Steerable PCA for Rotation-Invariant Image Recognition

Vonesch, Cédric; Stauber, Frédéric; Unser, Michaël

doi:10.1137/15m1014930

Cited by 5 publications

(5 citation statements)

References 23 publications

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“…As guaranteed by (35) and (10) for space/frequency localized images, smaller values of T lead to smaller approximation errors, and we notice that our PSWFs-based method outperforms the FFBsPCA algorithm in terms of accuracy for T = 10 and T = 10 −3 . It is also important to mention that the number of PSWFs taking part in the approximation of each image does not increase significantly when lowering T (see (11)).…”

Section: Numerical Experimentsmentioning

confidence: 57%

“…whereĨ ϕ K,m (x) is the expansion via the steerable principal components (33), and E(ε, δ c , T ) is the approximation error term of (10) for the truncated series of PSWFs. In essence, (35) asserts that if the images are sufficiently localized in space and frequency, then for an appropriate truncation parameter T , the error in expanding I m (x) using the steerable principal components computed fromÎ m (x) is close to the smallest possible error given by (34).…”

Section: Fast Pswfs Coefficients Approximationmentioning

confidence: 99%

“…Recently, [38] and [37] utilized Fourier-Bessel basis functions to expand the images, followed by applying PCA in the domain of the expansion coefficients, thus accounting for all (infinitely many) rotations. We also mention [35], where the authors present an accurate algorithm to obtain the steerable principal components of templates whose analytical form is known in advance.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 3 we present a fast algorithm for approximating the expansion coefficients from Section 2 up to an arbitrary precision. In Section 4 we formalize the procedure of steerable-PCA for the continuous setting (similarly to [35]), and combine it with our PSWFs-based approximation scheme. Section 5 then summarizes all relevant algorithms, and analyses in detail the computational complexities involved.…”

Section: Introductionmentioning

confidence: 99%

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Steerable Principal Components for Space-Frequency Localized Images

Landa¹,

Shkolnisky²

2017

SIAM J. Imaging Sci.

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As modern scientific image datasets typically consist of a large number of images of high resolution, devising methods for their accurate and efficient processing is a central research task. In this paper, we consider the problem of obtaining the steerable principal components of a dataset, a procedure termed “steerable PCA” (steerable principal component analysis). The output of the procedure is the set of orthonormal basis functions which best approximate the images in the dataset and all of their planar rotations. To derive such basis functions, we first expand the images in an appropriate basis, for which the steerable PCA reduces to the eigen-decomposition of a block-diagonal matrix. If we assume that the images are well localized in space and frequency, then such an appropriate basis is the prolate spheroidal wave functions (PSWFs). We derive a fast method for computing the PSWFs expansion coefficients from the images' equally spaced samples, via a specialized quadrature integration scheme, and show that the number of required quadrature nodes is similar to the number of pixels in each image. We then establish that our PSWF-based steerable PCA is both faster and more accurate then existing methods, and more importantly, provides us with rigorous error bounds on the entire procedure.

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Section: Numerical Experimentsmentioning

confidence: 57%

Section: Fast Pswfs Coefficients Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Steerable Principal Components for Space-Frequency Localized Images

Landa¹,

Shkolnisky²

2017

SIAM J. Imaging Sci.

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“…, N − 1}, respectively), but it also allows these transforms to be generalized to other signal domains. This, in turn, makes possible the analysis of new applications such as steerable principal component analysis (PCA) [68] where the domain is the rotation angle on [0, 2π), an imaging system with a pupil of finite size [11], line-of-sight (LOS) communication systems with orbital angular momentum (OAM)-based orthogonal multiplexing techniques [70], and many other applications such as those involving rotations in three dimensions [6,Chapter 5].…”

Section: Time-limited Toeplitz Operators On Locally Compact Abelian G...mentioning

confidence: 99%

Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation

Zhu,

Wakin

2017

Preprint

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Toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for linear, time-invariant (LTI) systems. Due to the fact that any practical system can access only signals of finite duration, time-limited restrictions of Toeplitz operators are naturally of interest. To provide a unifying treatment of such systems working on different signal domains, we consider timelimited Toeplitz operators on locally compact abelian groups with the aid of the Fourier transform on these groups. In particular, we survey existing results concerning the relationship between the spectrum of a time-limited Toeplitz operator and the spectrum of the corresponding non-time-limited Toeplitz operator. We also develop new results specifically concerning the eigenvalues of time-frequency limiting operators on locally compact abelian groups. Applications of our unifying treatment are discussed in relation to channel capacity and in relation to representation and approximation of signals.

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