public class AlgorithmColocalizationRegression
extends AlgorithmBase
implements UpdateVOISelectionListener

This algorithm creates a 2D histogram from 2 colors of a single image or from 2 black and white images and uses an orthogonal line fit of the histogram data to generate a correlation line thru the histogram. Upper right region rectangles with lower left corners on the correlation line are used as colocalization regions, regions where both of the 2 colors are significantly above background. The colocalization frame is tied to a version of the source image or images, so that only pixels located in the colocalization region are displayed. When not in free range mode, a user movable point along the correlation line allows for the display of source image pixels and statistics associated with different upper right colocalization regions. In free range mode, the point can be moved anywhere in the histogram buffer.

If neither registration or background subtraction are selected, then the original source image or images are used in the histogram buffer. If registration is selected and background subtraction is not selected, then the registered image or images are used in the histogram buffer. If registration is not selected and background subtraction is selected, then the subtracted image or images are used in the histogram buffer. If both registration and subtraction are selected, then the background averages are obtained from the original source image or images. Next, registration is performed. Finally, background subtraction is performed on the registered image or images and the subtracted image or images are used for the histogram buffer. If the maximum data range is decreased, then bin1 or bin2 will be decreased if necessary so as not to exceed the actual number of channels in the data range.

Note that the background selection voi is always blue and the voi to limit analysis to a region of the image is assumed to be a different color.

An optional registration may be performed before colocalization. In this registration both images or colors are moved halfway to a common center point. The registration is always done on an entire image. AlgorithmRegOAR2D is used with the cost function being the only registration parameter the user can vary in the dialog box. Correlation ratio is the default, but the user can also select least squares, normalized cross correlation, or normalized mutual information. The colocalization will be performed on the registered image or images rather than on the original image.

The amount of colocalization can be measured with the linear correlation coefficient. Let the 2 channels be R and G, the average values be Rav and Gav. Then the linear correlation coefficient = num/denom with num = sum over i of ((R(i) - Rav) * (G(i) - Gav)) denom = square root[(sum over i of (R(i) - Rav)*(R(i) - Rav)) * (sum over i of (G(i) - Gav)* (G(i) - Gav))] This is equivalent to the more traditional form of the equation: with num = Nsum(R(i)G(i)) - sum(R(i))sum(G(i)) denom = sqrt[Nsum(R(i)*R(i)) - (sumR(i))*(sumR(i))] * sqrt[Nsum(G(i)*G(i)) - (sumG(i))*(sumG(i))] The linear correlation coefficient ranges from -1 to 1 with 0 the value for random patterns and 1 the value for 100% colocalization. Negative values of the coefficient are not used for colocalization since that would indicate an anticolocalized situation with a bright pixel in one channel implying a dim pixel in another channel. Note that the medical literature on colocalization refers to the traditional linear correlation coefficient as Pearson's coefficient.

The P-value statistic measures the portion of the linear correlation coefficient values of images with 1 randomly scrambled channel whose values are less than the linear correlation coefficient value of the unscrambled image. 200 randomizations are performed resulting in a fairly accurate significance test with a maximum P-value of 0.995. A P-value of 0.95 or more is somewhat arbitrarily used to indicate the existence of true colocalization. The P-value option was not found to be useful and is currently disabled.

Pixel intensity values are spatially correlated because of the point spread function. Therefore, the number of independent samples in an image is not the number of pixels in the image but rather the number of objects the size of a point spread function that will fit into the image. A normalized autocovariance is calculated for each channel(although much of the medical literature erroneously calls it an autocorrelation). The channel whose autocovariance is smallest is divided up into squares or cubes with sides equal to the full width at half maximum of the smallest autocovariance. These squares or cubes are randomly placed in forming 200 randomly colocalized images.

