Digital Image Fundamentals and Fourier Analysis

Image sampling, quantization and resolution

Digitizing an image has two steps. Sampling fixes the spatial positions at which the scene is measured, which sets the number of pixels and the ground size of each pixel, that is, the spatial resolution. Quantization maps the continuous brightness at each sample onto a finite set of integer levels. With n bits per pixel, which is the radiometric resolution, the number of gray levels is L = 2^{n}, so a 5-bit image has 32 levels.

The uncompressed storage of an image without header is the number of rows times the number of columns times the number of bands times the bits per sample, \text{bits} = R \times C \times B \times n. Dividing by 8 gives bytes. For example, an 8-bit single-band image of 256 \times 256 pixels needs 65\,536 bytes. Kilobyte conventions matter in answer keys: taking 1 kB as 1000 bytes and taking 1 kB as 1024 bytes give different results for the same data volume.

• Header and interleaving — a header adds a fixed number of bytes, while BIL, BIP and BSQ only change the order in which band values are written, so the total size is unchanged apart from the header.

The bit depth of a quantizer follows from the number of distinct codes, n = \log_2 N_{\text{codes}}. A voltage range of 0 to 5 V changes the size of each quantization step but not the number of bits. Going the other way, a digital number DN of an 8-bit quantizer over a 0 to 5 V range is converted back to voltage by assuming full scale corresponds to the maximum code 255, so V = \frac{DN}{255} \times 5.

For a fixed pixel count, the ground area covered by an image is proportional to the square of the pixel size, so an image with smaller pixels covers less ground and shows finer details. When the across-track and along-track pixel dimensions are unequal, the image has aspect ratio distortion, which is a geometric and not a radiometric distortion. Sampling below the Nyquist rate causes aliasing, in which a fast pattern is recorded as a slower or reversed one, as in the wagon-wheel effect. For a checkerboard whose squares have side s pixels, the spatial period is 2s; with a sampling interval of 1 pixel, the Nyquist criterion requires 2s \geq 2, so squares smaller than 1 pixel alias. Finally, the total bits of an image equal N^{2} k for N \times N pixels at k bits per pixel, so the number of gray levels follows from L = 2^{k}.

Spatial filter kernels

A spatial filter replaces each pixel with a weighted sum of its neighbourhood, and the weights form the kernel or mask. A 3 × 3 mean (box) kernel has all weights equal to 1/9, so the output at the centre is the arithmetic mean of the nine pixel values in the window. Averaging kernels with positive, equal weights smooth the image and act as low-pass filters.

Kernels are matched to their function by their coefficient pattern. A uniform positive kernel is an averaging low-pass filter. A horizontal Sobel mask is a first-derivative edge detector. A four-neighbour Laplacian kernel has a centre value and negative neighbours that sum to zero, so it is a second-derivative high-pass operator. A sharpening kernel with a large positive centre and negative neighbours, whose coefficients sum to 1, enhances high-frequency detail.

• Linear operation — one that satisfies superposition and scaling. Convolution and moving averages are linear, because they are weighted sums with fixed weights.
• Median filter — a non-linear, order-statistic filter, because the median of a sum is generally not the sum of the medians.

Repeated convolution with a normalized kernel composes the kernels. Convolving \frac{1}{4}[1\ 2\ 1] with itself gives \frac{1}{16}[1\ 4\ 6\ 4\ 1], whose coefficients sum to 16 so that unit gain is preserved. A check that the coefficients sum to the normalizing factor is therefore a quick test of unit gain.

Frequency domain filtering

In a centred Fourier spectrum the origin is placed at the middle of the frequency plane, and distance from the centre measures spatial frequency. The central zone holds the low-frequency components, which describe average brightness and slowly varying background, while the periphery holds the high frequencies that describe edges and fine texture.

A Gaussian lowpass transfer function is H(u,v) = \exp\left(-\frac{D^{2}(u,v)}{2D_0^{2}}\right), where D_0 is the cutoff frequency. It attenuates high frequencies and therefore smooths the image, including scan-line noise. The corresponding highpass filter is the complement H_{HP} = 1 - H_{LP}, which is zero at the origin and approaches unity at high frequencies, so it passes edges.

To suppress low-frequency content, a filter must weight high frequencies more strongly than the low ones. The answer key identifies kernels with a strong centre weight balanced by negative surrounding weights as the ones that reduce low-frequency content relative to detail, while kernels of uniform positive weights are low-pass.

The convolution theorem links the two domains. Convolution of two images in the spatial domain corresponds to multiplication of their Fourier transforms in the frequency domain, and the converse holds as well. Cross-correlation instead corresponds to multiplication by a conjugate transform.

Visual image interpretation elements

Visual interpretation uses elements such as tone or colour, size, shape, texture, pattern, height, shadow, site and association. When two features have similar tone, size and texture, a distinguishing element must be found. For example, a meandering natural river and a straight, regularly shaped canal of similar width are separated mainly by shape.

• Projection — a mathematical rule for mapping the curved Earth onto a plane. It is a property of the map base, not a visible feature, so it is not an interpretation key.
• Pattern, texture and association — the spatial arrangement of objects, the apparent roughness of tone, and the relation of a feature to other recognisable features are all standard keys.

Band responses help identify surface classes. The shortwave infrared band responds strongly to the moisture content of soil and vegetation, because water absorbs strongly at those wavelengths. The near-infrared band separates land from water, since water absorbs it strongly while land reflects it. The blue band separates water, which reflects little blue light, from snow, which reflects strongly. Clouds and snow are both bright in the visible bands, including green, so the green band does not separate them.

RGB additive color display

On an additive RGB display, the red, green and blue channels emit light that adds together. Each channel is driven by a value from 0 to 255 for 8-bit data. When the three channel values are equal, the pixel appears as a neutral tone, and a value near mid-scale gives a medium gray, whereas all channels at 0 give black and all at 255 give white.