Image registration using log-polar mappings for recovery of large-scale similarity and projective transformations
Summary (3 min read)
Introduction
- Therefore, the images differ by large rotation and scale.
- See [2] for a recent survey of image registration techniques.
- Similarity measures like the zero-mean normalized sum of squared differences (SSD) and correlation coefficient are invariant to the linear intensity changes.
- In Section II, the authors discuss related work on the standard Levenberg–Marquardt algorithm (LMA) and log-polar techniques.
II. PREVIOUS WORK
- The authors discuss related work on the LMA and the log-polar techniques.
- In Section II-A, the authors present a background of the Levenberg–Marquardt nonlinear least-squares optimization algorithm that is useful for achieving subpixel registration accuracy.
- The log-polar transform is described in Section II-B.
- In Section II-C, the authors discuss the Fourier–Mellin transform, its limitations, and a review of related work.
- Section II-D discusses a feature-based method that can register images subjected to large scale changes (i.e., ) and arbitrary rotation.
B. Log-Polar Transform
- The log-polar transformation is a nonlinear and nonuniform sampling of the spatial domain.
- The log-polar transform has received considerable attention.
- If the authors assume that and lie along the horizontal and vertical axes, respectively, then image shown in Fig. 2(a) will be mapped to image in Fig. 2(b) after a log-polar coordinate transformation.
- The log-polar mapping is an accepted model of the representation of the retina in the primary visual cortex in primates, also known as V1 [23]–[25].
- This bandwidth reduction helps us process a high resolution image only at the focus of attention while aware of a wider field of view.
D. Feature-Based Image Registration
- Feature-based image registration algorithms extract salient structures, such as points, lines, curves, and regions, from graylevel images and establish correspondences between features using invariant descriptors.
- In more recent feature-based work, registration for wide baseline applications has been reported in [48]–[52].
- These results are promising in that they accommodate larger deformations.
- These stages include corner detection, conversion to invariant descriptors, matching based on the Mahalanobis distance or k-d tree, and outlier removal using the RANSAC algorithm.
- Whereas their methods are designed to operate under textured regions, they may fail in smooth regions.
III. MODIFIED LMA
- The LMA solves the following system of equations in an iterative fashion: (5) where is the Hessian matrix and is the residual vector (6) (7).
- The authors can improve the standard Levenberg–Marquardt optimization algorithm outlined above by adding two modifications.
- The first modification includes the use of a multiresolution pyramid for both reference and target images.
- The second modification virtually eliminates the calculation of the Hessian matrix (7) which would otherwise have been computed in every iteration.
- The authors second modification is based on the work of [16], whereby registration was performed on medical images subjected to similarity transforms (rotation/scale/translation).
A. Multiresolution Pyramid
- The original image, sitting at the base of the pyramid, is downsampled by a constant scale factor in each dimension to form the next level.
- This is repeated from one level to the next until the tip of the pyramid is reached.
- Second, the smoothness conditions imposed by successively bandlimiting the pyramid levels causes to be computed on smoother images.
- An example of computed on two different pyramid levels is shown in Fig.
- Thus, the relation between parameters is (11).
B. Modified Levenberg–Marquardt Algorithm
- In the standard LMA, the authors calculate the vector and Hessian matrix in each iteration.
- This is achieved by casting this problem into one where is transformed into , leaving unchanged from one iteration to the next.
- An important distinction between the standard and modified LMA methods lie in the manner in which the unknown parameters are updated in each iteration.
- Instead of minimizing (15a), the authors minimize (15c) with respect to the parameters .
- For further details about resampling, see [57].
IV. GLOBAL REGISTRATION USING LOG-POLAR TRANSFORM
- The authors have implemented a new algorithm for automatically finding the translation between both input images in the presence of large scale and rotation.
- The radius and the center of the template are optionally given by the user.
- The authors compute the base of the logarithm for log-polar transformation as follows: (20) where is the width of the input image ( diameter).
- The normalized correlation coefficient similarity measure is given as follows: (21) where is the average of image .
- Furthermore, the bulk of their computation is performed at the coarsest level where there are fewest pixels.
