Comparison of the cosine and Fourier transforms of Markov-1 signals

Hamidi, Mohammad; Pearl, Judea

doi:10.1109/tassp.1976.1162839

Cited by 102 publications

(23 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Digital Object Identifier 10.1109/TSP.2015.2447494 [2] is asymptotically equivalent to the KLT for the whole class of stationary processes [3], including the AR(1) model [4]; thus, for a Gaussian input, all these transforms result in a fully decoupled (independent) representation. However, this favorable independence-related property is extinguished for non-Gaussian processes.…”

Section: Introductionmentioning

confidence: 99%

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

Pad

Unser

2015

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

Abstract-The discrete cosine transform (DCT) is known to be asymptotically equivalent to the Karhunen-Loève transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes. Since being uncorrelated under the Gaussian hypothesis is synonymous with independence, it also yields an independent-component analysis (ICA) of such signals. In this paper, we present a constructive non-Gaussian generalization of this result: the characterization of the optimal orthogonal transform (ICA) for the family of symmetric--stable AR(1) processes. The degree of sparsity of these processes is controlled by the stability parameter with the only non-sparse member of the family being the classical Gaussian AR(1) process with . Specifically, we prove that, for , a fixed family of operator-like wavelet bases systematically outperforms the DCT in terms of compression and denoising ability. The effect is quantified with the help of two performance criteria (one based on the Kullback-Leibler divergence, and the other on Stein's formula for the minimum estimation error) that can also be viewed as statistical measures of independence. Finally, we observe that, for the sparser kind of processes with , the operator-like wavelet basis, as dictated by linear system theory, is undistinguishable from the ICA solution obtained through numerical optimization. Our framework offers a unified view that encompasses sinusoidal transforms such as the DCT and a family of orthogonal Haar-like wavelets that is linked analytically to the underlying signal model. Index Terms-Operator-like wavelets, independent-component analysis, auto-regressive processes, stable distributions.

show abstract

Section: Introductionmentioning

confidence: 99%

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

Pad

Unser

2015

IEEE Trans. Signal Process.

View full text Add to dashboard Cite

show abstract

“…Due to data-dependency, the PCA is not suitable for all saliency detection tasks when it is obtained from finite image samples. As has been mentioned, there exists an asymptotic equivalence between DCT and the PCA for Markov-1 signals (Hamidi and Pearl 1976;Rao and Yip 1990). This means that the PCA for digital images approaches the DCT as the number of training samples tends to infinity.…”

Section: Discussionmentioning

confidence: 91%

“…Ahmed et al (1974) proposed a discrete cosine transform (DCT) and compared its performance with the Karhunen-Loeve transform (also known as PCA) in image processing applications. After that, a number of studies (e.g., Shanmugam 1975;Hamidi and Pearl 1976;Clarke 1981; Uenohara and Kanade 1998) have mathematically proved the asymptotic equivalence between the DCT and the PCA for Markov-1 processes, which is commonly used to approximate image data. This means that PCA for Markov-1 signals approaches the DCT as the number of training samples tends to infinity.…”

Section: Extending To Pulsed Cosine Transformmentioning

confidence: 99%

“…This DCT-based attention model is referred to as pulsed cosine transform (PCT) in this paper. It can be shown that DCT is asymptotically equivalent to the PCA for signals coming from a first-order Markov model, which is a reasonable model for digital images (Hamidi and Pearl 1976;Oja 1992;Rao and Yip 1990). Therefore, PCT may offer better performance than P 2 CA in saliency detection.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bottom–up attention: pulsed PCA transform and pulsed cosine transform

Wang

Zhang

2011

Cogn Neurodyn

View full text Add to dashboard Cite

In this paper we propose a computational model of bottom-up visual attention based on a pulsed principal component analysis (PCA) transform, which simply exploits the signs of the PCA coefficients to generate spatial and motional saliency. We further extend the pulsed PCA transform to a pulsed cosine transform that is not only data-independent but also very fast in computation. The proposed model has the following biological plausibilities. First, the PCA projection vectors in the model can be obtained by using the Hebbian rule in neural networks. Second, the outputs of the pulsed PCA transform, which are inherently binary, simulate the neuronal pulses in the human brain. Third, like many Fourier transform-based approaches, our model also accomplishes the cortical center-surround suppression in frequency domain. Experimental results on psychophysical patterns and natural images show that the proposed model is more effective in saliency detection and predict human eye fixations better than the state-of-the-art attention models.

show abstract

“…In this case, the projection subspace is suboptimal, though very close to the PCA subspace when the source is a Markov process of 1st order [35]. In this case, the feature vector is: …”

Section: Dct Space Euclidean Metricmentioning

confidence: 99%

Integrating semantic analysis and scalable video coding for efficient content-based adaptation

Herranz

2007

Multimedia Systems

View full text Add to dashboard Cite

Scalable video coding has become a key technology to deploy systems where the adaptation of content to diverse constrained usage environments (such as PDAs, mobile phones and networks) is carried out in a simple and efficient way. Content-based adaptation and summarization are fields that aim for providing improved adaptation to the user, trying to optimize the semantic coverage in the adapted/ summarized version. This paper proposes the integration of content analysis with scalable video adaptation paradigm. They must be fitted in such a way that the efficiency of scalable adaptation is not damaged. An integrated framework is proposed for semantic video adaptation, as well as an adaptive skimming scheme that can use the results of semantic analysis. They are described using the MPEG-21 DIA tools to provide the adaptation in a standard framework. Particularly, the case of activity analysis is described to illustrate the integration of semantic analysis in the framework, and its use for online content summarization and adaptation. Overall efficiency is achieved by means of computing activity using compressed domain analysis with several metrics evaluated as measures of activity.

show abstract

Comparison of the cosine and Fourier transforms of Markov-1 signals

Cited by 102 publications

References 4 publications

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

Bottom–up attention: pulsed PCA transform and pulsed cosine transform

Integrating semantic analysis and scalable video coding for efficient content-based adaptation

Contact Info

Product

Resources

About