Computing Steerable Principal Components of a Large Set of Images and Their Rotations

Ponce, Colin; Singer, Amit

doi:10.1109/tip.2011.2147323

Cited by 17 publications

(18 citation statements)

References 22 publications

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“…There are quite a few papers that deal with the problem of fast and accurate rotational alignment of images; see, e.g., [22, 14, 6]. In the experiments reported here we rotationally align the images using a method that we recently developed that uses a steerable basis of eigenimages [21, 32]. Using this method, we computed the

(\begin{matrix} 40, 000 \\ 2 \end{matrix})

rotational alignments in about 30 minutes using all eight cores (here the computation ran in parallel).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors

Singer¹,

Zhao²,

Shkolnisky³

et al. 2011

SIAM J. Imaging Sci.

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The cryo-electron microscopy (cryo-EM) reconstruction problem is to find the three-dimensional structure of a macromolecule given noisy versions of its two-dimensional projection images at unknown random directions. We introduce a new algorithm for identifying noisy cryo-EM images of nearby viewing angles. This identification is an important first step in three-dimensional structure determination of macromolecules from cryo-EM, because once identified, these images can be rotationally aligned and averaged to produce “class averages” of better quality. The main advantage of our algorithm is its extreme robustness to noise. The algorithm is also very efficient in terms of running time and memory requirements, because it is based on the computation of the top few eigenvectors of a specially designed sparse Hermitian matrix. These advantages are demonstrated in numerous numerical experiments.

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(\begin{matrix} 40, 000 \\ 2 \end{matrix})

rotational alignments in about 30 minutes using all eight cores (here the computation ran in parallel).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors

Singer¹,

Zhao²,

Shkolnisky³

et al. 2011

SIAM J. Imaging Sci.

Self Cite

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“…From (11) and (23), the computational complexity of the last step is essentially O(L 3 ), which thus governs the computational complexity of the entire procedure described in this section. We end this discussion with the observation that the complexity of evaluating (27) can be further reduced by exploiting the rapid decay of the functions R c N,n (r) with the radius r (see Figure 1b for an illustration), due to which a substantial number of the terms involving R c N,n (r ℓ ω ) can be discarded for certain sets of the radii r ℓ ω and indices {(N, n)}. To exemplify this point, for T = 1, L = 150 and c = πL, we have that about 30% of the values R c N,n (r ℓ ω ) are below 10 −12 , and therefore can be safely discarded from (27).…”

Section: Fast Pswfs Coefficients Approximationmentioning

confidence: 99%

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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“…The idea of approximating a single template in a steerable basis has been studied independently by Perona [1,2] and Hel-Or & Teo [8], using continuous-domain formalisms; others have favored purely discrete formulations [3,9]. Note that [9] covers the case of multiple templates, but only in combination with a discrete set of rotations.…”

Section: Existing Work and Contributions Of This Papermentioning

confidence: 99%

“…Note that [9] covers the case of multiple templates, but only in combination with a discrete set of rotations. Here we consider a completely isotropic formulation in the continuous domain.…”

Section: Existing Work and Contributions Of This Papermentioning

confidence: 99%

“…We establish the connection between the isotropic PCA problem that underlies [9] and our steerability-constrained PCA. Moreover our functional perspective puts the emphasis on the rotation invariance of the principal-component subspaces and on their application to the efficient directional analysis of images.…”

Section: Existing Work and Contributions Of This Papermentioning

confidence: 99%

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Design of steerable filters for the detection of micro-particles

Vonesch

Stauber

Unser

2013

2013 IEEE 10th International Symposium on Biomedical Imaging

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This paper presents two contributions. We first introduce a continuous-domain version of Principal-Component Analysis (PCA) for designing steerable filters so that they best approximate a given set of image templates. We exploit the fact that steerability does not need to be enforced explicitly if one extends the set of templates by incorporating all their rotations. Our results extend previous work by Perona to multiple templates.We then apply our framework to the automatic detection and classification of micro-particles that carry biochemical probes for molecular diagnostics. Our continuous-domain PCA formalism is particularly well adapted in this context because the geometry of the carriers is known analytically. In addition, the steerable structure of our filters allows for a fast FFT-based recognition of the type of probe.

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Computing Steerable Principal Components of a Large Set of Images and Their Rotations

Cited by 17 publications

References 22 publications

Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors

Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors

Steerable Principal Components for Space-Frequency Localized Images

Design of steerable filters for the detection of micro-particles

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