OTRAS TRANSFORMADAS
ESPECTRALES DISCRETAS
Transformada Discreta de Fourier
Directa:
N 1
F (u ) 

 j
2
un
N
f (n )e
u = 0,1,2, ..., N-1
n0
Inversa:
f (n) 
1
N
N 1
 F (u ) e
u 0
j
2
un
N
n= 0,1,2, ... , N-1
Transformada Discreta de Fourier (1D)
N 1
F (u ) 

 j
f (n )e
2
un
N
n0
N 1
F (u ) 

f (n) g (n, u )
n0


F (u )  g ( n , u ) f ( n )
nucleo


F (u )  g ( n , u ) f ( n )
nucleo
=


F (u )  g ( n , u ) f ( n )
nucleo
=
Transformada Discreta de Fourier (1D)
g (n, u )  e
PARTE REAL
 j
2  un
N
PARTE IMAGINARIA
Transformada Walsh
k 1
g (n, u ) 
 (  1)
i0
b i ( n ) b k 1  i ( u )
Núcleo de la Transformada Walsh para N=32
(1 en blanco, -1 en negro)
Transformada Walsh

http://www.cs.tut.fi/~dabov/fwt/
Transformada Hadamard
k 1
 bi ( n ) bi ( u )
g ( n , u )  (  1) i  0
Transformada Hadamard (otra forma de
definición)
Y en general :
Fast Walsh-Hadamard Transform (N=8)
Transformada Discreta
Coseno
g ( n , u )  cos
( 2 n  1) u 
2N
para n  0 ,1,... N  1
u  1, 2 ,... N  1
Transformada Discreta Coseno
Nucleo de la Transformada Walsh-Hadamard (2D)
Núcleo de la Transformada Discreta Coseno (2D)
Transformada Hartley
 2

g ( x , u )  cas 
ux 
 N

cas ( )  cos(  )  sen ( )
 2

 2

g ( x , u )  cos 
ux   sen 
ux 
 N

 N

http://www.de.ufpe.br/~rjdsc/research/papers/Cintra__A_Transformada_Aritmetica_de_Hartley.pdf
Núcleo de la Transformada Hartley (N=16)
Transformada Hotelling


Se trata de una transformada basada
en información estadística.
Para efectos de compresión de datos,
es aún mejor que la DCT, es decir,
comprime la mayor cantidad de
energía en el menor número de
coeficientes


Los vectores que forman la nueva
base son los EIGENVECTORES de la
población.
Los eigenvectores definen las
direcciones de máxima distribución de
los datos, en consecuencia son los
vectores óptimos de representación.

Sea S el conjunto de K
vectores ¨n¨dimensionales :
 x1 
 
x
 2

x  . 
 
. 
x 
 n
Se define la media como:



K
 
mx  Ex
 
mx 
1
K

x
k 1
k
La matriz de covarianza de la población, se define como:





T 
C x  E  ( x  m x )( x  m x ) 


Dado que Cx es una matriz real y simétrica, siempre es posible
encontrar un conjunto de “n” eigenvectores ortonormales
e1 , e 2 , .... e n
correspond ientes a los n eigenvalor es :
 1 ,  2 , ...  n
Sea A la matriz formada por los “n” eigenvectores ordenados
por su correspondiente eigenvalor en magnitud decreciente, se
define la Transformada Hotelling como:



y  A (x mx)
Propiedades del nuevo conjunto transformado:

  
my  E y  0
 
C y  AC x A
T
 1

. 2

C y  .
.


0 ...

.
0

0




 n 
Se puede reconstruir el conjunto original aplicando
la transformación inversa:



x  A y  mx
T
Aplicación de Transformadas Espectrales
en Compresión de Datos
Comparación entre transformadas
•Se sub-divide la imagen en blocks de 8X8 pixels
•Se aplica la transformada a cada uno de los blocks
•Se descarta el 50% de los coeficientes
•Se aplica transformada inversa a cada block
•Se reconstruye la imagen
•Se calcula el error rms (raíz cuadrático medio)
error rms (raíz cuadrático medio)
e rms 
M 1 N 1



f ( x, y )  f ( x, y )



