A new class of shift-invariant operators
|
|
- Maria Holt
- 5 years ago
- Views:
Transcription
1 1 A new class of shift-invariant operators Janne Heiilä Machine Vision Group Department of Electrical and Information Engineering P.O. Box 4500, University of Oulu, Finland Tel.: , Fax: jth@ee.oulu.fi Abstract This paper proposes a class of operators with a shift invariance property. These operators are derived from two-dimensional complex moment invariants based on the observation that there is a duality between rotation invariance and shift invariance. A general form of the shift invariants belonging to this class is presented, which shows that polyspectral invariants such as the power spectrum and the bispectrum are members of the class. Methods for computing shift invariants for one-dimensional and two-dimensional signals are also presented. The examples given in the paper suggest that the higher order operators can preserve the original signal waveform better than the autocorrelation. Index Terms Discrete Fourier transform, translation invariance, moment invariants, power spectrum, bispectrum. EDICS Category: 1-TFSR I. INTRODUCTION In signal analysis, a usual manoeuvre is to compare the incoming signal waveform with a set of prototypes, and to select that prototype which has the highest resemblance. Before this comparison can be made the signals need to be aligned in order to compensate for the possible delay or shift between them. Another possibility is to convert the signals into a shift-invariant representation, when alignment is not necessary. Formally, if r(t) is a delayed version of s(t) so that r(t) =s(t ), (1)
2 2 where is an arbitrary delay, a shift-invariant operator S{ } satisfies the following equation S{r(t)} = S{s(t)}. (2) A well-nown operator for attaining shift invariance is the autocorrelation ϱ (l) = N 1 =0 r r +l (3) defined here for a discrete-time sequence {r } N 1 0. Another commonly used shift-invariant operator is the power spectrum which is the Fourier transform of the autocorrelation F{ϱ (l)} n = F{r } n F{r } n = F{r } n 2, (4) where F{ } denotes the discrete Fourier transform (DFT). A straightforward extension of the autocorrelation is the higher order statistics (HOS) with N th order autocorrelations or cumulants which also have the shift invariance property [1]. Higher order autocorrelation features have been used in pattern recognition, for example, in [2]. The Fourier transforms of the cumulants are called polyspectra [3]. Bispectrum is a special case of polyspectra and it is defined by B n1,n 2 = F{r } n1 F{r } n2 F{r } n 1+n 2. (5) As a result a two-dimensional shift-invariant representation of the one-dimensional signal is obtained. Shift and scale invariant features derived from bispectrum have been proposed in [4] for 1-D signals and in [5] for 2-D signals. Bispectrum in system identification has been discussed in [6], and several algorithms for reconstructing the signal from bispectrum have been suggested in the literature, e.g. in [7] and [8]. Trispectrum is another instance of the polyspectra with a 3-D representation of a 1-D signal [3]. Another operator for attaining shift invariance is R-transform [9], which has a fast FFT-lie algorithm, and only additions, subtractions, and absolute value operations are needed. Wagh and Kanetar [10] defined a class of shift-invariant transforms where R-transform also belongs. In that class, M-transform is another transform that has the same invariance property. In some cases shift invariance can be achieved by normalizing the data by using the first order moments. This is usually carried out by shifting the signal so that its centroid becomes zero. However, this procedure is very sensitive to noise, because only the first order statistics of the data is considered. In this paper, a new class of shift-invariant operators is proposed based on the relationship between complex moments, moment invariants and shift invariants.
3 3 II. DERIVATION OF THE OPERATORS Let us start from the complex moment c pq of the order p + q which is defined for a two-dimensional probability mass function f(m, n) as follows c pq = (m + jn) p (m jn) q f(m, n) (6) m= n= where j is the imaginary unit. In finite domain (6) can be expressed as c pq = N 1 =0 and in polar coordinates the equivalent representation is c pq = (x + jy ) p (x jy ) q f(x,y ), (7) N 1 =0 r p+q e j(p q)θ f(r,θ ). (8) Assuming that θ =2π/N and f(r,θ )=1/N, the complex moment becomes c pq = 1 N N 1 =0 r p+q e 2πj(p q) N, (9) which we can recognize as the (p q)th bin of the DFT calculated for the sequence {r p+q } N 1 =0. With the notation used in the previous section the moment is expressed as c pq = F{r p+q } p q. (10) It should be noticed that when using the discrete phase angles θ instead of continuous values we assume that r is sampled uniformly, which is the most common case with 1-D time series. Another important observation is that {r } can be an arbitrary discrete time sequence with r R, and it can also have negative values, although it is a usual convention to interpret r as a positive distance from the origin of the complex plane to the point x + jy. However, we can also have another interpretation where r is a negative value and in that case the phase angle is shifted by 180, i.e., r e jθ = r e j(θ+π). Assuming that p and q are integers, it is straightforward to prove that (9) holds also with this interpretation: c pq = 1 N ( r ) p+q e j(p q)(θ+π) = 1 N e jπ(p+q) r p+q e j(p q)θ e jπ(p q) = 1 N e 2jπp r p+q e j(p q)θ = 1 N rp+q e j(p q)θ = c pq. (11) It is evident from (6)-(9) that rotation invariance in the 2-D space implies shift invariance for the 1-D signal {r }. This gives us a basis for deriving the new class of shift-invariant operators. If we can find complex moment invariants to 2-D rotations, we will also get 1-D shift invariants based on these equations.
