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Compressive Sensing Based on Local Regional Data in Wireless Sensor Networks

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### Compressive Sensing Based on Local Regional Data in Wireless Sensor Networks

Hao Yang , Liusheng Huang , HongliXu , Wei Yang

2012 IEEE Wireless Communications and Networking Conference: Mobile and Wireless Networks

Outline

- Introduction
- Compressive Sensing Theory
- Compressive Sensing for Local Region
- Spatial-Temporal Correlation
- Random Distribution
- Gaussian Distribution
- Experiments
- Conclusion

Introduction (1/3)

- Reducing energy consumption of sensors is an especial important problem to prolong the lifetime of wireless sensor networks.
- Compressive Sensing (CS), as a novel and effective signal transform technology, has been proposed in different fields, such as medical and image [1,2] .

[1] J. Haldar, and D. Hernando, and ZP Liang, “Compressed Sensing in MRI with non-Fourier encoding” , IEEE Trans. Med. Imag, vol. 30, pp. 893–903, 2011.

[2] J. Wu, and F. Liu, and LC Jiao, and X.Wang, “Compressive Sensing SAR Image Reconstruction Based on Bayesian Framework and Evolutionary Computation”, IEEE Trans. Image Processing, no. 99, pp. 1-1, 2011.

Introduction (2/3)

- Compressive sensing is a data acquisition technique that aims to measure sparse and compressible signals which can be recovered exactly from very few measurements.
- Nyquist rate
- a lower bound for the sample rate for alias-free signal sampling
- Compressive Sampling/Sensing
- a novel sensing/sampling against the common wisdom in data acquisition

Introduction (3/3)

- It is difficult to promote the advantage of the technology in actual applications in WSNs due to the characteristic of itself.
- The effect of using CS technique depends immensely on whether the feature (sparsity) of the signal is known beforehand.
- In this paper, a CS approach based on local regional data is presented and verified in actual environments.

Compressive Sensing Theory (1/4)

- Compressive sensing (also known as compressed sensing, compressive sampling, or sparse sampling)
- Compressive sensing is the reconstruction of sparse signals from very few samples.

Compressive Sensing Theory (2/4)

- : an N-dimensional signal
- : measurement matrix , a matrix, where
- : , which is an M-dimensional vector and could be considered as M samples from through the measurement matrix .
- According to CS theory, it is probably that could be reconstructed from if this signal can be represented as a linear combination of only basis vectors ().

Compressive Sensing Theory (3/4)

- Furthermore, the matrix must satisfy Restricted Isometry Property (RIP) in order to recover exactly.
- For any subset I of , if equation (1) can be satisfied, then C could be recovered accurately with high probability by solving

Compressive Sensing Theory (4/4)

- There are three aspects that need to be considered in the theory of CS.
- A special sparse transform is needed.
- A measurement matrix should be irrelevant to the sparse transform.
- A reliable recovery algorithm should be designed to reconstruct the signal C.

Distributed Compressive Sensing (DCS)

- Distributed Compressed Sensing (DCS) is proposed to exploit both intra- and inter-signal correlation structures.
- In a typical DCS scenario, a number of sensors measure signals (of any dimension) that are each individually sparse in some basis and also correlated from sensor to sensor.
- This theory rests on a concept, which is called Joint Sparsity Model (JSM).

[9] M.F. Duarte, and S. Sarvotham, and D. Baron, and M.B. Wakin, and R.G. Baraniuk, “Distributed Compressed Sensing of Jointly Sparse Signals”, Asilomar Conf. Signals, Sys., Comput, pp. 1537-1541, 2005.

Joint Sparsity Model (JSM)

- Consider signals, and there exists a sparse basis Ψ, in which can be sparsely represented.
- all signals share a common sparse component while each individual signal contains a sparse individual component.

Joint Sparsity Model (JSM)

- with , and , the sparsityof signal z is K, while the sparsity of signal is .
- Thus, the entire signals X have the same common component and different individual components .

common component

individual component

Compressive Sensing for Local Region (1/2)

- Sensors usually gather data from their surroundings and they may be similar with each other if they are closed in geographically.
- In the other words, both time and space are not independent in actual environments.
- Owning to synchronized tendency of changing among the relative sensors, it is possible that relative sizes of data gathered by sensors in a local region would not alter frequently.

