Patent Application
20110276612
The present invention relates to a method, a device, a computer program and a computer program product for determining a representation of a signal and in particular a compressed representation of a signal.
Traditional capture and processing of analogue signals consists of two steps: sample, then compress. If the signal is bandlimited, then sample at a little over the Nyquist frequency, namely at twice the frequency range. Lossy compression, for example JPEG for an image, then throws away lots of redundant information. Compressive sampling is a new and promising method to simultaneously sample and compress a signal.
In E. J. Candès and M. Wakin: “An introduction to compressive sampling”, IEEE Signal Processing Magazine, 25:21-30, March 2008, a method for compressive sampling is described. A signal, which is sparse under a given basis, is sampled incoherently under a measurement basis. Measurements are taken uniformly at random. A number of measurements is much smaller than a length of the signal.
It is a challenge to provide a method, a device, a computer program and a computer program product for determining a representation of a signal that is simple and that allows for an efficient representation of the signal.
According to a first aspect of the invention, a method for determining a representation of a signal and a corresponding device, computer program and computer program product are provided. The method comprises selecting a predetermined number of row vectors from a predetermined measurement matrix. The predetermined measurement matrix is predetermined dependent on a product of a predetermined Hadamard matrix or generalized Hadamard matrix and a predetermined representation matrix. The predetermined representation matrix represents a predetermined basis for the signal. The method further comprises determining a respective inner product of the signal and each of the predetermined number of selected row vectors resulting in a predetermined number of measurements. These predetermined number of measurements form the representation of the signal. The device is operable to perform this method. The computer program product comprises a computer readable medium embodying program instructions executable by a computer to perform this method. The device particularly comprises the computer or is designed as a computer. The computer program comprises program instructions being adapted to perform this method. The computer program may be executable by the device. The method may be implemented in hardware, that is as a circuitry specifically designed or adapted to perform the method.
The advantage is that by using the predetermined measurement matrix that is predetermined dependent on the product of the predetermined Hadamard matrix or generalized Hadamard matrix and the predetermined representation matrix the predetermined measurement matrix is maximally incoherent with respect to the predetermined representation matrix. This enables the predetermined number of measurements to be predetermined smaller than compared to using another measurement matrix without maximal incoherence with respect to the predetermined representation matrix. The representation of the signal may thus be efficient. Further, the method is simple. By this, the device, the computer program and the computer program product may also be simple.
The term “predetermined” means in the context of this application that the corresponding entity may be determined outside the described method, that is, the entity is predetermined with respect to the described method but is not necessarily a constant. For example, the term “predetermined number” may thus include that the number is determined dependent on the signal and can thus be different for different signals.
According to a second aspect of the invention, a method for determining a representation of a signal and a corresponding device, computer program and computer program product are provided. The method comprises determining a first transform dependent on the signal by multiplying a predetermined representation matrix representing a predetermined basis for the signal with the signal or by applying a first transform algorithm based on the predetermined basis for the signal to the signal. The method further comprises determining a second transform dependent on the first transform by multiplying a predetermined Hadamard matrix or generalized Hadamard matrix with the first transform or by applying as a second transform algorithm a Hadamard transform or fast Hadamard transform to the first transform. The method further comprises selecting a predetermined number of measurements from the second transform. These predetermined number of measurements form the representation of the signal. The device is operable to perform this method. The computer program product comprises a computer readable medium embodying program instructions executable by a computer to perform this method. The device particularly comprises the computer or is designed as a computer. The computer program comprises program instructions being adapted to perform this method. The computer program may be executable by the device. The method may be implemented in hardware, that is as a circuitry specifically designed or adapted to perform the method.
