exact transform is acceptable if the induced distortion is limited. This strategy is based on finding a compromise between the transformation accuracy and the resource allocated to compute the transformation. Wavelets and Multi Resolution Processing:. Wavelets act as a base for representing images in varying degrees of resolution. Images subdivision means dividing images into smaller regions for data compression and for pyramidal representation.