4 4 Moment invariants have been widely studied in pattern recognition. Hu [11] introduced seven invariants of the second and third order φ 1 φ 7 that are invariant to translation, rotation, and scale changes. Flusser [12] showed that Hu s invariants are partly dependent, and he proposed another set of invariants ψ 1 ψ 6 based on the second and third order moments. With the notation used above we can rewrite these moment invariants into the following form: ψ 1 = φ 1 = c 11 = F{r 2 } 0, ψ 2 = φ 4 = c 21 c 12 = F{r 3 } 1F{r 3 } 1, ψ 3 = φ 6 =Re(c 20 c 2 12 )=Re{F{r2 } 2 (F{r 3 } 1 )2 }, ψ 4 = Im(c 20 c 2 12) =Im{F{r 2 } 2 (F{r 3 } 1) 2 }, ψ 5 = φ 5 =Re(c 30 c 3 12 )=Re{F{r3 } 3 (F{r 3 } 1 )3 }, ψ 6 = φ 7 =Im(c 30 c 3 12) =Im{F{r 3 } 3 (F{r 3 } 1) 3 }, (12) where Re( ) means the real part and Im( ) the imaginary part of a complex number. The first moment invariant ψ 1 is the DFT coefficient defined at frequency 0. As it can be seen from (9) this invariant equals to the average of r 2, which is clearly shift-invariant. Comparing ψ 2 with (4) reveals that this invariant is the power spectrum component of {r 3} with n =1. The invariants ψ 5 and ψ 6 are actually the real and the imaginary parts of the trispectrum [3] of {r 3 }. On the other hand, the invariants ψ 3 and ψ 4 cannot be explained by the shift invariance of the power spectrum or the polyspectra, because two different sequences {r 2} and {r3 } are involved. This suggests that there is some more general theory behind these moment invariants as well as the corresponding shift invariants. Recently, Flusser [12] proposed a general framewor for constructing rotation invariants from complex moments. He showed that m I = c di p i,q i (13) i=1 is invariant to rotation if m d i (p i q i )=0, (14) i=1 where m 1, i =1,...,m, and d i, p i and q i are non-negative integers. In order to generalize our discussion about the shift invariants we need to review the proof of these formulas from [12]. Let f be a rotated version of a 2-D image f so that f (r,θ )=f(r,θ +α) where α is the angle of rotation. The complex moments of f denoted by c pq can be expressed by the term of c pq in the following
5 5 manner: c pq = ej(p q)α c pq. (15) In order to construct a rotation invariant descriptor we need to eliminate α from its expression. This can be achieved by multiplying moments of different order in such a way that the condition in (14) is satisfied. However, profound inspection of these formulas reveals that satisfying the condition (14) does not necessarily require that the moments involved in the product (13) have the same radial distance r. The only requirement is that each pair of moments must satisfy (15). Based on the duality between the rotation and shift invariance, we can extend this result to construct shift invariants. It was already pointed out that r can be an arbitrary signal, but now it is evident that the signals involved in the expression of the invariant does not have to be identical, but they only need to be shifted by the same amount. If we want to construct a shift invariant for a single sequence, we can utilize this result by introducing a set of real 1-D functionals τ i : R Rand use τ i (r ) instead of r. For example, ψ 3 and ψ 4 in (12) use the functionals τ 1 (r) =r 2 and τ 2 (r) =τ 3 (r) =r 3. From (10) we notice that the set of functionals in all moment invariants have the same form τ i (r) =r p+q. In general, the functionals can be any real-valued and position invariant mappings of r. Henceforth, we will call the functionals τ i shaping functions. Assuming that we have a 1-D real valued discrete time sequence {r }, we can now write the following general form of a shift-invariant operator: m Ψ ω1,ω 2,...,ω m {r } = F{τ i (r )} ωi, (16) i=1 where ω 1,...,ω m are integer parameters and they must satisfy the constraint m ω i =0. (17) i=0 These equations specify a new class of shift-invariant operators. Because the shaping function τ i can have different forms as discussed above, (16) represents an unlimited number of shift invariants. Notice that because of the constraint (17) the parameter space spanned by (ω 1,ω 2,...,ω m ) has only m 1 degrees of freedom. We can immediately see that the power spectrum (4) and the bispectrum (5) are members of this class. For the power spectrum m =2, τ 1 (r) =τ 2 (r) =r, and ω 1 = ω 2 = n, where n =0,...,N 1. For the bispectrum m =3, τ 1 (r) =τ 2 (r) =τ 3 (r) =r, ω 1 = n 1, ω 2 = n 2, and ω 3 = n 1 n 2.
6 6 III. COMPUTATIONAL ASPECTS In the basic form the operator (16) produces an (m 1)-dimensional representation of the 1-D signal. In many cases, it is more desirable to have a representation with the same dimensionality as with the original signal. In this section, the dimensionality problem is solved by considering only linear slices of the multidimensional representation. The sequence of the invariants is also transformed bac to the spatial domain. Next, we will assume only 1-D and 2-D input signals, but the generalization of the method for N-D signals is straightforward. A. 1-D signals Let {r } be an arbitrary real-valued 1-D discrete-time signal with N samples. From (16) we can see that there are various options to construct the invariants for {r }. We can select different values for m and for the parameters ω 1,...,ω m. Also, the shaping functions may change for each different set of parameter values. Because of this vast amount of possibilities we need to limit ourselves to a more compact set of the invariants. The principle is that we only compute a 1-D sequence of the invariants with the length of mn samples. This length is consistent with the requirement set for the power spectrum and the autocorrelation as well as for the other polyspectra to avoid the wraparound error. The second criterion for selecting the invariants is that only a single linear 1-D slice of the (m 1)-dimensional parameter space is used. This guarantees that the sequence containing the invariants is symmetric and consequently its inverse DFT becomes real-valued. In principle, slices of different orientations could be used, but in this paper we select the diagonal slice so that ω 1 = ω 2 =... = ω m 1 = n where n =0,...,mN 1. As it is shown in [13] diagonal slice of the bispectrum can be used for reconstructing the signal, which indicates that it contains all the necessary information of the signal waveform. The third constraint is that the shaping function τ i is not changed for the different values of ω 1,...,ω m. This is just for a practical reason, because otherwise it would not be possible to utilize the fast Fourier transform (FFT) algorithms for computing the invariants. For notational convenience, let R n,i F{τ i (r )} n. In general, we would need to compute mn samples of the DFT for each i =1,...,m. However, when using the diagonal slice ω 1 = ω 2 =...= ω m 1 = n, the first m 1 DFTs in (16) are the same assuming that τ 1 (r) =τ 2 (r) =,...,= τ m 1 (r). Based on the constraint (17) the last DFT to be computed becomes R (m 1)n,m = R(m 1)n,m. In other words, we need the DFT samples from the bins n and (m 1)n, where n =0,...,mN 1. The first set of samples are directly obtained from the sequence {R n,1 }. For the second set we need to permute the samples.