This method, device, computer program and computer program product according to the second aspect essentially are equivalent to the method, device, computer program and computer program product, respectively, according to the first aspect. The advantage is that by determining the first transform based on the predetermined basis for the signal and by determining the second transform based on the predetermined Hadamard matrix or generalized Hadamard matrix or the Hadamard transform or fast Hadamard transform the predetermined number of measurements are taken with respect to a basis that is maximally incoherent with respect to the predetermined basis for the signal. This enables the predetermined number of measurements to be predetermined smaller than compared to using another measurement basis without maximal incoherence with respect to the predetermined basis for the signal. The representation of the signal may thus be efficient. Further, the method is simple. By this, the device, the computer program and the computer program product may also be simple.
According to a preferred embodiment of the first or second aspect, a Fourier matrix is used as the generalized Hadamard matrix. The advantage is that efficient and/or fast implementations exist for handling Fourier matrices. Thus, the method can be implemented very efficiently, in software as well as in hardware. A further advantage is that the Fourier matrix exists in any size and is therefore suited for processing signals of any length.
According to a further preferred embodiment of the first or second aspect, the selection of the predetermined number of the row vectors or of the measurements comprises selecting the predetermined number of the row vectors and of the measurements, respectively, uniformly at random from all row vectors of the predetermined measurement matrix and from all elements of the second transform, respectively. The advantage is that this is simple.
According to a further preferred embodiment of the first or second aspect, the predetermined representation matrix or predetermined basis for the signal is being selected such that the signal is k-sparse or with respect to a predetermined threshold approximately k-sparse under the predetermined basis for the signal with a sparsity being smaller than a length of the signal. The advantage is that the predetermined number of measurements may be predetermined the smaller the smaller the sparsity of the signal is with respect to the predetermined basis for the signal. Preferably, the sparsity is less than ten percent of the length of the signal and is further preferably less than five percent of the length of the signal. The representation of the signal may thus be especially efficient.
The invention and its embodiments will be more fully appreciated by reference to the following detailed description of presently preferred but nonetheless illustrative embodiments in accordance with the present invention when taken in conjunction with the accompanying drawings.
The figures are illustrating:
The predetermined representation matrix B represents a predetermined basis for the signal s. Preferably, the predetermined representation matrix B is a unitary n×n matrix. Preferably, the signal s is sparse or approximately sparse under the predetermined basis. Particularly, the signal s is k-sparse or approximately k-sparse under the predetermined basis with a sparsity k that is smaller than the length n of the signal s and that preferably is much smaller than the length n of the signal s. The signal s has a k-sparse expansion under the predetermined basis if all but the sparsity k of its values are zero. The signal s has an approximately k-sparse expansion under the predetermined basis if all but the sparsity k of its values have a magnitude of less than a predetermined threshold. This predetermined threshold may be predetermined application dependent. The predetermined threshold generally is predetermined such that the loss of information due to ignoring values below the predetermined threshold is acceptable with regard to the application. Preferably, most but a few values of the expansion of the signal under the predetermined basis are zero or have a magnitude of less than the predetermined threshold. Preferably, the sparsity k amounts to less than ten percent of the length n of the signal s and further preferably amounts to less than five percent of the length n of the signal s. The signal s may then be compressed effectively. The predetermined representation matrix B preferably is not an identity matrix, that is, a product of any Hadamard matrix or generalized Hadamard matrix H and the predetermined representation matrix B preferably always differs from the Hadamard matrix or generalized Hadamard matrix H used in the calculation of the product. The same applies to a transform corresponding to the predetermined representation matrix B, for example a first transform b. An output of this transform preferably always differs from its input.
The predetermined number m preferably is predetermined to be smaller than the length n of the signal s and is preferably predetermined to be much smaller than the length n of the signal s. Preferably, the predetermined number m is predetermined dependent on the sparsity k, that is, the smaller the sparsity k the smaller the predetermined number m and vice versa. As a rule of thumb the predetermined number m may be predetermined to amount to about four times the sparsity k. However, the predetermined number m may be predetermined to be smaller or greater than four times the sparsity k. The predetermined number m may be predetermined dependent on a reconstruction algorithm used for reconstructing the signal s from a predetermined number m of measurements y1, . . . , ym that form the representation y of the signal s and/or dependent on requirements with respect to the application.
