Kohonen Map for Medical Image Compression Applications

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<Contributed to MeLi Wiki by Professor Anke Meyer-Baese, Department of Scientific Computing, Florida State University>


Concept of Unsupervised Learning

The main features of neural systems are their massive parallel processing architectures and the capabilities of learning from the presented inputs. They can be utilized to perform a specific task only by means of adequately adjusting the connection weights, i.e., by training them with the presented data. For each type of artificial neural network, there exists a corresponding learning algorithm by which we can train the network in an iterative updating manner. Those learning algorithms fit into two main categories: supervised learning and unsupervised learning [1].

For supervised learning, not only the input data but also the corresponding target answers are presented to the network. Learning is done by the direct comparison of the actual output of the network with known correct answers. This is also referred to as learning with a teacher. In contrast, if only input data without the corresponding target answers are presented to the network for learning, we have unsupervised learning.


Self-Organizing Feature or Kohonen Map

A self-organizing feature map or Kohonen map is implemented by a 1D or 2D lattice of neurons. The neurons become specialized to various input patterns or classes of input patterns while the network performs an unsupervised competitive learning process. The weights of the neurons that have close locations on the lattice are trained to represent similar input vectors. We obtain a network that preserves neighborhood relations in the input space, and thus preserves the topology of the input space. This map is known as a self-organizing feature map [2].

Mathematically, the self-organizing map (SOM) determines a transformation from a high-dimensional input space onto a one- or two-dimensional discrete map. The transformation takes place as an adaptive learning process such that when it converges the lattice represents a topographic map of the input patterns. The training of the SOM is based on a random presentation of several input vectors, one at a time. Typically, each input vector produces the firing of one selected neighboring group of neurons whose weights are close to this input vector.

The most important components of such a network are [2][3]:

1. A 1D or 2D lattice of neurons that encodes an input pattern <math>\mathbf{x}</math> of an arbitrary dimension into a specific location on it, as shown in Figure 1(a).

2. A method that selects a ``winning neuron'' based on the similarity between the weight vector <math>\mathbf{w}</math> and the input vector <math>\mathbf{x}</math>:

<math>i(\mathbf{x})=arg\min_j\left \| \mathbf{x}(n)-\mathbf{w}_j(n) \right \|, j=1,2,\cdots,N</math>

3. An interactive network that activates the winner and its neighbors simultaneously. A neighborhood <math>\Lambda_{i(\mathbf{x})}(n)</math> which is centered on the winning neuron is a function of the discrete time <math>n</math>.

Figure 1(b) illustrates such a neighborhood, which first includes the whole array and then shrinks gradually to only one ``winning neuron, represented by the black circle.

4. An adaptive learning process that reinforces all neurons in the close neighborhood of the winning neuron, and inhibits all those that are farther away from the winner. The synaptic weight vectors of all neurons (winners or not) are adjusted, using the update equation


<math>\mathbf{w}_j(n+1)=\begin{cases} 
\mathbf{w}_j(n)+\eta (n)[\mathbf{x}(n)-\mathbf{w}_j(n)]& \text{ if } j\in \Lambda_{i(\mathbf{x})}(n) \\ 
\mathbf{w}_j(n)& \text{ else }
\end{cases}</math>


Figure 1. Kohonen neural network; and (b) neighborhood <math>\Lambda_i</math>, of varying size, around "winning" neuron <math>i</math>, identified as a black circle.


Application of Kohonen Map for Medical Image Compression

Image compression based on Kohonen maps has been become an useful tool for many applications [4]. Data compression in medical imaging is desirable and often essential for improving cost and time for storage and communication.

For example, a typical digitized mammogram with <math>4500 \times 3200</math> pixels with 50-micron spot size and and 12-bit-per-pixel depth is approximately 38 megabytes of data. Extensive medical studies using digitized data format can easily require unacceptably long transmission times and can lead to major data management problems in local disk storage. A solution solely based on advancing technologies for transmission and storage is not sufficient.

Data compression is desirable and often essential for improving cost and time for storage and communication.

The typical digital compression system has three main steps: (1) signal decomposition based on a Fourier and wavelet transform, (2) coefficient quantization, and (3) lossless or entropy coding such as Huffman or arithmetic coding. Decompression is defined as the inverse operation.

A compression scheme applied in digital mammography is shown below , where the transmission time of four images per patient per screening is critical.


Figure 2 Kohonen.jpg


Scheme of the compression mechanism illustrated for digital mammography [5]. Four images per patient are produced by a single screening.

Kohonen-type neural networks can be applied to compress digital mammograms.


Figure 3 Kohonen.jpg


Original mammogram (a) and (b) same mammogram compressed by the proposed algorithm, compression rate = 64, PSNR = 23.5dB [5].

Project

Draw some mammographic images of both malignant and benign lesions from the MIAS database http://peipa.essex.ac.uk/info/mias.html and apply the compression scheme as shown above based on Kohonen maps.

Choose different 2-D lattice sizes of 25, 100 and 225 neurons and perform compression on the mammographic images and determine the compression ratio for each lattice size. Plot the learning error over time and visualize the compression results.


Wiki Assessment

  1. Describe the differences between supervised and unsupervised clustering?
  2. What is the architecture of a Kohonen map?
  3. Describe the differences between neighborhood and topology-preserving. To which group does the Kohonen map belong?
  4. What is a “winning neuron”?
  5. Why is the Kohonen map suitable for image compression?
  6. Describe the compression and decompression mechanism?


References

  1. ↑ S. Theodoridis and K. Koutroumbas. “Pattern Recognition”, Academic Press, San Diego, 1998.
  2. ↑ 2.0 2.1 T. Kohonen, “Self-Organization and Associative Memory”, Springer Verlag, 1988.
  3. ↑ S. Haykin, “Neural Networks”, Prentice Hall, 1999.
  4. ↑ C. Amerijckx, M. Verleysen, P. Thissen and J. Legat,” Image Compression by Self-Organized Kohonen Map”, IEEE Transactions on Neural Networks, 9:503-507, 1998.
  5. ↑ 5.0 5.1 A. Meyer-Baese, “Pattern Recognition for Medical Imaging”, Elsevier Science/Academic Press, 2003.
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