7 7 Here, we can utilize the conjugate symmetry and periodicity of the DFT by considering the sequence as infinite length with a period of mn samples. The sample indices needed are obtained from the equation l =(m 1) n mod mn, n =0,...mN 1. (18) Next, a set of invariants {Ψ n,n,...,n, l {r }} n=0 mn 1 are computed using (16). Finally, we can return bac to the time domain by taing the inverse DFT: ρ = F 1 {Ψ n,n,...,n, l {r }}, =0,...,mN 1. (19) This will mae the resulting descriptor ρ comparable with the autocorrelation function. Notice that {ρ } is a real-valued sequence, because {Ψ} is symmetric. However, {ρ } is not necessarily symmetric, although autocorrelation is always a symmetric function. B. 2-D signals Let {r 1, 2 1 =0,...,N 1 1, 2 =0,...,N 2 1} be an N 1 by N 2 array, and {R n1,n 2,i n 1 = 0,...,mN 1 1,n 2 =0,...,mN 2 1} the corresponding 2-D DFT array, where R n1,n 2,i = F{τ i (r 1, 2 )} n1,n 2. In order to apply the shift-invariant operator to the 2-D array, we need to perform a similar permutation for both indices as in the 1-D case so that l 1 = (m 1)n 1 mod mn 1, n 1 =0,...mN 1 1, l 2 = (m 1)n 2 mod mn 2, n 2 =0,...mN 2 1. (20) Again the shift invariants are computed based on (16) for each n 1 and n 2. Finally, the array of the invariants is converted into the spatial domain by using the 2-D inverse DFT. IV. NUMERICAL EXAMPLES This section gives some examples how different shift invariants from the new class wor in practice. Both 1-D and 2-D cases are considered and the methods described in the previous section are applied. For brevity, we only use two third order operators denoted by ρ(b) and ρ(c) and compare them with the well-nown autocorrelation denoted by ρ(a). The operator ρ(b) can be characterized by the following attributes m =3, τ 1 (r) =τ 2 (r) =τ 3 (r) =r, which is basically the inverse DFT of the 1-D diagonal slice of the bispectrum. The operator ρ(c) is characterized by m =3, τ 1 (r) =τ 2 (r) =r and τ 3 (r) = t r, where the time derivative t r can be approximated in the discrete domain by t r r +1 r. Notice that this approximation is position invariant although it depends on the indices and +1.
8 8 Fig D examples. From left to right: originals, autocorrelations, and two third order descriptors. In the first example illustrated in Fig. 1, four 1-D signals are shown in the leftmost column. The first two signals are mutually shifted, the third one is a mirrored version of the second signal and the fourth one was obtained by adding zero mean Gaussian noise to the second signal. The corresponding autocorrelations ρ(a) are in the second column, and the responses of the operators ρ(b) and ρ(c) are shown in the last two columns, respectively. As we can see, all three operators are shift-invariant. However, the autocorrelation is also invariant to the signal mirroring which maes it impossible to determine if the signals has been flipped or not, whereas this information is still available from the other two operators. The reason for this is that the autocorrelation does not preserve the phase-spectrum of the signal unlie the other two operators. As it is well-nown the phase-spectrum contains important shape information. From the last signal we notice that ρ(a) and ρ(b) suppress the noise, whereas ρ(c) seems to be pass the noise with less attenuation. This indicates that ρ(c) is more sensitive to high-frequency components, which might be a useful property when small details must be recognized. In the second example, the 2-D versions of the same operators ρ(a), ρ(b) and ρ(c) are applied with
9 9 Fig D examples. From left to right: originals, autocorrelations, and two third order descriptors. four intensity images. In the case of ρ(c) the shaping function is now the spatial difference along the vertical image axis i.e. τ 3 (x, y) =r(x, y +1) r(x, y). The original images are shown in Fig. 2 on the left. The first and third images represent two letters G and P. The second image is a shifted and noisy version of the first one and the last image is a mirror reflected version of the third one. Again, the autocorrelations are in the second column, and the responses of the operators ρ(b) and ρ(c) are in the following two columns. The same observations can be made as in the 1-D case. We can also notice some resemblance between the original images and the responses of the last two operators. This suggests that ρ(b) and ρ(c) could be better operators for pattern recognition purposes than the autocorrelation. V. CONCLUSION This paper has presented a new class of shift-invariant operators that can be computed efficiently with the discrete Fourier transform. Within the new class, an unlimited number of operators can be constructed
