Image Sampling

• sensors in Digital Camera is discrete and in a grid as shown above
• the world around us is continuous
• digital images are discrete samples of the real world
• every digital image begins with a sampling process that converts the continuous visual world into discrete pixels

• the above image shows the results of image sampling and quantization
• if we have smaller sampling intervals, we will have samples leading to a higher spatial resolution

Image Resolution

Sampling Frequency

• sampling frequency describes how densly the image is sampled
• high sampling frequency preserves more image details with less aliasing

$$f_s=\frac{1}{△x}$$

Aliasing

Aliasing occurs when the sampling frequency is lower than the signal frequency.

Nyquist Sampling Theorem

To reconstruct a signal correctly (from digital back to contiuous), the sampling frequency must be at least twice its highest frequency.

$$f_s>=2f_{max}$$

Anti-Aliasing

Sampling Pipeline

graph TD
A[Continuous Image] --> B[Low Pass Filter used to remove High frequencies] --> C[Sampling] --> D[Digital Image]

Image Interpolation

Interpolation estimates unknown pixels using existing neighbouring pixels.

Method	Neighborhood Size	Visual Quality	Computational Speed	Common Artifacts
Nearest Neighbor	$1\times 1$ (1 pixel)	Lowest	Fastest	Blocky, jagged edges (pixelation)
Bilinear	$2\times 2$ (4 pixels)	Moderate	Fast	Slight blurriness / soft edges
Bicubic	$4\times 4$ (16 pixels)	High	Moderate	Minor haloing
Lanczos	$6\times 6$ or $8\times 8$ (36 to 64 pixels)	Highest	Slowest	Slight ringing near sharp contrast edges

Nearest Neighbor: Copies the value of the single closest pixel. It requires minimal computation but creates severe "staircase" or pixelated artifacts when upscaling.

Bilinear Interpolation: Calculates a weighted average of the $4$ closest neighboring pixels using linear interpolation in both horizontal and vertical directions. It eliminates blocky pixels but introduces a soft blur.

Bicubic Interpolation: Uses a $4\times 4$ grid ($16$ surrounding pixels) fitted to cubic polynomials. It preserves sharper edges and smoother gradients than bilinear interpolation at the cost of higher processing time.

Lanczos Interpolation: Uses a sinc-based windowed filter over a larger neighborhood ($6\times 6$ or $8\times 8$). It delivers the sharpest detail preservation and highest image fidelity, making it the preferred choice for high-quality image resizing.