Algorithms for calculating multichannel image histograms using hierarchical data structures
A.Y. Denisova, V.V. Sergeyev


Samara National Research University, Samara, Russia,
Image Processing Systems Institute f RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Samara, Russia

Full text of article: Russian language.

In the article we offer a novel approach to calculating multichannel image histograms using hierarchical data structures. The proposed methods result in a histogram-tree which has many advantages over existing data structures traditionally used for multidimensional histograms. The suggested data structure requires a significantly smaller memory space for storage in comparison with a histogram that is presented as a frequency table for all possible brightness values (histogram-hypercube). The histogram-tree provides a dramatic decrease in speed in comparison with the list of unique brightness values and their frequencies (histogram-list). Moreover, the histogram-tree allows one to approximate the probabilities of brightness values which are not presented in the analyzed image in contrast with the histogram-list.
The possibility of histogram approximation is very important for practical applications. Such approximation for histogram-tree is constructed by means of tree reduction. Nodes with frequencies lower than a given threshold are removed from the histogram-tree. The reduced histogram-tree benefits from the decrease of the required memory space while maintaining the accuracy of the original probability density distribution representation.
We propose two algorithms for constructing a histogram-tree. The “depth” algorithm operates with image pixels sequentially. For each pixel it constructs appropriate branches and nodes until the maximum depth of the tree is achieved. This algorithm has a recursive realization and is more time efficient than the other one. The “across” algorithm performs a multiple scan of the image, each time filling just one tree level. This algorithm allows one to operate with images with a larger number of channels than the “depth” algorithm because it is more memory efficient in the runtime when approximating a histogram. Both algorithms build the histogram-tree significantly faster when compared with the construction of the histogram-list.
The theoretical estimates and experimental results described in the article demonstrate that computing histogram of multichannel images using histogram-tree has many advantages over the histogram-hypercube and histogram-list.

multichannel images, histogram, linked list, hierarchical data structure, non- balanced tree.

Denisova AY, Sergeev VV. Algorithms for calculating multichannel image histogram using hierarchical data structures. Computer Optics 2016; 40(4): 535-542. DOI: 10.18287/2412-6179-2016-40-4-535-542.


  1. Ioannidis Y. The history of histograms (abridged). Proceedings of the 29th international conference on Very large data bases-Volume 29. VLDB Endowment; 2003: 19-30.
  2. Narendra PM, Goldberg M. A non-parametric clustering scheme for LANDSAT. Pattern Recognition 1977; 9(4): 207-215. DOI: 1016/0031-3203(77)90005-X.
  3. Asmus VV, Bunichev AA, Pyatkin VP. The Cluster Analysis and Classification with Training of Multispectral Data of Earth Remote Sensing [in Russian]. Journal of Siberian Federal University: Engineering & Technologies 2009; 2(1): 23-31.
  4. Sidorova VS. Multidimensional histogram and separating of the vector feature space on the unimodal clusters [in Russian]. International conference Graphicon, 2005; 2005: 267-274.
  5. Mason M, Duric Z. Using histograms to detect and track objects in color video. Applied Imagery Pattern Recognition Workshop, AIPR 2001 30th. IEEE, 2001: 154-159. DOI: 10.1109/AIPR.2001.991219.
  6. Wharton SW. A generalized histogram clustering scheme for multidimensional image data. Pattern Recognition 1983; 16(2): 193-199. DOI: 10.1016/0031-3203(83)90022-5.
  7. Ventzel ES. The Theory of possibility [In Russian]. Moscow: “Vysshaya shkola” Publisher; 1999.
  8. Russinovich , Solomon D. Internal structure of Microsoft Windows: Windows Server 2003, Windows XP and Windows [In Russian]. Saint-Petersburg: “Piter” Publisher; 2005.
  9. MODIS Level 1B Product User’s Guide. Members of the MODIS Characterization Support Team. NASA/Goddard Space Flight Center, Greenbelt, MD 20771. July 20, 2012.
  10. Denisov VI, Lemeshko BU, Postovalov SN. Applied statistics. The rules of coherence of experimental distribution with theoretical distribution. Guidelines. Part 1. χ2 tests [In Russian]. Novosibirsk: “NGTU” Publisher, 1998.

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