Generally, a generalized Hadamard matrix is defined as a n×n matrix satisfying H.H*=nI and with all its entries being roots of unity. In this context, H* represents a conjugate transpose of the generalized Hadamard matrix H and I represents an identity matrix. A complex number w is called a rth root of unity if wr=1. Generally, a Hadamard matrix is a special case of the generalized Hadamard matrix with all its entries being real values, that is +1 or −1. Generally, a generalized Hadamard matrix exists for all lengths n of the signal s. Further, it is hypothesized that a Hadamard matrix exists for the length n of the signal s being 2 or a multiple of 4. For the length n being a power of two the predetermined Hadamard matrix H may be easily constructed with a Sylvester construction for which also a fast implementation exists. Alternatively, a Paley construction may be used when the length n of the signal s amounts to q+1 with q being any prime power congruent to 1 or 3 modulo 4, that is, q=p1 for some positive integer 1 and prime p and q=1 (mod 4) or q=3 (mod 4). The predetermined generalized Hadamard matrix H may for example be a Fourier matrix with elements wij, with rows i and columns j=0, . . . , n−1, where the complex w=cos (2π/n)+i·sin (2π/n) is a i·j-root of unity. However, preferably, the predetermined generalized Hadamard matrix H is not a Fourier matrix if the predetermined representation matrix B is the identity matrix.
In a step S3a a predetermined measurement matrix M is provided. The predetermined measurement matrix M represents a predetermined measurement basis for taking the predetermined number m of measurements y1, . . . , ym that form the representation y of the signal s. The predetermined measurement matrix M is predetermined dependent on the predetermined Hadamard matrix or generalized Hadamard matrix H and the predetermined representation matrix B. Particularly, as shown as a step S3b, the predetermined measurement matrix M is predetermined dependent on a product of the predetermined Hadamard matrix or generalized Hadamard matrix H and the predetermined representation matrix B. By this, the predetermined measurement matrix M is maximally incoherent with respect to the predetermined representation matrix B. This holds for any predetermined representation matrix B. This allows for the minimal predetermined number m of measurements y1, . . . , ym and thus for the representation y of the signal s being very efficient. Generally, if the predetermined measurement matrix M is a unitary n×n matrix, the predetermined measurement matrix M is maximally incoherent to another unitary n×n matrix, that is the predetermined representation matrix B, if and only if sqrt(n)·B·MT is a generalized Hadamard matrix, with sqrt(n) representing a square root of the length n of the signal s and MT representing a transpose of the predetermined measurement matrix M. A coherence between the predetermined representation matrix B and the predetermined measurement matrix M may be defined as μ(B,M)=sqrt(n)·max |<uj,vi>| for 1≦i, j≦n with rows i, j, that is, the coherence is a maximum absolute value of all inner products of row vectors uj of the predetermined representation matrix B and row vectors vi of the predetermined measurement matrix M. In other words, the coherence measures the largest correlation between any two row vectors uj, vi of the predetermined representation matrix B and predetermined measurement matrix M. The maximal incoherence then is achieved when the coherence is smallest.
A step S4 may be provided to provide the signal s. In a step S5, a predetermined number m of row vectors v1, . . . , vm of the predetermined measurement matrix M are selected from all available row vectors vi of the predetermined measurement matrix M. In a step S6, an inner product of the signal s and each of the predetermined number m of the row vectors v1, . . . , vm is respectively determined. Each of these inner products forms one of the predetermined number m of measurements y1, . . . , ym that form the representation y of the signal s. Because the predetermined number m preferably is smaller and particularly much smaller than the length n of the signal s, the representation y of the signal s generally is a compressed representation y of the signal s. The method ends with a step S7.
The selection of the predetermined number m of the row vectors v1, . . . , vm of the predetermined measurement matrix M may be performed randomly and preferably uniformly at random. However, the selection may alternatively be performed differently, for example dependent on requirements of the reconstruction algorithm.
