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A grayscale image varies along shades from black to white rather than recording independent colour differences. A numerical grayscale image assigns one shade sample to each pixel; an encoding determines how that sample maps to a displayed shade. A colour raster image can also look grayscale, for example when its three sRGB components are equal.

The convention in our lossy block averaging example is black at the integer code 0 and white at 255. Its arithmetic means act on those stored codes. This is a fully specified numerical experiment, but it does not claim that code 128 means half the physical light of code 255. PNG explicitly allows samples that are not proportional to light: its grayscale interpretation depends on the indicated transfer curve, and can be device-dependent when that information is absent.^[ref]

Two different questions therefore need different operations. To average stored codes, average the integers. To average relative luminance for neutral sRGB shades, decode the codes to linear light, average there, and encode the result again. Equal weights on black and white give encoded value $1/2$ by the first method, but approximately $0.73536$ by the second. Rounded to eight-bit codes these are 128 and 188. This example concerns ideal neutral shades and the specified sRGB conversion; it does not assert a universal grayscale conversion for every colour encoding.

The upper panels hold the spatial grid fixed and change sample depth: four levels produce broad constant bands, while 256 levels make smaller numerical steps. The lower panels hold the colour interpretation fixed and compare encoded sRGB values with relative luminance. All panels are calculated synthetic examples, not measured display data. Grayscale is useful when shade structure is the quantity of interest: a synthetic scalar image signal can be inspected without introducing separate colour components.

Image compression applies data compression to pictures, usually represented by raster images. Nearby pixels often have related shades; an encoder can exploit that structure. A flat region invites run-length encoding, while lossy block averaging trades fine detail for fewer stored samples. The choice depends on whether exact pixel values matter and on the patterns in the image.

Lossless source coding recovers the encoded pixel values exactly, including sharp text or numerical image data. Lossy compression preserves an approximation and can remove features that later image processing might need. A visually smooth result is therefore a different promise from exact recovery. Neither method has a universal compression ratio: the chosen file format, its file header and the particular image all matter.

PNG is a real format with lossless source coding of its encoded image data. Its specified compression uses Deflate on reversibly filtered rows, rather than our toy RLE format. The lossy block averaging example is also a teaching format, not a description of PNG. A lossy preprocessing step followed by saving as PNG preserves the already changed pixels, not the earlier original.^[ref]

Lossy compression reduces a representation while permitting irreversible changes to the original. The encoder can map distinct inputs to the same output, so its decoder cannot recover both originals exactly. The useful question becomes how much storage is saved for an acceptable amount of distortion: mean squared error is one numerical criterion, but it does not by itself measure how noticeable a visual change is.

For example, storing the arithmetic mean of four grayscale pixels retains their average shade while discarding their arrangement. Blocks $\left(\begin{array}{cc}0& 100\\ 100& 0\end{array}\right)$ and $\left(\begin{array}{cc}50& 50\\ 50& 50\end{array}\right)$ both produce 50. The former has detail and the latter is uniform, yet the decoder receives the same value. Lossy block averaging makes this ambiguity, its bit saving and its mean squared error exact.

Nearest-neighbor interpolation assigns each requested location the value of its nearest available sample, with a specified convention for ties. In lossy block averaging, samples are placed at block centres and each is copied into its four original pixel positions. This enlarges the raster image without recovering discarded detail.

A pixel is one picture element at an indexed position in a raster image. It can contain a single grayscale value or several colour components. The word can also refer to a physical display element; the indexed element of the stored image is the meaning used here. One stored pixel need not occupy exactly one physical display element when the image is resized. Microsoft's bitmap documentation likewise distinguishes an image's grid and its numerical colour representation.^[ref]

Separate position from value: raster sampling and reconstruction concerns the spatial grid, while sample depth concerns how many bits encode each component. Four times as many pixels can describe a finer grid without improving the quantization of any value. Conversely, more bits per sample can distinguish more shades without locating an edge more precisely.

In the binary example under run-length encoding, one bit per pixel selects black or white. In the grayscale example under lossy block averaging, eight bits per pixel select one of 256 codes. Calling them codes keeps their numerical values distinct from the physical light that a display emits.

Quantization maps numerical values to a specified discrete collection of representable values. In lossy block averaging, $Q(m)=\lfloor m+1/2\rfloor$ maps block arithmetic means in $[0,255]$ to eight-bit integers. Thus $Q(40.25)=40$ and $Q(40.5)=41$. Different means can share a stored value, making this step irreversible.