The image signal processor: demosaicing, white balance and noise

Before any clever multi-frame trick or neural network can run, raw sensor numbers must be cleaned, coloured and made linear. These early stages are handled mostly by the image signal processor (ISP), and their quality sets a ceiling on everything that follows. PHOT-201 introduced the Bayer pattern and the idea of demosaicing; this lesson looks inside the algorithms. You will model sensor noise mathematically, run a bilinear demosaicing algorithm on a synthetic image and measure its errors, estimate white balance automatically with two classic methods, and see how a colour correction matrix maps sensor colours to a standard colour space. These are the same building blocks phone and camera engineers tune, and understanding them explains many artefacts photographers see.

The early raw stages

A raw readout is not yet a clean measurement of light. The ISP applies several corrections in order.

• Black level subtraction: even with no light, each pixel reads a small offset value, often 64 or 256 on a 10- or 12-bit scale. This pedestal is subtracted so that zero light reads zero.
• Defective pixel correction: a few pixels are permanently stuck bright (hot) or dark (dead). The ISP keeps a map of them and replaces each with an estimate from neighbours of the same colour.
• Lens shading correction: small lenses deliver less light to the corners than the centre (vignetting) and can shift colour toward the edges. A per-channel gain map, calibrated at the factory for each module, flattens this.
• Linearization: the sensor response is made proportional to light, correcting any nonlinearity near saturation.
• Clipping detection: pixels at or near the saturation value are flagged, because their true value is unknown and later stages, such as HDR merging, must treat them specially.

These steps sound mundane, but errors in them cause familiar problems: coloured corners in phone photos of blank walls come from imperfect lens shading correction; magenta highlights come from mishandled clipping in one channel; a faint pattern in deep shadows can come from an inaccurate black level.

Modelling sensor noise

Noise in a raw pixel has two main parts. Shot noise depends on the signal: its variance equals the number of photoelectrons. Read noise comes from the electronics and is roughly constant regardless of signal. Engineers combine them in a Poisson-Gaussian model, in which the variance of a pixel value \( x \) is a linear function of its mean:

Here \( a \) depends on the sensor gain (ISO setting) and \( b \) on read noise. The model is calibrated for each sensor and ISO by photographing flat grey targets and plotting variance against mean. Denoising algorithms, burst merging and HDR all use this noise model to decide how much to trust each pixel. A pixel in a bright area has high absolute noise but high SNR; a pixel in deep shadow has low absolute noise but low SNR because read noise dominates.

That last point matters for burst photography. Each frame contributes its own read noise, so merging \( n \) short frames has \( n \) times the read noise variance of one long frame with the same total exposure. Modern sensors have very low read noise, which is part of why burst methods work so well; lesson 6 returns to this trade-off in night modes.

Demosaicing algorithms

A Bayer colour filter array records red at one pixel in four, blue at one in four and green at two in four. Demosaicing estimates the two missing colours at every pixel. The simplest method, bilinear interpolation, averages the nearest neighbours of each colour. At a red pixel, for example, the missing green is the average of the four green neighbours above, below, left and right, and the missing blue is the average of the four diagonal blue neighbours. Bilinear interpolation is fast and works well in smooth areas, but at sharp edges it averages across the edge, producing colour fringes and zipper artefacts, a jagged alternating pattern along edges.

Better algorithms are edge-aware. They estimate the direction of local edges, horizontal or vertical, and interpolate along rather than across them. Others exploit the fact that colour differences (red minus green, blue minus green) vary more smoothly than colours themselves, so they interpolate green first at full resolution and then interpolate the colour differences. Well-known methods include adaptive homogeneity-directed (AHD) demosaicing and the many variants in raw converters. Since about 2016, learned demosaicing using convolutional neural networks has outperformed hand-designed methods on benchmarks, and joint demosaicing and denoising networks treat both problems together, because noise and missing samples interact.

import numpy as np
rng = np.random.default_rng(420)
h, w = 32, 32
img = np.zeros((h, w, 3))
img[:, :16] = [0.8, 0.3, 0.2]          # warm left half
img[:, 16:] = [0.2, 0.4, 0.9]          # cool right half: a sharp vertical edge
img += rng.normal(0, 0.005, img.shape)  # a little texture
# RGGB Bayer mosaic: keep one channel per pixel
mask = np.zeros((h, w, 3), bool)
mask[0::2, 0::2, 0] = True              # R
mask[0::2, 1::2, 1] = True              # G
mask[1::2, 0::2, 1] = True              # G
mask[1::2, 1::2, 2] = True              # B
mosaic = np.where(mask, img, 0.0)
def conv3(x, k):
    p = np.pad(x, 1, mode="reflect")
    return sum(k[i, j] * p[i:i + h, j:j + w] for i in range(3) for j in range(3))
k_g = np.array([[0, 1, 0], [1, 4, 1], [0, 1, 0]]) / 4
k_rb = np.array([[1, 2, 1], [2, 4, 2], [1, 2, 1]]) / 4
out = np.stack([conv3(mosaic[..., 0], k_rb), conv3(mosaic[..., 1], k_g), conv3(mosaic[..., 2], k_rb)], -1)
err = np.abs(out - img).mean(axis=2)
print(f"mean error away from the edge: {err[:, 2:12].mean():.4f}")
print(f"mean error in the 4 columns at the edge: {err[:, 14:18].mean():.4f}")
print(f"edge error is {err[:, 14:18].mean() / err[:, 2:12].mean():.0f} times larger")

mean error away from the edge: 0.0032
mean error in the 4 columns at the edge: 0.0581
edge error is 18 times larger

The output shows the core weakness of bilinear demosaicing: error near the edge is many times larger than in flat areas, and because each channel is interpolated from different positions, the errors differ by channel and appear as colour fringes. Edge-aware and learned methods attack exactly this region.

