Direct Integral Theory

2020 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Integral Theory ◽  
Direct Integral

scholarly journals On the structure of polynomially normal operators

1984 ◽  
Vol 30 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Compact Operator ◽  
Normal Operator ◽  
Integral Theory ◽  
Direct Integral ◽  
Direct Sum ◽  
Essentially Normal ◽  
Normal Operators

We present some results concerning the structure of polynomially normal operators. It is shown, among other things, that if Tn is normal for some n > 1, then T is quasi–similar to a direct sum of a normal operator and a compact operator and if p(T) is normal with T essentially normal, then T can be written as the sum of a normal operator and a compact operator. Utilizing the direct integral theory of operators we finally show that if p(T) is normal and T*T commutes with T + T*, then T must be normal.

scholarly journals Low-Rank and Sparse Based Deep-Fusion Convolutional Neural Network for Crowd Counting

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Siqi Tang ◽  
Zhisong Pan ◽  
Xingyu Zhou
Keyword(s):  
Neural Network ◽  
Convolutional Neural Network ◽  
State Of The Art ◽  
Regression Method ◽  
Low Rank ◽  
Counting Method ◽  
Direct Integral ◽  
Crowd Counting ◽  
Counting Methods ◽  
Density Map

This paper proposes an accurate crowd counting method based on convolutional neural network and low-rank and sparse structure. To this end, we firstly propose an effective deep-fusion convolutional neural network to promote the density map regression accuracy. Furthermore, we figure out that most of the existing CNN based crowd counting methods obtain overall counting by direct integral of estimated density map, which limits the accuracy of counting. Instead of direct integral, we adopt a regression method based on low-rank and sparse penalty to promote accuracy of the projection from density map to global counting. Experiments demonstrate the importance of such regression process on promoting the crowd counting performance. The proposed low-rank and sparse based deep-fusion convolutional neural network (LFCNN) outperforms existing crowd counting methods and achieves the state-of-the-art performance.

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