where ( ) = 1/(2 )1/2 ej in .NET

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where ( ) = 1/(2 )1/2 ej
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If the admissibility condition is met, then a wavelet series decomposes a signal x(t) onto a basis of continuous-time wavelets, or synthesis wavelets, i,k (t), as shown: x(t) =
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The wavelet coef cients, Ci,k are de ned as Ci,k = x(t)
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The wavelet function (t) is constructed by a scale function (t), which is a solution of a two-scale difference equation: (t) =
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The sequence {hn } is called the low-pass lter or scaling sequence and is constrained to obey the regularity condition. This condition guarantees the numeric stability of a wavelet decomposition of any function in the Hilbert Space (L2 (R)). A scaling function with a valid {hn } is an admissible scaling function and can be used to construct the wavelet as (t) =
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where {gk } is a high-pass lter sequence. The signal decomposition can be done using orthogonal wavelets, in which case the synthesis wavelets are time-reversed versions of the analysis wavelets [10, 22].
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The DWT is similar to the wavelet series but is applied to discrete signals x[n]. It achieves a multiresolution decomposition of x[n] on I octaves given by i = 1, . . . , I and x[n] =
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ai,k gi [2n k] +
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The DWT computes wavelet coef cients ai,k and scaling coef cients bi,k for i = 1, . . . , I, which are given by ai,k =
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x[n] gi [2n k]
and bi,k =
x[n] hi [2n k],
where gi [2n k] is the discrete wavelet sequence and hi [2n k] is the scaling sequence. The wavelet sequence gi [2n k] serves as a high-pass lter, and the scaling sequence hi [2n k] serves as a low-pass lter. The decomposition of the signal into different frequency bands is achieved by successive high- and low-pass ltering operations. The output of the high-pass lter is referred to as the detail signal, and the output of the low-pass lter is referred to as the approximation signal. The resulting sets of wavelet coef cients compose what is called an Ith-level wavelet decomposition of the signal. The higher-numbered decomposition levels represent lower-frequency bands, as the signal is low-pass ltered repeatedly.
EEG was recorded from electrode locations O1, O2, P3, P4, and Fz relative to reference electrode Cz according to the international 10 20 placement system. Stimuli were presented using a demonstration copy of Presentation, Version 9.81 (Neurobehavioral Systems Inc., Albany CA). Presentation is a fully programmable stimulus delivery software system designed for behavioral and physiological experiments. The software used to acquire the data was AcqKnowledge Version 3.5.7 by Biopac Systems Inc. Data were acquired at a rate of 500 samples/second (500 Hz). All analysis and signal processing was performed of ine. Subjects were randomly presented images of known and unknown faces with no occlusion, known and unknown faces with various degrees of noise overlaid, and nally images of composite known and unknown faces. Subjects were instructed to click a mouse button when they were presented with an image of a familiar face. Stimuli were presented for 800 ms and were separated by a 600-ms off sequence consisting of a 400 300 pixel black rectangle with a centered white xation cross
A Multiresolution Analysis
on an 800 600 gray background. The known and unknown images were converted to gray scale, cropped to 400 300 pixels, and centered on an 800 600 pixel gray background with a centered white xation cross using Matlab Version 6.5. Unknown images were taken from the Psychological Image Collection at Stirling (PICS) University database [23], the Aleix database [24], and the Yale database [22]. Mean luminance for all stimuli and off stimulus was approximately equal. The rst trial ( Chk ) stimulus was a circular checkerboard display used as a base to verify valid data collection. The second trial ( Just Faces ) consisted of a set of 53 images of unknown faces taken from the previously mentioned databases and one image that the subject was familiar with (known). The known image was displayed a total of 53 times and was interspersed randomly with the unknown images. Although the unknown images were all unique, they were treated as one overall unknown image and contrasted during analysis with the known image. The third, fourth, and fth trials ( Noise01, Noise02, and Noise03 ) consist of a set of 54 unknown images and one known image. Gaussian white noise with variance = 0.4, 0.3, and 0.05, respectively, was added to each unknown and known image using Matlab s imnoise command. An example is provided in Figure 19.1. The sixth trial ( Mix01 ) consists of a set of four images, both face02, both face03, both face04, and both face05 (see Figure 19.2). The images are created by replacing vertical and horizontal lines in the known image with vertical and horizontal lines in the unknown image to achieve a pseudo-morphing effect. The rst image of the set, both face02.bmp, is created by replacing every other vertical and horizontal line in the known image with the corresponding vertical and horizontal lines in the unknown image. The second image of the set, both face03.bmp, is created by replacing every third vertical and horizontal line in the known image with the corresponding vertical and horizontal lines in the unknown image. The resulting