10 10 by using different shaping functions. These operators can also be extended to multidimensional signals in a straightforward manner. The power spectrum and the bispectrum as well as the other polyspectra can be seen as members of this class. Since the power spectrum does not preserve the phase information, it does not always provide a sufficient basis for signal analysis, while the higher order operators could give a better description of the original signal by retaining the phase information. The examples given in the paper indicate that these higher order operators transformed bac to the time or spatial domain preserve the characteristics of the original signal better than autocorrelation. This property is liely to be useful in pattern recognition applications, where shift invariance is in a ey role. REFERENCES [1] J. A. McLaughlin and J. Raviv, Nth-order autocorrelations in pattern recognition, Information and Control, vol. 12, pp , [2] T. Kurita, N. Otsu, and T. Sato, A face recognition method using higher order local autocorrelation and multivariate analysis, in Proc. 11th IAPR International Conference on Pattern Recognition (ICPR 92), The Hague, Netherlands, Aug. 1992, pp [3] J. M. Mendel, Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications, Proc. IEEE, vol. 79, no. 3, pp , [4] V. Chandran and S. L. Elgar, Pattern recognition using invariants defined from higher order spectra - one-dimensional inputs, IEEE Trans. Signal Processing, vol. 41, no. 1, pp , [5] V. Chandran, B. Carswell, B. Boashash, and S. L. Elgar, Pattern recognition using invariants defined from higher order spectra - 2-d image inputs, IEEE Trans. Image Processing, vol. 6, no. 5, pp , [6] G. B. Giannais and J. M. Mendel, Identification of non-minimum phase systems using higher-order statistics, IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp , [7] B. Sadler and G. B. Giannais, Shift and rotation invariant object reconstruction using the bispectrum, Journal of the Optical Society of America A, vol. 9, pp , [8] A. P. Petropulu and H. Pozidis, Phase reconstuction from bispectrum slices, IEEE Trans. Image Processing, vol. 46, no. 2, pp , [9] H. Reitboec and T. P. Brody, A transformation with invariance under cyclic permutation for applications in pattern recognition, Information and Control, vol. 15, no. 2, pp , [10] M. Wagh and S. Kanetar, A class of translation invariant transforms, IEEE Trans. Acoust., Speech, Signal Processing, vol. 25, no. 2, pp , [11] M. K. Hu, Visual pattern recognition by moment invariants, IEEE Trans. Inform. Theory, vol. 8, pp , [12] J. Flusser, On the independence of rotation moment invariants, Pattern Recognition, vol. 33, pp , [13] S. A. Dianat and M. R. Raghuveer, Fast algorithms for phase and magnitude reconstruction from bispectra, Optical Engineering, vol. 29, no. 5, pp , 1990.
A Method for Blur and Similarity Transform Invariant Object Recognition
A Method for Blur and Similarity Transform Invariant Object Recognition Ville Ojansivu and Janne Heikkilä Machine Vision Group, Department of Electrical and Information Engineering, University of Oulu,
More informationAffine Normalization of Symmetric Objects
Affine Normalization of Symmetric Objects Tomáš Suk and Jan Flusser Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08 Prague 8, Czech
More informationScience Insights: An International Journal
Available online at http://www.urpjournals.com Science Insights: An International Journal Universal Research Publications. All rights reserved ISSN 2277 3835 Original Article Object Recognition using Zernike
More informationBispectral resolution and leakage effect of the indirect bispectrum estimate for different types of 2D window functions
Bispectral resolution and leakage effect of the indirect bispectrum estimate for different types of D window functions Teofil-Cristian OROIAN, Constantin-Iulian VIZITIU, Florin ŞERBAN Communications and
More informationSYSTEM RECONSTRUCTION FROM SELECTED HOS REGIONS. Haralambos Pozidis and Athina P. Petropulu. Drexel University, Philadelphia, PA 19104
SYSTEM RECOSTRUCTIO FROM SELECTED HOS REGIOS Haralambos Pozidis and Athina P. Petropulu Electrical and Computer Engineering Department Drexel University, Philadelphia, PA 94 Tel. (25) 895-2358 Fax. (25)
More informationImage Recognition Using Modified Zernike Moments
Sensors & Transducers 204 by IFSA Publishing S. L. http://www.sensorsportal.com Image Recognition Using Modified ernike Moments Min HUANG Yaqiong MA and Qiuping GONG School of Computer and Communication
More informationAlgorithms for Computing a Planar Homography from Conics in Correspondence
Algorithms for Computing a Planar Homography from Conics in Correspondence Juho Kannala, Mikko Salo and Janne Heikkilä Machine Vision Group University of Oulu, Finland {jkannala, msa, jth@ee.oulu.fi} Abstract
More informationA New Efficient Method for Producing Global Affine Invariants
A New Efficient Method for Producing Global Affine Invariants Esa Rahtu, Mikko Salo 2, and Janne Heikkilä Machine Vision Group, Department of Electrical and Information Engineering, P.O. Box 45, 94 University
More informationUNIFORMLY MOST POWERFUL CYCLIC PERMUTATION INVARIANT DETECTION FOR DISCRETE-TIME SIGNALS
UNIFORMLY MOST POWERFUL CYCLIC PERMUTATION INVARIANT DETECTION FOR DISCRETE-TIME SIGNALS F. C. Nicolls and G. de Jager Department of Electrical Engineering, University of Cape Town Rondebosch 77, South
More informationFourier transforms and convolution
Fourier transforms and convolution (without the agonizing pain) CS/CME/BioE/Biophys/BMI 279 Oct. 26, 2017 Ron Dror 1 Outline Why do we care? Fourier transforms Writing functions as sums of sinusoids The
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 12 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial,
More information1. Groups Definitions
