Image Resampling | Sahithyan's Notes

Changing image size parameters requires resampling to create new pixel values: adding pixels when upsampling, replacing pixels when downsampling.

Resampled pixels must:

Preserve image features, including fine detail
Reduce noise
Not introduce artefacts

Scaling by a Factor of 2

Enlarge by replicating each row and column
Reduce by removing every other row and column
Repeat progressively for larger scaling factors

Backward Mapping

Each pixel $(x, y)$ in the new grid maps back to a source position.

s_x = \frac{x}{k_x} \qquad s_y = \frac{y}{k_y}

k_x = \frac{\text{new width}}{\text{source width}} \qquad k_y = \frac{\text{new height}}{\text{source height}}

Here:

$k_x, k_y$: horizontal and vertical scale factors
$(s_x, s_y)$: source position, generally non-integer

A non-integer $(s_x, s_y)$ falls between source pixels, so no stored value sits there. The output value is built from the 2×2 block of source pixels enclosing it, with corners $(\lfloor s_x \rfloor, \lfloor s_y \rfloor)$ and $(\lceil s_x \rceil, \lceil s_y \rceil)$ , using either nearest neighbour or interpolation. Corners outside the source are clipped to the border.

Nearest Neighbour Method

Replaces each new pixel with the value of the closest corner of the enclosing block, using any distance metric.

Preserves most fine detail. Still prone to checkerboard and aliasing effects.

Nearest neighbour copies stored values exactly. An indexed image can therefore be resampled without first converting palette indices into RGB values.

def nearest(src, sw, sh, dw, dh):
    kx, ky = dw / sw, dh / sh
    dst = [[0] * dw for _ in range(dh)]
    for y in range(dh):
        for x in range(dw):
            sx, sy = x / kx, y / ky

            # int(round(sx)) - nearest integer
            # min with sw - 1 to clip to border
            nx = min(sw - 1, int(round(sx)))
            ny = min(sh - 1, int(round(sy)))
            dst[y][x] = src[ny][nx]
    return dst

Interpolation

An alternative to direct scaling: reconstruct, or estimate, the continuous intensity function from discrete samples, then resample it at the required resolution.

Linear interpolation
Assumes the variation between samples is a straight line.
Cubic interpolation
Uses a higher-order polynomial, giving a smoother variation.

In 2-D, these become bilinear and bicubic interpolation.

Bilinear Interpolation

The new pixel value is a weighted sum of the 4 surrounding source pixels, using the fractional offsets of the source position.

f_x = s_x - \lfloor s_x \rfloor \qquad f_y = s_y - \lfloor s_y \rfloor

\begin{aligned} v = \ & v_{00}\,(1 - f_x)(1 - f_y) \\ {}+{} & v_{10}\,f_x\,(1 - f_y) \\ {}+{} & v_{01}\,(1 - f_x)\,f_y \\ {}+{} & v_{11}\,f_x\,f_y \end{aligned}

Here:

$f_x, f_y$: fractional distances from the top-left neighbour, in $[0, 1)$
$v_{00}$: top-left neighbour value, at $(\lfloor s_x \rfloor, \lfloor s_y \rfloor)$
$v_{10}$: top-right, $v_{01}$ : bottom-left, $v_{11}$ : bottom-right
$v$: interpolated value, rounded to an integer

Each colour channel is interpolated separately.

Smoother than nearest neighbour, with no checkerboard effect
Blurs fine detail, since every output is an average
Reduces aliasing on downscaling

Nearest neighbour versus bilinear upscaling — A small region upscaled 8 times. Left: Nearest neighbour, with blocky cells. Right: Bilinear, smooth but blurred. Generated from a photo by Hannes Röst, CC BY-SA 3.0.

def clip(v, lo, hi):
    return max(lo, min(hi, v))

def resample(src, sw, sh, dw, dh, mode):
    kx, ky = dw / sw, dh / sh
    dst = [[0] * dw for _ in range(dh)]
    for y in range(dh):
        for x in range(dw):
            sx, sy = x / kx, y / ky

            x0 = clip(int(sx), 0, sw - 1)
            y0 = clip(int(sy), 0, sh - 1)
            x1 = clip(x0 + 1, 0, sw - 1)
            y1 = clip(y0 + 1, 0, sh - 1)

            fx, fy = sx - x0, sy - y0
            dst[y][x] = round(
                src[y0][x0] * (1 - fx) * (1 - fy)
                + src[y0][x1] * fx * (1 - fy)
                + src[y1][x0] * (1 - fx) * fy
                + src[y1][x1] * fx * fy
            )
    return dst

For a colour image, run the same kernel on each channel.

Multi-Scale Pyramids

A series of progressively lower-resolution images derived from one source, each level holding a different band of detail.

Gaussian Pyramid

The stack of Gaussian-filtered, downsampled images.

G_{n+1} = \operatorname{downsample}(H * G_n)

Here:

$G_0$: the original image
$G_{n+1}$: $G_n$ filtered with a Gaussian low-pass filter, then downsampled by 2

Each level halves the width and height. The Gaussian filter removes the high frequencies that would otherwise alias during downsampling.

Gaussian Pyramid Filter

Use the 5×5 Gaussian low-pass filter. Apply its 1-D kernel horizontally, then vertically.

After filtering, retain every second row and column:

G_{n+1}(x,y) = \sum_{i=-2}^{2}\sum_{j=-2}^{2} H(i,j)\,G_n(2x-i,2y-j)

Here:

$H$: 2-D Gaussian-pyramid filter
$G_n$: input pyramid level
$G_{n+1}$: filtered and downsampled pyramid level

Laplacian Pyramid

The detail lost at each downsampling step, stored as a stack of difference images.

L_n = G_n - U(G_{n+1})

Here:

$G_n, G_{n+1}$: consecutive levels of the Gaussian pyramid above
$U(G_{n+1})$: $G_{n + 1}$ upsampled back to level $n$ ‘s resolution
$L_n$: band-pass detail at level $n$

$U(G_{n+1})$ is a blurred version of $G_n$ , since a Gaussian filter and downsampling made $G_{n+1}$ before interpolation back up. Subtracting it leaves only the high-frequency detail that step discarded.

The top level keeps the full Gaussian image $G_N$ , not a difference.

Reconstruction runs bottom-up.

G_n = L_n + U(G_{n+1})

Starting from $G_N$ , each finer level is recovered exactly.

Properties

Both pyramids hold about $\tfrac{4}{3}$ as many pixels as the original. As they form a geometric series:

1 + \tfrac14 + \tfrac{1}{16} + \dots = \frac{1}{1 - \tfrac14} = \tfrac43

That is the raw pixel count, not the encoded size. Only the Laplacian pyramid compresses below the original.

Each $L_n$ is near-zero over smooth regions
Entropy coding then stores $L_n$ in far fewer bits than $G_n$
The extra $\tfrac13$ in pixel count is outweighed by the drop in bits per pixel

Uses:

Progressively encoded compression, coarse levels sent first
Image blending across scales
Image indexing and search

Worked Example

Bilinear Value

The values at $(0,0)$ , $(1,0)$ , $(0,1)$ and $(1,1)$ are $10$ , $30$ , $50$ and $90$ . Find the value at $(0.25, 0.5)$ .

Solution Reveal Hide

r_0 = 10(1-0.25) + 30(0.25) = 15

r_1 = 50(1-0.25) + 90(0.25) = 60

P(0.25, 0.5) = 15(1-0.5) + 60(0.5) = 37.5

The interpolated value is $37.5$ .