V. EXPERIMENTAL RESULTS
- An analytical evaluation of the robustness of image registration algorithms is an elusive task.
- Performance is highly dependent on the content of the input images.
- Many proposed image registration algorithms in the literature have limited their published results to the use of a few reference images and their synthetically generated target images.
- The reference and target images are taken by a camera with optical zoom.
- In Section V-B, the authors test the robustness of their algorithm with a large suite of 10 000 image pairs.
A. Uncalibrated Test Images
- A Canon PowerShot G3 digital camera with 4 optical zoom was used to capture a set of test images taken from natural and man-made scenes.
- The authors method uses all pixels and does not depend on any specific feature set.
- Images were acquired with (a) no magnification and (b) 4 magnification with unknown rotation about the optical axis.
- In order to quantify registration accuracy, the authors compute the correlation coefficient in the overlapping area.
- All of the non-SIFT methods failed to register the image pairs in Fig.
B. Calibrated Test Images
- It is not feasible to capture a very large set of images with a variety of image content and transformation parameters.
- The authors generated 10 000 target images from these random parameters.
- First, the authors used their log-polar module to recover the global rotation, scale, and translation parameters.
- The authors have tested their registration algorithm to create image mosaics by stitching together low resolution frames from several overlapping images.
- In order to best expose any misalignment, the authors applied unweighted averaging upon the overlapping areas.
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...LPT [ 3 , 4] is a well known tool for image processing for its rotation and scale invariant properties....
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References
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Frequently Asked Questions (13)
Q2. What have the authors stated for future works in "Image registration using log-polar mappings for recovery of large-scale similarity and projective transformations" ?
Additional future work will accelerate correlation. It is worthwhile to examine whether this process may be accelerated by positioning the sliding window on areas of high information content only. Entropy, variance, or other statistically discriminating techniques can be used to quantify information content. Recent success with scale-invariant interest points ( e. g., SIFT ) suggest that the log-polar windows should be centered at these extracted positions.
Q3. What is the purpose of the log-polar transform?
The log-polar transform has two principal advantages: 1) rotation and scale invariance and 2) the spatially varying sampling in the retina is the solution to reduce the amount of information traversing the optical nerve while maintaining high resolution in the fovea and capturing a wide field of view.
Q4. What is the purpose of the INRIA method?
The INRIA method computes interest points at different scales, calculating at each scale a set of local descriptors that are invariant to rotation, translation, and illumination.
Q5. What measures have been used to test the similarity of a polar image?
The authors have extensively tested several similarity measures, including normalized correlation coefficient, phase correlation, and mutual information.
Q6. Why does the image in Fig. 3(b) defie recovery?
Although the Fourier–Mellin transform is able to correctly register the synthetic image shown in Fig. 3(c), the image in Fig. 3(b) defies recovery because of the lack of similarity in its spectra compared to that of the reference image.
Q7. What is the main contribution of this work?
An important contribution of this work is that the authors introduce a new method based on the log-polar transform in the spatial domain that works robustly with real images.
Q8. What is the new method for finding scale, rotation, and translation parameters?
The new method is based on multiresolution log-polar transformations to simultaneously find the best scale, rotation, and translation parameters.
Q9. How did the authors show the importance of the log-polar module?
In order to show the importance of the log-polar module, the authors ran the LMA without the estimated initial parameters from the log-polar module.
Q10. How many pairs of images are shown in Fig. 1?
The correlation coefficient values for the thirty pairs mentioned above are all above 0.9, which is very good considering camera noise and artifacts introduced by warping to produce the target images.
Q11. How does the LMA solve the system of equations?
The LMA solves the following system of equations in an iterative fashion:(5)where is the Hessian matrix and is the residual vector(6)(7)The authors can improve the standard Levenberg–Marquardt optimization algorithm outlined above by adding two modifications.
Q12. What was the first report of the authors?
They were, however, reported recently in [40] and [41], where the authors showed that rotation and scale introduce aliasing in the low frequencies.
Q13. Why does the Fourier–Mellin transform help register two images?
Note that artificial black backgrounds can help register two images because it ensures that the authors consider the same underlying content.