1. Groups Definitions 1 1. Groups Definitions A group is a set S of elements between which there is defined a binary operation, usually called multiplication. For the moment, the operation will be denoted
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationSIMPLE GABOR FEATURE SPACE FOR INVARIANT OBJECT RECOGNITION
SIMPLE GABOR FEATURE SPACE FOR INVARIANT OBJECT RECOGNITION Ville Kyrki Joni-Kristian Kamarainen Laboratory of Information Processing, Department of Information Technology, Lappeenranta University of Technology,
More informationGaussian-Shaped Circularly-Symmetric 2D Filter Banks
Gaussian-Shaped Circularly-Symmetric D Filter Bans ADU MATEI Faculty of Electronics and Telecommunications Technical University of Iasi Bldv. Carol I no.11, Iasi 756 OMAIA Abstract: - In this paper we
More informationA STATE-SPACE APPROACH FOR THE ANALYSIS OF WAVE AND DIFFUSION FIELDS
ICASSP 2015 A STATE-SPACE APPROACH FOR THE ANALYSIS OF WAVE AND DIFFUSION FIELDS Stefano Maranò Donat Fäh Hans-Andrea Loeliger ETH Zurich, Swiss Seismological Service, 8092 Zürich ETH Zurich, Dept. Information
More informationFrequency-domain representation of discrete-time signals
4 Frequency-domain representation of discrete-time signals So far we have been looing at signals as a function of time or an index in time. Just lie continuous-time signals, we can view a time signal as
More informationL29: Fourier analysis
L29: Fourier analysis Introduction The discrete Fourier Transform (DFT) The DFT matrix The Fast Fourier Transform (FFT) The Short-time Fourier Transform (STFT) Fourier Descriptors CSCE 666 Pattern Analysis
More informationThe Expectation-Maximization Algorithm
The Expectation-Maximization Algorithm Francisco S. Melo In these notes, we provide a brief overview of the formal aspects concerning -means, EM and their relation. We closely follow the presentation in
More informationGradient-Adaptive Algorithms for Minimum Phase - All Pass Decomposition of an FIR System
1 Gradient-Adaptive Algorithms for Minimum Phase - All Pass Decomposition of an FIR System Mar F. Flanagan, Member, IEEE, Michael McLaughlin, and Anthony D. Fagan, Member, IEEE Abstract Adaptive algorithms
More informationAccurate Orthogonal Circular Moment Invariants of Gray-Level Images
Journal of Computer Science 7 (5): 75-722, 20 ISSN 549-3636 20 Science Publications Accurate Orthogonal Circular Moment Invariants of Gray-Level Images Khalid Mohamed Hosny Department of Computer Science,
More informationarxiv:astro-ph/ v1 17 Oct 2006
Statistics of Fourier Modes in Non-Gaussian Fields Taahio Matsubara Department of Physics, Nagoya University, Chiusa, Nagoya, 464-860, Japan, and Institute for Advanced Research, Nagoya University, Chiusa,
More informationA Subspace Approach to Estimation of. Measurements 1. Carlos E. Davila. Electrical Engineering Department, Southern Methodist University
EDICS category SP 1 A Subspace Approach to Estimation of Autoregressive Parameters From Noisy Measurements 1 Carlos E Davila Electrical Engineering Department, Southern Methodist University Dallas, Texas
More informationTransform Representation of Signals
C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central
More informationThe statistics of ocean-acoustic ambient noise
The statistics of ocean-acoustic ambient noise Nicholas C. Makris Naval Research Laboratory, Washington, D.C. 0375, USA Abstract With the assumption that the ocean-acoustic ambient noise field is a random
More informationBlur Insensitive Texture Classification Using Local Phase Quantization
Blur Insensitive Texture Classification Using Local Phase Quantization Ville Ojansivu and Janne Heikkilä Machine Vision Group, Department of Electrical and Information Engineering, University of Oulu,
More informationRisi Kondor, The University of Chicago
Risi Kondor, The University of Chicago Data: {(x 1, y 1 ),, (x m, y m )} algorithm Hypothesis: f : x y 2 2/53 {(x 1, y 1 ),, (x m, y m )} {(ϕ(x 1 ), y 1 ),, (ϕ(x m ), y m )} algorithm Hypothesis: f : ϕ(x)
More informationEECS490: Digital Image Processing. Lecture #26
Lecture #26 Moments; invariant moments Eigenvector, principal component analysis Boundary coding Image primitives Image representation: trees, graphs Object recognition and classes Minimum distance classifiers
More informationLecture Notes 5: Multiresolution Analysis
Optimization-based data analysis Fall 2017 Lecture Notes 5: Multiresolution Analysis 1 Frames A frame is a generalization of an orthonormal basis. The inner products between the vectors in a frame and
More informationProof of convergence: KKT criterion
Efficient l point triangulation through polyhedron collapse Supplementary material March 6, 205 Part I Proof of convergence: KKT criterion Introduction In the paper Efficient l point triangulation through
More informationSolution of Kirchhoff's Time-Domain Integral Equation in Acoustics
Solution of Kirchhoff's Time-Domain Integral Equation in Acoustics Dr. George Benthien and Dr. Stephen Hobbs March 29, 2005 E-mail: george@gbenthien.net In this paper we will loo at the time-domain equivalent
More informationDesigning Information Devices and Systems I Spring 2016 Official Lecture Notes Note 21
EECS 6A Designing Information Devices and Systems I Spring 26 Official Lecture Notes Note 2 Introduction In this lecture note, we will introduce the last topics of this semester, change of basis and diagonalization.