Use linear least squares to calculate the best line fit for secondBuffer = a*firstBuffer + b. Use an orthogonal line fit where the minimized distance measurements are orthgonal to the proposed line. This is an orthogonal regression as opposed to a traditional direct regression of dependent y variable green on independent red variable x which would minimize the vertical distances between the points and the fitted line. An inverse regression of red on green would minimize the horizontal distances between the points and the fitted line. All 3 lines pass through the point (averagex, averagey), but the 3 lines only coincide if the correlation coefficient equals -1 or 1. Use of a direct regression only makes sense if one variable is dependent and one variable is independent. Since red and green dyes used in colocalization are best thought of as 2 independent variables, the orthogonal regression is most appropriate. Note that orthogonal regression is also called total least squares.

In the first iteration points which have a dataless background in both images are excluded from the regression line calculations. If buffer[i]

In the dialog the user can select to use points from the entire image or only points from the VOI region or only points allowed by a mask file. If VOI region is selected, then when a mouseReleased is performed on the VOI, the calculation is performed again. This allows the calculation to be performed with the same contour VOI in different positions. If the mask file radio button is selected, then the Choose mask file button will be enabled. When the Choose mask file button is pressed, a dialog for selecting a mask file will appear.

The second iteration thresholding works the same as the first iteration thresholding except for the line determination. In the first line determination iteration points are only excluded if they are below both background1 and background2, but in the second iteration line determination points are excluded if they are below either latPositive1 or lastPositive2. (lastPositive1,lastPositive2) is the last point going from top to bottom on the first iteration regression line before a L shaped subthreshold region with a zero or negative linear correlation coefficient is found. In the second iteration the L shaped subthreshold region determined from the first iteration is excluded from the regression line calculations. The default is to not perform a second iteration. The second iteration thresholding is not applied to the linear correlation coefficients, % colocalization area, % first color colocalization, or the % second color colocalization. The second iteration thresholding is just used for line generation with exclusion of points in the L shaped region.

The slope and offset of the orthogonal regression line can be obtained from the eigenvector of the smallest eigenvalue of the matrix: (count-1)*sample y variance -(count-1)*sample xy covariance -(count-1)*sample xy covariance (count-1)*sample x variance The eigenvector has the components (directionx, directiony), where the slope of the line equals = directiony/directionx offset = averagey - slope*averagex However, the method used here is to obtain the slope from the equation: slope = num/denom num = vary - varx + sqrt((vary - varx)*(vary - varx) + 4*covarxy*covarxy) denom = 2*covarxy Again offset = averagey - slope*averagex

Let n be the normal vector to the orthogonal regression line and let theta be the angle from the x axis to the normal. theta = arctan(slope) n = averagex*cos(theta) + averagey*sin(theta) The minimized residue, the mean square error, is given by MSE = varx*cos(theta)*cos(theta) + covarxy*sin(2*theta) + vary*sin(theta)*sin(theta) = (varx + 2*slope*covarxy + slope*slope*vary)/(1 + slope*slope)

To avoid including outliers in the regression line the intital regression line calculation may be iterated using only points whose distance from the line is is within +-2 times the square root of the mean square error. This will include 95% of the points. The iteration is performed until the fractional changes in the slope and offsets are

Note that there are 2 different types of regression line iterations. The first is the 1 or 2 iterations with the first iteration excluding only points with both buffer[i]

A histogram buffer is generated with bin1 counts along the x axis and bin2 counts along the y axis. bin1 and bin2 are user selectable in the dialog. leftPad, rightPad, bottomPad, and topPad spaces are left around the data area in histBuffer to allow room for axis lines and labels. For bin1 and bin2 in color images the default bin number = color max - color min + 1.

Calculate the linear correlation coefficients for all pixels whose values are either below threshold in buffer or are below a*threshold + b in secondBuffer. The values in buffer range from min1 to max1 and in secondBuffer range from min2 to max2. Calculate along the color that has the greatest range along the line segment. The line segment's (color1,color2) point just above the point where the linear correlation coefficient becomes negative or zero is taken in a nonrigorous fashion as the threshold point separating colocalized data from noncolocalized data.

