## Introduction

The conventional wisdom in modern structural biology is that as the resolution increases, structures become self-evident. Recent advances in microscopy have increased the theoretical light/electron microscopy (EM) resolution to 50 nm/1.5 Å (Reimer, 1997; Westphal *et al*., 2003), so that images of unprecedented clarity should be obtainable. However, at these high resolutions, practical limitations, such as avoiding radiation damage, place severe limitations on the data collection process. The consequence of this is that obtainable signal-to-noise ratios are often significantly less than 1.0, owing to a combination of shot noise and detector noise. This is an especially significant problem in cryo-EM tomography of unstained frozen specimens, where typically 100–200 tilted views need to be collected from the same sample and total doses need to kept below about 30 e^{−}Å^{−2}. The resulting three-dimensional (3D) reconstructions are quite noisy, which makes it a challenge to define accurately the shape and location of desired objects within the tomogram.

Non-specific staining is another source of ‘noise’ that further complicates interpretation of both EM and light microscopy data. For example, uranyl acetate is a popular EM stain that forms complexes primarily with phosphates on DNA, RNA and phosphoproteins. Images that are acquired using these stains are biased by the properties of the stain and thus do not necessarily represent a true picture of the underlying structure. Interpretation difficulties are compounded if non-specifically stained structures are packed densely, which is typical for many biological samples.

Light microscopy has related problems. Although the stains (especially fluorescent proteins) are very specific and provide strong contrast, experimental protocols often demand the collection of thousands of images from a single (often live) sample. In these cases, avoiding phototoxicity and bleaching of the fluorophore become paramount. As with the EM data, the result is a drastic reduction of the signal-to-noise ratio of each recorded image. Additional sources of noise, such as autofluorescence, background pools of unassembled fluorescent proteins and instrument noise, can contribute to the challenge of identifying and quantifying 3D cellular structures.

Many researchers believe that a simple solution to these problems is to construct software that can filter the image and ‘bring out’ the essential structure. Indeed, much effort by many groups, including ours, has been expended to develop filter methods to abstract structures and reduce noise (Böhm *et al*., 2000; Nicholson & Glaeser, 2001; Frangakis *et al*., 2002; Rath & Frank, 2004), with the best current method probably being anisotropic diffusion (Frangakis & Hegerl, 2001). Another approach is to locate known objects within the 3D reconstructions. Typically, such matched-filter correlation approaches can find objects under conditions of very high noise; however, there is much utility in developing hybrid methods that have the noise performance of matched filters yet do not require a priori knowledge of the search object.

Our approach is to develop a filter that preferentially highlights objects of defined size-classes within 3D volumes. This suggests using a wavelet transform. A wavelet transform is a convolution of a kernel (shape function) and the data. It differs from a Fourier transform in that the wavelet kernel is non-zero only over a finite spatial extent (chosen typically to equal the size of the feature of interest), whereas the Fourier kernel has infinite extent. It is this property of the wavelet kernel that makes it better than Fourier methods at defining frequency content as a function of spatial location. The wavelet transform shows how strongly the data are correlated to the kernel at each location in the data.

For ease of use and interpretation, we want the wavelet transform to have the following characteristics: (i) we want 2D or 3D data that are filtered with a wavelet of size *n* voxels to highlight preferentially those regions that have physical dimensions with a characteristic size of *n* voxels; and (ii) the wavelet transform should be invariant to rigid body rotations, i.e. rotating the wavelet transform of an object should be equivalent to the wavelet transform of the rotated object.

In general, wavelet methods can be quite complex, particularly for 3D data. Consequently, it would be an improvement to have a fast, efficient filter for 2D or 3D data whose only parameter is the characteristic spatial size of the structure of interest, and whose output is a spatial image of correlation strength. The 3D wavelet-based filter described here is a realization of these ideas. This filter, which highlights objects of a defined size-class, differs from a previous implementation of 3D wavelets (Stoschek & Hegerl, 1997) that used data-dependent thresholding at every wavelet size to denoise tomographic data globally.

We demonstrate the utility of our filter on synthetic data by showing how it can extract a pair of mathematically constructed helices from a noisy background, even at low signal-to-noise ratios. Four additional examples show how the filter works with typical noisy biological data: (i) EM data of positively stained microtubules, (ii) EM data of a negatively stained γ-tubulin ring complex (γTuRC), (iii) EM data of unstained microtubules preserved in vitreous ice and (iv) light microscopic images of *Caenorhabditis elegans* meiotic cells. Although this filter was developed primarily for biological applications, it is generally applicable to any 3D (or 2D) data.