More informationNormalizing bispectra
Journal of Statistical Planning and Inference 130 (2005) 405 411 www.elsevier.com/locate/jspi Normalizing bispectra Melvin J. Hinich a,, Murray Wolinsky b a Applied Research Laboratories, The University
More informationA Review of Higher Order Statistics and Spectra in Communication Systems
Global Journal of Science Frontier Research Physics and Space Science Volume 13 Issue 4 Version 1.0 Year Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc.
More informationInvariant Pattern Recognition using Dual-tree Complex Wavelets and Fourier Features
Invariant Pattern Recognition using Dual-tree Complex Wavelets and Fourier Features G. Y. Chen and B. Kégl Department of Computer Science and Operations Research, University of Montreal, CP 6128 succ.
More informationVector-attribute filters
Vector-attribute filters Erik R. Urbach, Niek J. Boersma, and Michael H.F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands April 2005 Outline Purpose Binary
More information3-D Projective Moment Invariants
Journal of Information & Computational Science 4: * (2007) 1 Available at http://www.joics.com 3-D Projective Moment Invariants Dong Xu a,b,c,d,, Hua Li a,b,c a Key Laboratory of Intelligent Information
More informationMULTICHANNEL SIGNAL PROCESSING USING SPATIAL RANK COVARIANCE MATRICES
MULTICHANNEL SIGNAL PROCESSING USING SPATIAL RANK COVARIANCE MATRICES S. Visuri 1 H. Oja V. Koivunen 1 1 Signal Processing Lab. Dept. of Statistics Tampere Univ. of Technology University of Jyväskylä P.O.
More informationMULTI-RESOLUTION SIGNAL DECOMPOSITION WITH TIME-DOMAIN SPECTROGRAM FACTORIZATION. Hirokazu Kameoka
MULTI-RESOLUTION SIGNAL DECOMPOSITION WITH TIME-DOMAIN SPECTROGRAM FACTORIZATION Hiroazu Kameoa The University of Toyo / Nippon Telegraph and Telephone Corporation ABSTRACT This paper proposes a novel
More informationDiscrete Simulation of Power Law Noise
Discrete Simulation of Power Law Noise Neil Ashby 1,2 1 University of Colorado, Boulder, CO 80309-0390 USA 2 National Institute of Standards and Technology, Boulder, CO 80305 USA ashby@boulder.nist.gov
More informationThe complex projective line
17 The complex projective line Now we will to study the simplest case of a complex projective space: the complex projective line. We will see that even this case has already very rich geometric interpretations.
More information2 Voltage Potential Due to an Arbitrary Charge Distribution
Solution to the Static Charge Distribution on a Thin Wire Using the Method of Moments James R Nagel Department of Electrical and Computer Engineering University of Utah, Salt Lake City, Utah April 2, 202
More informationSignals and Spectra - Review
Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs
More informationEigenface-based facial recognition
Eigenface-based facial recognition Dimitri PISSARENKO December 1, 2002 1 General This document is based upon Turk and Pentland (1991b), Turk and Pentland (1991a) and Smith (2002). 2 How does it work? The
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K R MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND First Printing, 99 Chapter LINEAR EQUATIONS Introduction to linear equations A linear equation in n unknowns x,
More informationMulti-Frame Factorization Techniques
Multi-Frame Factorization Techniques Suppose { x j,n } J,N j=1,n=1 is a set of corresponding image coordinates, where the index n = 1,...,N refers to the n th scene point and j = 1,..., J refers to the
More informationMinimum Repair Bandwidth for Exact Regeneration in Distributed Storage
1 Minimum Repair andwidth for Exact Regeneration in Distributed Storage Vivec R Cadambe, Syed A Jafar, Hamed Malei Electrical Engineering and Computer Science University of California Irvine, Irvine, California,
More informationTime-Delay Estimation *
OpenStax-CNX module: m1143 1 Time-Delay stimation * Don Johnson This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 1. An important signal parameter estimation
More informationRESEARCH ON COMPLEX THREE ORDER CUMULANTS COUPLING FEATURES IN FAULT DIAGNOSIS
RESEARCH ON COMPLEX THREE ORDER CUMULANTS COUPLING FEATURES IN FAULT DIAGNOSIS WANG YUANZHI School of Computer and Information, Anqing Normal College, Anqing 2460, China ABSTRACT Compared with bispectrum,
More informationVectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1. x 2. x =
Linear Algebra Review Vectors To begin, let us describe an element of the state space as a point with numerical coordinates, that is x 1 x x = 2. x n Vectors of up to three dimensions are easy to diagram.
More informationCMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation
CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is
More informationFeature Extraction Using Zernike Moments
Feature Extraction Using Zernike Moments P. Bhaskara Rao Department of Electronics and Communication Engineering ST.Peter's Engineeing college,hyderabad,andhra Pradesh,India D.Vara Prasad Department of
More informationReal-time Blind Source Separation
Real-time Blind Source Separation A Feasibility Study using CUDA th semester project, AAU, Applied Signal Processing and Implementation Spring 9 Group 4 Søren Reinholt Søndergaard Martin Brinch Sørensen
More informationCo-prime Arrays with Reduced Sensors (CARS) for Direction-of-Arrival Estimation
Co-prime Arrays with Reduced Sensors (CARS) for Direction-of-Arrival Estimation Mingyang Chen 1,LuGan and Wenwu Wang 1 1 Department of Electrical and Electronic Engineering, University of Surrey, U.K.
