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Pixel (Xmax, ymax)

Figure 7.1 A digital image is an organized array of pixels. Each pixel occupies a specific column and line. An image that is M pixels wide by N pixels deep contains a total of M x N pixels. In computer applications, the upper left pixel occupies the (0, 0) location, and the lower right occupies (xmax, ymax).

### 7.1.1 Pixel Coordinates

These specify the location of a pixel in an image. Pixel coordinates in raw images, calibration frames, and calibrated but otherwise unprocessed images refer to a corresponding location—a specific photosite—on the detector.

The coordinate assigned to the first sample in the first line read from a CCD is (0, 0), the second sample is (1, 0), then (2, 0), and so on to pixel (xmax, 0). On the CCD, the array is shifted down one line, and the first sample in the second line is (0, 1), the second sample of the second line is (1, 1), and so on. The last pixel in the last line read from the CCD is (xmax, ymax). The number of samples per line, M, is xmax + 1 , the number of lines in an image, N, is jmax + 1 , and the total number of pixels in the image is MN, or (xmax + 1 ) (ymax + 1 ).

In the literature of CCD imaging, the nomenclature for labeling the coordinate axes varies considerably. When referring to the entire image, the horizontal, column, or sample, axis is most often called the x-axis. The vertical, row, or line, axis is most often called the y-axis. When referring to a small portion of an image, such as the region immediately surrounding pixel (x, y), offsets in the x-axis are designated i, and offsets in the y-axis are designated j. Thus, pixels in the neighborhood surrounding pixel (x, y) will be named (x±i, y±j), where the variables i and j run through some range of values.

There are two conventions for displaying images: the Cartesian convention in which the y-axis increases upward, and the computer convention in which the y-axis increases downward. In both cases, the x-axis increases from left to right. Although books on image processing sometimes use the Cartesian display convention, virtually all software for small computers uses the computer display convention.

Although pixels have integer coordinates, image processing routines often must interpolate to find values "between" pixels. Because the numerical value of each pixel contains the integrated light of the whole photosite, the center of the pixel is the location (.x, _y), and the four "corners" of the pixel are (x-0.5, _y-0.5), (jc+0.5, j-0.5), (jc-0.5, j+0.5), and (x+0.5, y+0.5).

• Tip: AIP4Win provides tools for measuring pixel positions: the Image

Display Control and the Magnifying Glass Tool. The Image Display Control gives the pixel coordinates and pixel value under the cursor in the current image whenever the left mouse button is down. The Magnifying Glass Tool shows an enlarged region surrounding the cursor position, with a readout of the pixel coordinates and pixel value.

### 7.1.2 Pixel Value

Pixel value is the numerical value of a pixel, and it can be an integer or floatingpoint number. CCD cameras produce output in which the pixels have integer values; after processing, CCD images may contain integer or floating-point values. Pixel values can express almost any unit of measurement. They can represent the raw digital output of a CCD in analog-to-digital units (ADUs), but can also represent photoelectrons, radiometric units, units of energy (ergs cm-2 sec-1), or magnitudes per square arc second.

The unit of measurement found in raw CCD images, calibration frames, and calibrated but otherwise unprocessed images is the analog-to-digital unit, or ADU. ADUs are directly related to the statistically useful number of photoelectrons through the conversion factor, g. After several processing steps, this linear relationship may have become distorted, so it is usually desirable to regard pixel values in processed images as dimensionless numbers.

Although images from CCD cameras begin with integer pixel values, calibration and image processing soon convert these to non-integer values. If the noise level in the image is greater than one ADU, it is possible to work with integer pixel values with minimal loss of information. However, treating pixel values as floating-point numbers reduces loss to negliable levels and gives the user access to a wide dynamic range.

• Tip: AIP4Win treats all pixel values as floating-point values. The max imum value that AIP4Win allows is 1x10 , and the minimum is

-1x10 . The smallest allowed pixel value ±1x10 ; smaller values are treated as zero or i(not a number." Over this wide dynamic range, numerical values are carried with a precision of approximately one part in 8,400,000. 