Compressive Sensing for Local Region (2/2)

- Spatial-Temporal Correlation
- Information Relation (IR)
- Alternation Relation (AR)
- Spatial-Temporal Correlation Model (STCM)
- Compressive Methods
- Random Distribution
- Gaussian Distribution

Information Relation (IR)

- If sensor and sensor are in local regions, it is probability that the data gathered by both sensor and are correlative.
- Suppose denotes gathered data by sensor at time , “” denotes IR between two sensors, then

Alternation Relation (AR)

- Suppose sensor and sensor are in local regions, if denotes variation data of sensor from time to time and it changes smoothly, let “” also denotes AR between two sensors, then
- In this case, the value of sensor and sensor at time is = + and

Spatial-Temporal Correlation Model (STCM)

- Suppose sensor and sensor are in local regions, and they satisfy the assumptions of IR and AR , then if they probably satisfy:
- The reason that this inequality may be satisfied is that , and in local regions.
- This case usually satisfies in actual environment. That is, the relative sizes of sensors in real situation have transitivity in time sequence.
- In addition, it is normal that the relative sizes of sensors have transitivity in space position. That is, if , , denote the data of sensor i, j, k, and and , then .
- Therefore, the relative sizes of sensors in local regions are stable based on STCM in real environments.

Random Distribution (1/5)

- Suppose the data are collected by sensors which are distributed randomly at the same time as a signal .
- Every sensor should transmit their own values with no additional coefficients just once.

Random Distribution (2/5)

- disperse signal ( ), and .
- In the first step for compressing, it is obvious to choose the minimum of the data and make every element of the vector subtracts the common value which is equal to .
- The deviation caused by recovery algorithm could be reduced.

Random Distribution (3/5)

- In order to decrease the cost of sensors, a compressive matrix is designed, which called cm_1.
- ( =).
- the rank of the cm_1 can be guaranteed to be full rank and furthermore the rank of S would be same with the rank of X.

Random Distribution (4/5)

- In the second step for compressing, an increasing rate which called relation_rate is proposed in order to abstract more common value based on the above conclusion.

Random Distribution (5/5)

- Based on the above process of compressing, the elements of the signal F are probably close to zero.
- When the usage of CS, matrix cm_1 and cm_2 could be multiplied by original measurement matrix to get a new measurement matrix.

- X

Gaussian Distribution (1/2)

- In generally, there exists a kind of the data whose distributions usually obey the Gaussian distribution in real applications.

Gaussian Distribution (2/2)

- According to the probability density function of the standard normal distribution, the common component s(i) can be estimated as below.
- In formula (13), norminv() denotes the inverse function of the accumulation function of the probability and i/N is the accumulation probability. It is worthwhile that the accumulation probability should be multiplied a modified coefficient in case of its value reaches 1.
- In the usage of CS, the process in this situation is the same as the above description.

Experiments

- In actual experiments, sensors are placed both indoors and outdoors.
- Temperature and illumination in the local regions are observed.
- In the usage of CS technique in the paper, Discrete Cosine Transform (DCT), Fast Fourier transform (FFT) and Discrete Wavelet Transform (DWT) will be used as sparse transform matrixes [10-12] .

Experiments - Actual Environment

- The number of IRIS sensors in this experiments is 32, which gather temperature and illumination 3 times in every minute.
- For comparing to effects in simulation environments, data are transformed every 2, 5, 10, 20 minutes and reconstructed together as one signal based on l1 -algorithm by the sink sensor after receiving data eight times and the number of samples M is 15. That is, N=32*8=256.

Experiments - Simulation Environments

- Random distribution.

Experiments - Simulation Environments

- Gaussian distribution

Analysis of Efficiency

- : the relative error of based on traditional reconstruction algorithm.
- Based on DCS theory:
- The common component A can be abstracted easily in our algorithms, therefore the error rate is from the individual component B.
- According to our algorithms, the relative error is :

Conclusion

- In this paper, a compressive sensing algorithm is present based on spatial-temporal correlation for actual applications in WSNs.
- The paper analyzes the feature of data in local regions and proposes a reasonable model, STCM, to emerge the relationship among data of sensors.
- Two algorithms targeted to two kinds of data distributions are designed to optimize measurement matrix of CS.
- Experiments in actual environments and simulation environment shows our algorithms outperforms other traditional methods.
- Just once extra transmission for every sensor.

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