In a step S13 the first transform b is determined. Two alternatives for this are presented in
In a step S14 a second transform h is determined. Two alternatives for this are presented in
In steps S13 and S14 the first alternative of step S13 may be combined with the first or the second alternative of step S14. Similarly, the second alternative of step S14 may be combined with the first or the second alternative of step S14.
In a step S15 the predetermined number m of measurements y1, . . . , ym are selected from the second transform h. The selection of the predetermined number m of the measurements y1, . . . , ym of the second transform h may be performed randomly and preferably uniformly at random. However, the selection may alternatively be performed differently, for example dependent on requirements of the reconstruction algorithm.
The methods described with reference to
In one embodiment, the at least one signal source SRC may be located within the at least one processing unit CPU and, for example, may be designed as a counter. The values of the respective counter, that is the signal s, for example, may be used for a performance monitoring of the respective processing unit CPU and/or device DEV. Compressive sampling may be used for efficient gathering of performance information in a multicore system, that is a system with at least two processing cores per processing unit CPU or with at least two processing units CPU with at least one processing core. Modern processing units CPU contain multiple interacting parts making predicting an actual average of clock cycles per instruction for an execution of a particular piece of code difficult to achieve. This has led to the introduction of an additional unit to the processing unit CPU, a performance monitor unit that samples a performance of the rest of the processing unit CPU. Modern performance monitor units may contain hundreds of counters for indicating bus usage, caches misses, page faults, branch misses etc, as well as complex logic for determining which of the many pipelined instruction and predicted branches actually causes the counter to be increased. The performance monitor units on the distinct processing units CPU need to supply information to a task scheduler. Also, the logic of the performance monitor unit must be kept simple so as to reduce the amount of the core they occupy, that the software that extracts data from them must not consume too many cycles and the distribution of the data must not use too much memory bandwidth, particularly because the data must be gathered and processed on some centralized task scheduler running on one of the processing units CPU. One would expect data from the performance monitor unit to be heavily structured, as the behavior of threads in a recent pass is likely to resemble a present pass. Hence it should be compressible. Compression would be difficult to achieve using standards compression techniques without increasing the amount of storage on the performance monitor unit and introducing delays inappropriate with the needs of task scheduling. If the nature of the signal s is known it is possible to sample only the most important aspects of it to get an immediately compressed representation y of the signal s. The average clock cycles per instruction of the respective processing unit CPU, for example, represents the average number of cycles over some time period required to execute an instruction. Although the actually average clock cycles per instruction is discrete the rate at which it changes is so much faster than any realistic sampling period that it can be considered as a continuous function of time. Compressive sampling may be used to determine the compressed representation y of this actually average clock cycles per instruction as the signal s allowing for a simple performance monitor unit and low requirements for storing and bandwidth.
Another application deals with processing data handled in a streaming fashion, for example in streaming databases etc., and performing queries in these. This can be a daunting operation when there are huge amounts of data at stake. This is a problem that arises, for example, when handling network data information, where a typical problem is to return a list of “heavy-hitters”, the top flows responsible for a large amount of the traffic. Another important area is the financial markets such as storage and analysis of stock and FX rates. In any streaming database system there is the data gathering—the what and how to keep something—and the query mechanism—searching through what has been gathered. Compressive sampling, or variations thereof, is a promising technique for determining the what to keep and how to search.
Compressive sampling may further have many imaging applications, for example in magnetic resonance imaging, or short: MRI, and general camera sensor devices, or applications in audio processing. Use of compressive sampling may thus enable development of new cameras, medical image devices, security scanners or audio devices comprising new algorithms for recording and sampling audio. Further applications may be possible with respect to analogue I/O, in communication systems and networking and in particular sensor networks. Other applications may also be possible. Use of compressive sampling may also enable a reduction in power consumption of devices DEV, particularly because sampling may be performed in a computational efficient manner yet with an optimally minimal amount of samples needed.
| Number | Date | Country | Kind |
|---|---|---|---|
| 08105710.1 | Oct 2008 | EP | regional |
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/IB09/54807 | 10/29/2009 | WO | 00 | 6/6/2011 |
