Robust automated registration in the spatial domain Stéfan van der Walt, Ben Herbst Department of Electrical and Electronic Engineering, University of Stellenbosch, South Africa stefan@sun.ac.za Abstract Image registration algorithms have a great number of applications. Recently, many such methods have been developed, based on invariant localised interest points. These methods are fast, accurate and generally robust – ideal in almost every way. Unfortunately, they are inad- equate under certain circumstances. Zokai & Wolberg suggested the log-polar transform (LPT) for use in cases involving large changes in scale and rotation. We demon- strate that in addition to these properties, the LPT is use- ful in cases where other local descriptors fail. Although registration using the LPT is normally computationally intensive, we suggests ways in which its computational cost may be significantly reduced. Stacking of images, often used in astronomy, as well as panoramic stitching are used as examples. 1. Introduction Registration algorithms can be divided into two broad classes: those that operate in the spatial and frequency (i.e. Fourier) domains, respectively. In the spatial do- main, there are sparse methods including local descrip- tors, that depend on some form of feature extraction, and dense methods that operate directly on image values such as optical flow and correlation. The two classes generally differ in that the spatial methods are localised, whereas the frequency domain methods [15, 6, 8, 7] operate glob- ally. Attempts have been made to bridge this gap, by us- ing wavelet and other transforms to locate information- carrying energy [4]. These have been met with varying success. Each registration method has its own particular ad- vantages and disadvantages. Fourier methods, for exam- ple, are fast but inaccurate, suffer from resampling and occlusion effects [16, p. 1425], and only operate glob- ally. Iterative registration, on the other hand, is highly accurate but extremely slow, and prone to misregistration due to local minima in the minimisation space. These problems led to the development of methods based on localised interest points [1, 2, 10, 17, 18], such as the scale-invariant feature transform (SIFT) [13], the fast Speeded Up Robust Features (SURF) [9] and oth- ers [11]. All these methods depend on unique localised features, which are available in many images. There are, however, cases where it is very difficult to distinguish one feature from another without examining its spatial con- text. As an example, we will use frames recorded by a CCD mounted on a telescope pointing at a deep-space object. It is very difficult to find features to track in these images, because the stars (all potential features) are virtu- ally identical and rotationally invariant. Since local fea- tures fail, and global methods are slow and unreliable, we would like to find an algorithm that can bridge the gap. We will proceed to show that the log-polar transform (LPT) is an ideal candidate. While previously its use has been limited due to its high computational cost, we de- velop ways of reducing those costs and making the LPT behave more like local features. 2. The log polar transform The log-polar transform (LPT) spatially warps an image onto new axes, angle (θ) and log-distance (L). Using the centre of the image, (x c ,y c ) as reference, pixel coordi- nates (x, y) are written in terms of their offset from the centre, ¯ x = x − x c ¯ y = y − y c . For each pixel, the angle is defined by θ =  arctan ( ¯ y ¯ x ) ¯ x =0 0 ¯ x =0 with a distance of L = log b   ¯ x 2 +¯ y 2  . The base, b, which determines the width of the transform output, is chosen to be b = e ln(d)/w = d 1 w , where d is the distance from (x c ,y c ) to the corner of the image, and w is the width or height of the input image, whichever is largest.