Lossless compact histogram representation for multi-component images : application to histogram equalization. JOCELYN CHANUSSOTA
ALAIN CLEMENTB
A Signals & Images Laboratory LIS / INPG BP 46 - 38402 St Martin d’Heres Cedex Fr.
[email protected]
BERTRAND VIGOUROUXB
B Laboratoire d’Ingénierie des Systèmes Automatisés LISA - IUT d’Angers BP 42018 - 49016 Angers Cedex - France {alain.clement ; bertrand.vigouroux}@univ-angers.fr
Abstract— In this paper, the problem of histogram representation is addressed. In the case of multi-component images, this problem is not trivial : the theoretical “naïve” required memory space goes exponentially beyond todays technical capacities. To overcome this problem, we present a lossless compact representation of vector histograms. It is based on the use of a space filling curve to index the data space. The application of this representation to vector histogram equalization is then considered.
I.
INTRODUCTION
Many basic image processing algorithms for remote sensing are based on the use of the image histogram. The representation and handling of the histogram is easy and straightforward in most of the cases. But, when it comes to multi-component image processing, the memory space required for the storage of classical multi-dimensional image histograms increases dramatically. In most of the applications, this phenomenon prevents the use of the exact vector histogram. For instance, let us consider an image with the following characteristics : M.N pixels, P components, each value being coded on Q bits. The “naïve” histogram of such an image is a table with P dimensions containing 2(P.Q) cells. Each cell can contain a number ranging from 0 to M.N and thus requires log2(M.N) bits. Therefore, the memory size a priori required to encode the histogram is : 2(P.Q). log2(M.N) bits This number rapidly increases as the parameters of the image increase. In particular, there is an exponential growth of the memory requirement both with respect to P and Q. More and more multi-spectral images are used in satellite remote sensing, therefore P tends to increase (up to several tenths in the case of hyper-spectral images…). Furthermore, the sensors used for the acquisition are more and more precise, leading to an increase of Q. Finally, the size of the images also tend to grow (M.N is increasing). As a consequence, the classical “naïve” representation to encode an histogram cannot be used. For example, the histogram of a 3 component 1024.1024 image, each pixel being encoded with 16 bits would require a 7.108 Mo large memory !
2Q . P. log2(M.N) bits. This method does not take into account the inter-correlation between the different components and is therefore not satisfactory. (ii) The re-quantization of each spectral component on a lower number of levels enables the reduction of the memory size to 2P.q . log2(M.N) bits, with q