Aliasing, moiré and false colour

Fine repeating patterns, such as fabric weaves, screens and distant brickwork, can contain detail finer than the colour filter array samples. The result is aliasing: false patterns and false colours that were not in the scene. Many dedicated cameras have removed the optical low-pass filter that used to blur detail slightly to prevent this, relying on high pixel counts and software. Phones use very small pixels and typically bin pixels in quad-Bayer arrangements (groups of four same-colour pixels), which changes the sampling and requires special remosaicing algorithms when full resolution is wanted. Learned demosaicing reduces moiré considerably but cannot recover detail that was never sampled; it can only make a plausible guess. PHOT-201 covers pixel-shift modes, which avoid demosaicing by capturing each colour at each location.

White balance algorithms

The colour of light varies widely, from warm tungsten to cool shade, and the eye adapts to it. A camera must estimate the illuminant and apply per-channel gains so that neutral surfaces appear neutral. This is automatic white balance (AWB). Two classic assumptions are easy to compute. The grey world assumption says that the average colour of a scene is neutral grey, so the gains should equalize the channel means. The white patch (or max-RGB) assumption says that the brightest surface is white, so the gains should equalize the channel maxima. Both fail predictably: grey world fails when a scene is dominated by one colour, such as a forest or a red wall, and white patch fails when there is no white surface or when the brightest pixels are clipped or coloured light sources.

import numpy as np
rng = np.random.default_rng(420)
n = 5000
reflect = rng.uniform(0.05, 0.9, (n, 3))      # random surfaces
reflect[:50] = 0.95                            # a few white surfaces
light = np.array([1.0, 0.8, 0.55])             # warm illuminant (relative R, G, B)
seen = reflect * light
def gains_avg(x): m = x.mean(axis=0); return m[1] / m
def gains_max(x): m = np.percentile(x, 99.5, axis=0); return m[1] / m
true = light[1] / light
for name, g in [("average", gains_avg(seen)), ("max-RGB", gains_max(seen))]:
    print(f"{name:8s} gains R {g[0]:.2f} G {g[1]:.2f} B {g[2]:.2f}  error {np.abs(g - true).max():.3f}")
print(f"true gains R {true[0]:.2f} G {true[1]:.2f} B {true[2]:.2f}")
# a scene dominated by green foliage breaks the average method
fol = reflect * np.array([0.4, 1.0, 0.4])
g = gains_avg(fol * light)
print(f"foliage scene, average-method gains R {g[0]:.2f} B {g[2]:.2f} (true R {true[0]:.2f} B {true[2]:.2f})")

average  gains R 0.78 G 1.00 B 1.45  error 0.017
max-RGB  gains R 0.80 G 1.00 B 1.45  error 0.000
true gains R 0.80 G 1.00 B 1.45
foliage scene, average-method gains R 1.96 B 3.62 (true R 0.80 B 1.45)

In the program, "average" is the grey world method and "max-RGB" is the white patch method. Both recover the warm illuminant well on a varied scene, but grey world badly overcorrects the foliage scene, pushing it toward magenta. Modern phones use learned AWB: a neural network trained on thousands of scenes with known illuminants, often combined with face detection, an ambient light sensor and scene recognition. Google's approach, for example, has been described as a learning-based method trained on large labelled datasets. Even so, mixed lighting, such as a window and tungsten lamp in one room, has no single correct answer, and some pipelines now apply different white balance to different regions.

The colour correction matrix

After white balance, sensor colours still do not match a standard colour space, because the sensor's red, green and blue filters have different spectral responses from the human eye's cones and from the primaries of sRGB or Display P3. A colour correction matrix (CCM), a 3 by 3 matrix calibrated for each sensor, maps white-balanced camera RGB to linear standard RGB:

Each row usually sums to 1 so that neutral greys stay neutral. Large off-diagonal values, typically negative, increase saturation to compensate for overlapping filter responses, but they also amplify noise, because subtracting noisy channels from each other adds their variances. This is one reason low-light colour looks blotchy: the CCM amplifies chroma noise. Raw converters often use two CCMs calibrated under different illuminants and interpolate between them according to the estimated white balance. Professional colour management, covered in PHOT-202, starts from this point.

Tuning, artefacts and what photographers see

ISP tuning is a craft. Image quality engineers adjust dozens of parameters, from demosaicing thresholds to noise reduction strength at each ISO, and test them on standard charts and thousands of real scenes. Common artefacts trace back to these stages: maze-like patterns in fine texture come from demosaicing; colour shifts in corners come from lens shading; waxy skin comes from noise reduction applied before or during demosaicing; colour casts in mixed light come from AWB. Raw converters let you redo some stages with different algorithms, which is why converting the same raw file in two programs can give visibly different detail. When you evaluate a phone or camera for professional use, test fine fabric, foliage, mixed light and deep shadow, because these are where ISP choices show.