More informationConfidence and curvature estimation of curvilinear structures in 3-D
Confidence and curvature estimation of curvilinear structures in 3-D P. Bakker, L.J. van Vliet, P.W. Verbeek Pattern Recognition Group, Department of Applied Physics, Delft University of Technology, Lorentzweg
More information6 The Fourier transform
6 The Fourier transform In this presentation we assume that the reader is already familiar with the Fourier transform. This means that we will not make a complete overview of its properties and applications.
More informationENGIN 211, Engineering Math. Fourier Series and Transform
ENGIN 11, Engineering Math Fourier Series and ransform 1 Periodic Functions and Harmonics f(t) Period: a a+ t Frequency: f = 1 Angular velocity (or angular frequency): ω = ππ = π Such a periodic function
More informationElliptic Fourier Transform
NCUR, ART GRIGORYAN Frequency Analysis of the signals on ellipses: Elliptic Fourier Transform Joao Donacien N. Nsingui Department of Electrical and Computer Engineering The University of Texas at San Antonio
More informationLecture notes on Waves/Spectra Noise, Correlations and.
Lecture notes on Waves/Spectra Noise, Correlations and. References: Random Data 3 rd Ed, Bendat and Piersol,Wiley Interscience Beall, Kim and Powers, J. Appl. Phys, 53, 3923 (982) W. Gekelman Lecture 6,
More informationMultiscale Autoconvolution Histograms for Affine Invariant Pattern Recognition
Multiscale Autoconvolution Histograms for Affine Invariant Pattern Recognition Esa Rahtu Mikko Salo Janne Heikkilä Department of Electrical and Information Engineering P.O. Box 4500, 90014 University of
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationLecture 4: Probability and Discrete Random Variables
Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 4: Probability and Discrete Random Variables Wednesday, January 21, 2009 Lecturer: Atri Rudra Scribe: Anonymous 1
More informationRepresentation theory and quantum mechanics tutorial Spin and the hydrogen atom
Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition
More informationa 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts
More informationA Proposed Warped Choi Williams Time Frequency Distribution Applied to Doppler Blood Flow Measurement
9 Int'l Conf. Bioinformatics and Computational Biology BIOCOMP'6 A Proposed Warped Choi Williams Time Frequency Distribution Applied to Doppler Blood Flow Measurement F. García-Nocetti, J. Solano, F. and
More informationLv's Distribution for Time-Frequency Analysis
Recent Researches in ircuits, Systems, ontrol and Signals Lv's Distribution for Time-Frequency Analysis SHAN LUO, XIAOLEI LV and GUOAN BI School of Electrical and Electronic Engineering Nanyang Technological
More informationL6: Short-time Fourier analysis and synthesis
L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude
More informationChapter 4 Image Enhancement in the Frequency Domain
Chapter 4 Image Enhancement in the Frequency Domain Yinghua He School of Computer Science and Technology Tianjin University Background Introduction to the Fourier Transform and the Frequency Domain Smoothing
More informationPOINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES
November 1, 1 POINT VALUES AND NORMALIZATION OF TWO-DIRECTION MULTIWAVELETS AND THEIR DERIVATIVES FRITZ KEINERT AND SOON-GEOL KWON,1 Abstract Two-direction multiscaling functions φ and two-direction multiwavelets
More informationVarious Shape Descriptors in Image Processing A Review
Various Shape Descriptors in Image Processing A Review Preetika D Silva 1, P. Bhuvaneswari 2 1 PG Student, Department of Electronics and Communication, Rajarajeswari College of Engineering, Bangalore,
More informationSpatially adaptive alpha-rooting in BM3D sharpening
Spatially adaptive alpha-rooting in BM3D sharpening Markku Mäkitalo and Alessandro Foi Department of Signal Processing, Tampere University of Technology, P.O. Box FIN-553, 33101, Tampere, Finland e-mail:
More informationEFFICIENT CONSTRUCTION OF POLYPHASE SEQUENCES WITH OPTIMAL PEAK SIDELOBE LEVEL GROWTH. Arindam Bose and Mojtaba Soltanalian
EFFICIENT CONSTRUCTION OF POLYPHASE SEQUENCES WITH OPTIAL PEAK SIDELOBE LEVEL GROWTH Arindam Bose and ojtaba Soltanalian Department of Electrical and Computer Engineering University of Illinois at Chicago
More informationω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the
he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus
More informationAnalysis of Redundant-Wavelet Multihypothesis for Motion Compensation
Analysis of Redundant-Wavelet Multihypothesis for Motion Compensation James E. Fowler Department of Electrical and Computer Engineering GeoResources Institute GRI Mississippi State University, Starville,
More informationBlind channel deconvolution of real world signals using source separation techniques
Blind channel deconvolution of real world signals using source separation techniques Jordi Solé-Casals 1, Enric Monte-Moreno 2 1 Signal Processing Group, University of Vic, Sagrada Família 7, 08500, Vic
More informationMachine Learning. A Bayesian and Optimization Perspective. Academic Press, Sergios Theodoridis 1. of Athens, Athens, Greece.
