大规模平移、旋转和缩放的伪极坐标估计方法

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"Pseudopolar-Based Estimation of Large Translations, Rotations, and Scalings in Images" 这篇由Yosi Keller, Amir Averbuch和Moshe Israeli发表在2005年1月的IEEE Transactions on Image Processing上的论文，提出了一个基于伪极坐标（Pseudopolar, PP）的图像大位移估计方法，主要关注于图像注册中对大规模平移、旋转和缩放的无先验知识估计问题。传统的图像配准方法在处理大规模运动时可能会遇到挑战，因为这些操作可能超出算法的精确度或稳定性的范围。该论文提出了一种基于傅里叶变换的方法，通过使用伪极坐标傅里叶变换，显著提高了对极坐标和对数极坐标傅里叶变换的近似精度。这种方法的关键在于，它将旋转和缩放转换为平移操作，而平移可以通过相位相关法来估计。这种方法降低了计算复杂性，同时提高了配准算法的准确性、速度和鲁棒性。利用PP网格，该算法能够处理更大的尺度和任意旋转角度。与当前最先进的算法相比，它可以稳健地恢复高达4倍的缩放和任意旋转角，而现有算法的最大恢复缩放通常不超过2倍。值得注意的是，该算法仅依赖一维快速傅里叶变换（1D FFT），其整体计算复杂度远低于先前的工作。实验结果验证了该方法的有效性，显示了在处理大规模图像变换时的优越性能。这为图像处理和计算机视觉领域提供了新的工具，尤其是在需要精确估计大位移、旋转和缩放的应用中，例如遥感图像分析、医学影像配准和视频序列同步等领域。 "Pseudopolar-Based Estimation of Large Translations"这一技术是图像处理领域的一个重要进步，它提供了一种高效且准确的方法来处理图像中的大幅度变化，对于需要精确配准的场景具有广泛的应用潜力。

14 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 14, NO. 1, JANUARY 2005

Fig. 1. Flow of the state-of-the-art FFT-based image registration. 1) The

magnitudes of the polar DFT are approximated by interpolating the magnitudes

of the 2-D FFT. 2) The rotation



is recovered using 1-D phase-correlation

on the



axis. 3) One of the input images is rotated by angle



. 4) The

translation is recovered and the ambiguity in



is resolved by applying a 2-D

phase-correlation twice: using

rotated by both



and





In Section III-A, we present the fractional FFT. In Sec-

tion III-B, we describe PPFT and how to compute it using the

Fractional FFT. In Section III-C, we conclude the presentation

of the PPFT by presenting its geometric interpretation.

A. Fractional FFT

The fractional FFT (FRFT) [28], with its generalization given

by the chirp Fourier transform [29], is a fast

algo-

rithm that evaluates the DFT of a function on any equally spaced

set of

points on the unit circle. Speciﬁcally, the FRFT evalu-

ates the DFT on the points

(3.1)

where

is an arbitrary frequency spacing and is the length

of the input signal. For

, the FRFT evaluates the

standard DFT. The frequencies, at which the FRFT evaluates

the DFT, are illustrated in Fig. 2. We denote by

the cen-

tralized fractional FFT (CFRFT), which computes the FRFT

around the dc component. Thus, the CFRFT

“compresses”

the FFT of the input signal around the dc component, where

is the compression ratio. Fig. 2(b) is an example of CFRFT with

Fig. 2. Frequency samples of the FFT and Fractional FFT on the unit circle

using

samples and a spacing of

. (a) The FFT samples the Fourier

transform equally over

;



]

, where

=N

. (b) The FRFFT samples

the Fourier transform equally with any spacing

and initial phase

Fig. 3. Construction of the PP grid from the

and

subgrids.

Fig. 4. Computation of the PP subgrids. (a) The computation of the

subgrid.

The input image is 1-D FFT transformed in the

direction. Then, the fractional

FFT is applied in the

direction. (b) The computation of the

subgrid. The

input image is 1-D FFT transformed in the

direction. Then, the fractional

FFT is applied in the

direction.

B. Computing the PPFFT

We decompose the PP grid into two subgrids, denoted

and

, as shown in Figs. 3 and 4. We denote by and the

values of the DFT evaluated on the subgrids

and , respec-

tively. Next, we describe the algorithm that computes

. The

algorithm for

is easily obtained by switching the roles of

the

and axes and a thorough description of the algorithm

is given in [30].

1) Computing

: The computation of is based on

the separability of the 2-D DFT.

is computed by applying

the 1-D FFT in the

direction and then applying the CFRFT

in the

direction. A varying factor for the CFRFT is chosen

such that the

subgrid geometry is obtained.

a) Algorithm Flow:

1) Let

be the input image of size

. is zero-padded to size

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