**Flemming Friche Rodler**

**November 2001**

The main topic of this dissertation is the development and use
of methods for space efficient storage of data combined with fast retrieval.
By fast retrieval we mean that a single data element can be randomly selected
and decoded efficiently. In particular, the focus will be on compression of
volumetric data with fast random access to individual voxels, decoded from
the compressed data.

Volumetric data is finding widespread use in areas such as medical imaging, scientific visualization, simulations and fluid dynamics. Because of the size of modern volumes, using such data sets in uncompressed form is often only possible on computers with extensive amounts of memory. By designing compression methods for volumetric data that support fast random access this problem might be overcome. Since lossless compression of three-dimensional data only provides relatively small compression ratios, lossy techniques must be used. When designing compression methods with fast random access there is a choice between designing general schemes that will work for a wide range of applications or to tailor the compression algorithm to a specific application at hand. This dissertation we be concerned with designing general methods and we present two methods for volumetric compression with fast random access. The methods are compared to other existing schemes showing them to be quite competitive.

The first compression method that we present is suited for online compression. That is, the data can be compressed as it is downloaded utilizing only a small buffer. Inspired by video coding the volume is considered as a stack of slices. To remove redundancies between slices a simple ``motion estimation'' technique is used. Redundancies are further reduced by wavelet transforming each slice using a two-dimensional wavelet transform. Finally, the wavelet data is thresholded and the resulting sparse representation is quantized and encoded using a nested block indexing scheme, which allows for efficient retrieval of coefficients. While being only slightly slower than other existing schemes the method improves the compression ratio by about 50%.

As a tool for constructing fast and efficient compression methods that support fast random access we introduce the concept of lossy dictionaries and show how to use it to achieve significant improvements in compressing volumetric data. The dictionary data structure is widely used in computer science as a tool for space efficient storage of sparse sets. The main performance parameters of dictionaries are space, lookup time and update time. In this dissertation we present a new and efficient dictionary scheme, based on hashing, called CUCKOO HASHING. CUCKOO HASHING achieves worst case constant lookup time, expected amortized constant update time and space usage of three words per element stored in the dictionary, i.e., the space usage is similar to that of binary search trees. Running extensive experiments we show CUCKOO HASHING to be very competitive to the most commonly used dictionary methods. Since these methods have nontrivial worst case lookup time CUCKOO HASHING is useful in time critical systems where such a guarantee is mandatory.

Even though, time efficient dictionaries with a reasonable space usage exist, the space usage of these are still too large to be of use in lossy compression. However, if the dictionary is allowed to store a slightly different set than intended, new and interesting possibilities originate. Very space efficient and fast data structures can be constructed by allowing the dictionary to discard some elements of the set (false negatives) and also allowing it to include elements not in the set (false positives). The lossy dictionary we present in this dissertation is a variant of CUCKOO HASHING which results in fast lookup time. We show that our data structure is nearly optimal with respect to space usage. Experimentally, we find that the data structure has very good behavior with respect to keeping the most important elements of the set which is partially explained by theoretical considerations. Besides working well in compression our lossy dictionary data structure might find use in applications such as web cache sharing and differential files.

. In the second volumetric compression method with fast random access that we present in this dissertation, we look at a completely different and rather unexploited way of encoding wavelet coefficients. In wavelet based compression it is common to store the coefficients of largest magnitude, letting all other coefficients be zero. However, the new method presented allows a slightly different set of coefficients to be stored. The foundation of the method is a three-dimensional wavelet transform of the volume and the lossy dictionary data structure that we introduce. Comparison to other previously suggested schemes in the literature, including the two-dimensional scheme mentioned above, shows an improvement of up to 80% in compression ratio while the time for accessing a random voxel is considerably reduced compared to the first method

Available as *PostScript*,
*PDF*.

BRICS WWW home page