Originally every point along the color that has the greatest range along the line segment was included in the calculation. However, the time required was unacceptable for large images. Only those points necessary to find the transition point from negative to positive correlation coefficient are calculated. This is approximately the log to the base 2 of the number of points along the greatest color range. First the midpoint is calculated. If the midpoint linear correlation coefficient is positive, then the 1/4 point is calculated. If the midpoint linear correlation coefficient is negative, then the 3/4 point is calculated. Each new line segment increment to a new point is 1/2 the size of the previous line segment increment. The process stops when it finds a point with a negative or undefined linear correlation coefficient followed by a point with a positive correlation coefficient. When the mouse is dragged along the line, the histogram frame header display only changes when the mouse passes over a point for which the correlation and colocalizations have been calculated. On a mouse release the correlation and colocalizations will be performed at that point if they have not already been done.

Calculate the percent colocalization area, the percent red colocalization, and the percent green colocalization for each point along the correlation line. Each point along the correlation line has a red value = threshold and a green value = secondThreshold. Thus, each point defines an upper right rectanglular region characterized by red >= threshold and green >= secondThreshold. The percent colocalization area is given by 100 times the number of pixels belonging to the upper right rectangular region divided by the number of pixels in the entire image or the selected voi. The percent red colocalization is given by 100 times the sum of the red values in the upper right rectangular region divided by the sum of the red values in the image or the selected voi. The percent green colocalization is given by 100 times the sum of the green values in the upper right rectangular region divided by the sum of the green values in the image or the selected voi. Note that only pixels which equal or exceed at least 1 of the 2 user specified thresholds are included in any of the above sums.

In the default option for a color colocalization all pixel values > background are included in the sum in the denominator. In a user selected option only pixels whose values are >= the threshold for the color being colocalized will be included in the denominator sum. For example, in the threshold only option red colocalization is given by 100 times the sum of the red values in the upper right rectangular region divided by the sum of the red values >= red threshold in the image or selected voi.

Pressing the free region toggle button on the histogram frame allows the user to take the VOI point off the least squares fit line. These correlation and colocalization values are only calculated in response to a mouse release.

There are 2 nonrigorous aspects associated with this algorithm. First, a single line is likely derived from at least 4 populations - (no color1, no color2), (no color1, color2), (color1, no color2), (color1, color2) to describe the characteristics of only the (color1, color2) population. Second, a more traditional statistical approach would use data within a fixed perpindicular distance from the line as being validly described by the line.

The calculated data is passed to ViewJFrameColocalization. References: 1.) Protein-protein interaction quantified by microscopic co-localization in live cells by Sylvain V. Costes, Dirk Daelemans, Edward H. Cho, George Pavlakis, and Stephen Lockett

2.) Dynamics of three-dimensional replication patterns during the S-phase, analysed by double labelling of DNA and confocal microscopy by E.M.M.Manders, J.Stap, G.J.Brakenhoff, R.Van Driel, and J.A.Aten, Journal of Cell Science, Vol 103, 1992, pp. 857-862.

3.) Measurement of co-localization of objects in dual-colour confocal images by E.M.M.Manders, F.J.Verbeek, and J.A.Aten, Journal of Microscopy, Vol. 169, Pt. 3, March, 1993, pp. 375-382.

5.) Equations for slope and offset of the orthogonal least squares problem from A Critical Examination of Orthogonal Regression and an application to tests of firm size interchangeability by Gishan Dissanaike and Shiyun Wang

6.) Equations for slope and mean square error of the orthogonal least squares from Straight Line Extraction Using Iterative Total Least Squares Methods by Jan A. Van Mieghem, Hadar I. Avi-Itzhak, and Roger D. Melen, Journal of Visual Communication and Image Representation, Vol. 6, No. 1, March, 1995, pp. 59-68.