MN x  0 y  0 

1
2
Error raiz cuadrático
medio:
2.5
1.8
1.3
A Novel Technique for Audio Signals Watermarking in the Wavelet and Walsh Transform Domains
Akhaee, M.A. Ghaemmaghami, S. Khademi, N.
Sharif Univ. of Technol., Tehran;
This paper appears in: Intelligent Signal Processing and Communications, 2006. ISPACS '06.
International Symposium on
Publication Date: 12-15 Dec. 2006
On page(s): 171-174
Location: Yonago,
ISBN: 0-7803-9732-0
INSPEC Accession Number: 9505907
Digital Object Identifier: 10.1109/ISPACS.2006.364860
Current Version Published: 2007-05-29
Abstract
This paper presents a novel approach to audio signals watermarking in the wavelet or the Walsh transform
domain. The idea is to embed watermark data in the coefficients of some scales of the transform domain.
The overall bit rate of this method is about 90 bps. Due to low computational complexity of the suggested
approach, particularly in the Walsh domain, this algorithm can be implemented in real time. Experimental
results show robustness of the proposed method in low SNRs and also against some typical attacks, such as
MP3 compression, echo, filtering, etc. Subjective evaluation confirms transparency of the watermarked audio
signals
Frequency acquisition in a synchronous CDMA cordless phone usingfast Walsh transform
Chun Chian Lu
Comput. & Commun. Res. Lab., ITRI, Hsinchu ;
This paper appears in: Personal, Indoor and Mobile Radio Communications, 1995. PIMRC'95.
'Wireless: Merging onto the Information Superhighway'., Sixth IEEE International Symposium
on
Publication Date: 27-29 Sep 1995
Volume: 3, On page(s): 990Meeting Date: 09/27/1995 - 09/29/1995
Location: Toronto, Ont., Canada
ISBN: 0-7803-3002-1
References Cited: 8
INSPEC Accession Number: 5352899
Digital Object Identifier: 10.1109/PIMRC.1995.477092
Current Version Published: 2002-08-06
Abstract
We have designed a CDMA cordless phone for fast acquisition and simple tracking. The standard
acquisition proceeds through searching for the peak of correlation to get the delay offset, and correcting
the frequency offset. We use the fast Walsh transform to estimate the frequency offset. The final
correlations are tested within bounds so that the probability of false alarm is very small. The methods and
bounds are derived from the formulas of the Walsh transform of sine functions. The method of frequency
estimation is demonstrated by simulation its application in a spread spectrum receiver
A Walsh-Hadamard Transform LSI for Speech Recognition
Ohga, Hidefumi Yabuuchi, Hidekazu Tsuboka, Eiichi Mayumi, Kazuaki Adachi, Kozo Nishijima,
Osamu
Matsushita Electric Industrial Co., Ltd.;
This paper appears in: Consumer Electronics, IEEE Transactions on
Publication Date: Aug. 1982
Volume: CE-28, Issue: 3
On page(s): 263-270
Location: Rosemont, IL, USA,
ISSN: 0098-3063
Digital Object Identifier: 10.1109/TCE.1982.353920
Current Version Published: 2007-05-07
Abstract
Abstract- Speech recognition systems are coming to a practical stage thanks to the recent progress of
the semiconductor technology. We have developed a low cost speaker-dependent speech recognition
unit using Walsh-Hadamard transform (WIT). A WHT LSI has been developed to reduce the cost and
the space of the recognition unit, and a high rate of recognition has been obtained. The speech
recognition algorithm and the LSI are described in this paper.
Interictal spike detection using the Walsh transform
Adjouadi, M. Sanchez, D. Cabrerizo, M. Ayala, M. Jayakar, P. Yaylali, I. Barreto, A.
Dept. of Electr. & Comput. Eng., Florida Int. Univ., Miami, FL, USA;
This paper appears in: Biomedical Engineering, IEEE Transactions on
Publication Date: May 2004
Volume: 51, Issue: 5
On page(s): 868-872
ISSN: 0018-9294
INSPEC Accession Number: 7950387
Digital Object Identifier: 10.1109/TBME.2004.826642
Current Version Published: 2004-04-19
Abstract
The objective of this study was to evaluate the feasibility of using the Walsh transformation to detect interictal spikes
in electroencephalogram (EEG) data. Walsh operators were designed to formulate characteristics drawn from
experimental observation, as provided by medical experts. The merits of the algorithm are: 1) in decorrelating the
data to form an orthogonal basis and 2) simplicity of implementation. EEG recordings were obtained at a sampling
frequency of 500 Hz using standard 10-20 electrode placements. Independent sets of EEG data recorded on 18
patients with focal epilepsy were used to train and test the algorithm. Twenty to thirty minutes of recordings were
obtained with each subject awake, supine, and at rest. Spikes were annotated independently by two EEG experts. On
evaluation, the algorithm identified 110 out of 139 spikes identified by either expert (True Positives=79%) and missed
29 spikes (False Negatives=21%). Evaluation of the algorithm revealed a Precision (Positive Predictive Value) of 85%
and a Sensitivity of 79%. The encouraging preliminary results support its further development for prolonged EEG
recordings in ambulatory subjects. With these results, the false detection (FD) rate is estimated at 7.2 FD per hour of
continuous EEG recording.
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Otras Transformadas