The (auto)triple correlation l(3)(t 1 , t 2 ) is defined as the triple function integral, applied to the signal l(t) l(3)(t 1 , t 2 ) = ∫ l(t)l(t + t 1 )l(t + t 2 ) dt. The triple correlation l(3)(t 1 , t 2 ) is less popular than the standard (double) correlation l(2)(t 1 ) for several reasons: l(2)is sometimes easier to observe and to process, l(3)is small for many bipolar or complex signals, the mathematics associated with l(2)is better known. On the other hand, the triple correlation l(3)knows more about the signal l than does the ordinary autocorrelation l(2). Also l(3)is in some ways more sensitive, in other ways less sensitive to noise, to bias drifts, etc. Hence, there are situations, where it is quite favorable to evaluate one-dimensional signals or two-dimensional pictures by means of their triple correlations. We will review the underlying mathematical tools and report on our projects where triple correlations were employed for studying laser pulse shapes, sound quality, halftone print statistics, mobility of bacteria, and astronomical speckle interferometry. We will mention also how others have used the triple correlation for ocean waves, engine noises, intensity interferometry, and other optical signal processing tasks.

http://ieeexplore.ieee.org/iel5/5/31339/01457214.pdf

Triple correlations

Automatic contrast enhancement of low-light images based on local statistics of wavelet coefficients

The use of self-entropy as a focus measure in digital holography

The entropy or average information of an image can be determined approximately from the histogram of the image. The histogram shows the different grey level probabilities in the image. The entropy is useful, for example, for automatic image focusing: as the state of focusing of an image varies, so does its entropy. We present a method for measuring the entropy quite quickly and with reasonable accuracy. Our method is fast for two reasons: we have derived empirically a simple approximation formula for the entropy of images, and we were able to utilize an existing TV optical nonlinear component which performed the analogue calculation of our formula at TV speed. Our method can be applied to real-time focusing of two- or three-dimensional images in a TV system, for example in microscopy.

Measurement of the Entropy of an Image with Application to Image Focusing

Breast cancer is the most common cause of death in women and the second leading cause of cancer deaths worldwide. Primary prevention in the early stages of the disease becomes complex as the causes remain almost unknown. However, some typical signatures of this disease, such as masses and microcalcifications appearing on mammograms, can be used to improve early diagnostic techniques, which is critical for women’s quality of life. X-ray mammography is the main test used for screening and early diagnosis, and its analysis and processing are the keys to improving breast cancer prognosis. As masses and benign glandular tissue typically appear with low contrast and often very blurred, several computer-aided diagnosis schemes have been developed to support radiologists and internists in their diagnosis. In this article, an approach is proposed to effectively analyze digital mammograms based on texture segmentation for the detection of early stage tumors. The proposed algorithm was tested over several images taken from the digital database for screening mammography for cancer research and diagnosis, and it was found to be absolutely suitable to distinguish masses and microcalcifications from the background tissue using morphological operators and then extract them through machine learning techniques and a clustering algorithm for intensity-based segmentation.

Digital Image Processing Technique for Breast Cancer Detection

Histograms of images show frequencies of occurrence of different grey levels in an image. Uses of such histograms include image encoding, defocus measurements and texture discrimination. Our three aims are to simplify the determination of histograms by measuring only a few moments of the histogram, to achieve fast hybrid electro-optical measurement of the moments and other statistical features, and to apply these measurements to texture discrimination.

Ch. Thum

Papers

Measurement of the Entropy of an Image with Application to Image Focusing

Hybrid Measurements of Image Histograms, with an Application to Texture Discrimination