Machine Learning A Bayesian and Optimization Perspective Academic Press, 2015 Sergios Theodoridis 1 1 Dept. of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens,
More informationThe Quantum Heisenberg Ferromagnet
The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,
More informationMRI beyond Fourier Encoding: From array detection to higher-order field dynamics
MRI beyond Fourier Encoding: From array detection to higher-order field dynamics K. Pruessmann Institute for Biomedical Engineering ETH Zurich and University of Zurich Parallel MRI Signal sample: m γκ,
More information2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016
6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge
More informationNOISE ROBUST RELATIVE TRANSFER FUNCTION ESTIMATION. M. Schwab, P. Noll, and T. Sikora. Technical University Berlin, Germany Communication System Group
NOISE ROBUST RELATIVE TRANSFER FUNCTION ESTIMATION M. Schwab, P. Noll, and T. Sikora Technical University Berlin, Germany Communication System Group Einsteinufer 17, 1557 Berlin (Germany) {schwab noll
More informationA NEW BAYESIAN LOWER BOUND ON THE MEAN SQUARE ERROR OF ESTIMATORS. Koby Todros and Joseph Tabrikian
16th European Signal Processing Conference EUSIPCO 008) Lausanne Switzerland August 5-9 008 copyright by EURASIP A NEW BAYESIAN LOWER BOUND ON THE MEAN SQUARE ERROR OF ESTIMATORS Koby Todros and Joseph
More informationRecent Advances in SPSA at the Extremes: Adaptive Methods for Smooth Problems and Discrete Methods for Non-Smooth Problems
Recent Advances in SPSA at the Extremes: Adaptive Methods for Smooth Problems and Discrete Methods for Non-Smooth Problems SGM2014: Stochastic Gradient Methods IPAM, February 24 28, 2014 James C. Spall
More informationOn New Radon-Based Translation, Rotation, and Scaling Invariant Transform for Face Recognition
On New Radon-Based Translation, Rotation, and Scaling Invariant Transform for Face Recognition Tomasz Arodź 1,2 1 Institute of Computer Science, AGH, al. Mickiewicza 30, 30-059 Kraków, Poland 2 Academic
More informationROTATION, TRANSLATION AND SCALING INVARIANT WATERMARKING USING A GENERALIZED RADON TRANSFORMATION
ROTATION, TRANSLATION AND SCALING INVARIANT WATERMARKING USING A GENERALIZED RADON TRANSFORMATION D. SIMITOPOULOS, A. OIKONOMOPOULOS, AND M. G. STRINTZIS Aristotle University of Thessaloniki Electrical
More informationIMPROVEMENTS IN ACTIVE NOISE CONTROL OF HELICOPTER NOISE IN A MOCK CABIN ABSTRACT
IMPROVEMENTS IN ACTIVE NOISE CONTROL OF HELICOPTER NOISE IN A MOCK CABIN Jared K. Thomas Brigham Young University Department of Mechanical Engineering ABSTRACT The application of active noise control (ANC)
More informationObject Recognition Using Local Characterisation and Zernike Moments
Object Recognition Using Local Characterisation and Zernike Moments A. Choksuriwong, H. Laurent, C. Rosenberger, and C. Maaoui Laboratoire Vision et Robotique - UPRES EA 2078, ENSI de Bourges - Université
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationReview of Linear System Theory
Review of Linear System Theory The following is a (very) brief review of linear system theory and Fourier analysis. I work primarily with discrete signals. I assume the reader is familiar with linear algebra
More informationLecture Notes 1: Vector spaces
Optimization-based data analysis Fall 2017 Lecture Notes 1: Vector spaces In this chapter we review certain basic concepts of linear algebra, highlighting their application to signal processing. 1 Vector
More informationWavelets Marialuce Graziadei
Wavelets Marialuce Graziadei 1. A brief summary 2. Vanishing moments 3. 2D-wavelets 4. Compression 5. De-noising 1 1. A brief summary φ(t): scaling function. For φ the 2-scale relation hold φ(t) = p k
More informationA Comparison of Particle Filters for Personal Positioning
VI Hotine-Marussi Symposium of Theoretical and Computational Geodesy May 9-June 6. A Comparison of Particle Filters for Personal Positioning D. Petrovich and R. Piché Institute of Mathematics Tampere University
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationMusical noise reduction in time-frequency-binary-masking-based blind source separation systems
Musical noise reduction in time-frequency-binary-masing-based blind source separation systems, 3, a J. Čermá, 1 S. Arai, 1. Sawada and 1 S. Maino 1 Communication Science Laboratories, Corporation, Kyoto,
More informationFourier Series. (Com S 477/577 Notes) Yan-Bin Jia. Nov 29, 2016
Fourier Series (Com S 477/577 otes) Yan-Bin Jia ov 9, 016 1 Introduction Many functions in nature are periodic, that is, f(x+τ) = f(x), for some fixed τ, which is called the period of f. Though function
More informationBasic Concepts of. Feature Selection
Basic Concepts of Pattern Recognition and Feature Selection Xiaojun Qi -- REU Site Program in CVMA (2011 Summer) 1 Outline Pattern Recognition Pattern vs. Features Pattern Classes Classification Feature
More informationIOMAC'15 6 th International Operational Modal Analysis Conference
IOMAC'15 6 th International Operational Modal Analysis Conference 2015 May12-14 Gijón - Spain PARAMETER ESTIMATION ALGORITMS IN OPERATIONAL MODAL ANALYSIS: A REVIEW Shashan Chauhan 1 1 Bruel & Kjær Sound
More informationIsotropic harmonic oscillator
Isotropic harmonic oscillator 1 Isotropic harmonic oscillator The hamiltonian of the isotropic harmonic oscillator is H = h m + 1 mω r (1) = [ h d m dρ + 1 ] m ω ρ, () ρ=x,y,z a sum of three one-dimensional
More informationHIGHER-ORDER SPECTRA OF NONLINEAR POLYNOMIAL MODELS FOR CHUA S CIRCUIT
Letters International Journal of Bifurcation and Chaos, Vol. 8, No. 12 (1998) 2425 2431 c World Scientific Publishing Company HIGHER-ORDER SPECTRA OF NONLINEAR POLYNOMIAL MODELS FOR CHUA S CIRCUIT STEVE
More information