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In scientific image processing and image analysis, an image is something different than a regular digital photograph of a beautiful scene you shot during your latest vacation.
In the context of science, digital images are samples of information, sampled at vertex points of n-dimensional grids.
What are pixel values?
Human visual perception is very good at certain tasks, such as contrast correction, detecting subtle differences in bright colors, but notoriously bad with other things, such as discerning dark colors, or classifying colors without appropriate reference. It is therefore very important to keep in mind that the pixel values are numbers, not colors.
For example, when you recorded an image using a confocal microscope, the values you get at a certain coordinate are not color values, but relate to photon counts.
Pixels are not little squares
And voxels are not cubes!
Why (lossy) JPEGs should not be used in imaging
JPEG stands for Joint Photographic Experts Group who were the creators of a commonly used method of lossy compression for photographic images. This format is commonly used in web pages and supported by the vast majority of digital photographic cameras and image scanners because it can store images in relatively small files at the expense of image quality. There are also loss-less JPEG modes, but these in general are not widely implemented and chances are that most of the images are of the lossy type.
"Lossy compression" means that in the process of file size reduction certain amount of image information is discarded. In the case of JPEGs, this might not be readily obvious to an observer, but it will be important for image processing purposes.
"Loss-less or non-lossy compression" means that the file size is reduced, but the data stored is exactly the same as in the original.
In JPEG lossy compression, therefore, the stored image is not the same as the original, and hence not suitable for doing serious imaging work. While JPEG compression throws away information that the eye cannot detect easily, this results in considerable image artifacts (variable "blockiness" of the image in groups of 8x8 pixels). Any attempts to process or quantify those images will be affected in uncontrollable ways by the presence of the artifacts. This is particularly obvious in the hue channel of JPEG compressed colour images when converted to HSB colour space.
Example: a section of the famous Mandrill image. From left to right, you see the original (with an 8-bit colormap), the hue channel of the original, and the hue channel after saving as a JPEG with ImageJ's default options -- note in particular the vertical and horizontal artifacts:
While most digital cameras save in JPEG format by default, it is very likely that they also support some non-lossy format (such as TIFF or a custom RAW format). Use those formats instead. A format called JPEG2000 supported by various slide scanners and used in "virtual slide" products was created to improve image quality and compression rates, however both lossy and non-lossy versions the JPEG2000 format exist. Use only the non-lossy formats.
Multichannel images in particular are harmed by the JPEG format: since the multiple channels are misinterpreted as red, green and blue (while most channels integrate more than just one wavelength), JPEG will shift the colors imperceptibly to improve compression. In the worst case, this can lead to colocalization where there was none.
TIFF or PNG formats available in ImageJ/Fiji are non-lossy formats. TIFF preserves any calibrations applied to your images, but images are not compressed. PNG files are smaller as non-lossy compression is applied but they do not store calibration data.
Once an image has been saved as compressed JPEG there is no way of reverting to the original, therefore an old JPEG-compressed image saved again as TIFF or PNG still contains all the original JPEG compression artifacts.