The two great themes of spatial imaging are **convolution** (a measurement integrates over a region — the PSF) and **sampling** (a discrete grid can only represent frequencies below Nyquist, else they alias). This section's single claim is that **both recur, identically, on the *time* axis** — and together they organise the entire part. A frame is an *integral over the exposure* → **motion blur** is convolution *in time*. A video is a *sequence of temporal samples* → motion faster than the frame rate **aliases** (the wagon wheel) *in time*. And the question "do I think of motion as *particles I follow* or *a time-series at each fixed pixel*?" is the **Lagrangian vs Eulerian** distinction that separates flow/tracking from video magnification.

💡 **Big lesson (recurrence of L5 / L16 — Nyquist, and *prefilter before you downsample*, on the time axis):** the spatial sampling laws apply unchanged in **time**. **L5:** a frame rate $f_s$ can only faithfully capture temporal frequencies below $f_s/2$; faster motion **folds down** to a false low frequency — the **wagon-wheel effect** is temporal moiré. **L16:** the cure for aliasing is to **prefilter before sampling** — and in time the prefilter is *built into the camera*: integrating over the exposure window (which produces motion blur) is exactly the temporal **anti-alias filter**. So motion blur and temporal aliasing are not two problems but **the same tradeoff** — a longer exposure removes strobing by *blurring*, a shorter one gives sharp frames that *strobe*. (→ see Big lessons **L5** & **L16**, first placed in [[Linearity, aliasing and deblurring]] / [[Basic image processing and ISP]]; the **wagon-wheel** is L16's named temporal instance.)

The still-image part of this book taught one compression idea over and over: **don't store what the viewer won't miss, and don't store what you can predict.** JPEG ([[File formats and compression]]) throws away high-frequency chroma and finely-quantizes the DCT because perception won't notice. Video adds a second, far larger redundancy that stills can't touch: **the next frame looks almost exactly like this one.** At 30–60 fps the camera and the world barely move between frames; most of the picture is *already on screen*. Coding each frame independently (so-called **Motion-JPEG** — literally JPEG-per-frame) ignores this entirely and is wildly wasteful. Real codecs **predict each frame from its neighbours and code only the prediction error.** That is the whole subject.

💡 **Big lesson (correspondence on a budget):** **motion compensation is optical flow you can afford.** A codec needs, for every block, *where did this come from in a frame we already have?* — exactly the correspondence question of [[Optical flow]]. But it does not need a physically correct, dense, sub-pixel flow field; it needs a **coarse, block-constant** field that is **cheap to estimate** and, crucially, **cheap to transmit** — and it chooses that field to **minimise total bits**, not endpoint error. So decades before learned optical flow, video codecs were already estimating dense-ish correspondence at massive scale — just optimised for rate, not accuracy. The recurring move: *reuse what the decoder already has, and code only the difference.* (Recurs as the framing for [[Video magnification]], which keeps the difference instead of discarding it; and is the rate-aware cousin of the correspondence story in [[Optical flow]].)

You cannot see a heartbeat in a face, or a skyscraper sway, or a wall flex under a passing truck — these changes are real but **below the threshold of perception**, buried in pixel values that look constant. Yet the information is *there* in an ordinary video: the green channel of a cheek really does brighten and dim with each pulse; an edge really does move a fraction of a pixel as a structure vibrates. **Video magnification** is the family of methods that **pulls out these tiny temporal variations and scales them up** until they're plainly visible — turning a normal camera into an instrument for the invisible.

💡 **Big lesson (Eulerian vs Lagrangian — the cheaper view often wins):** there are two ways to amplify small motion, the same dichotomy from [[Motion blur and temporal sampling]]. The **Lagrangian** way *follows points*: estimate the optical-flow displacement $\delta(t)$ of each feature, multiply it, and re-render the frame with the exaggerated motion. It's intuitive but **fragile** — it inherits every failure of [[Optical flow]] (occlusion, aperture problem, sub-pixel error), and tracking *tiny* motions accurately is exactly where flow is weakest. The **Eulerian** way *stays put*: at each **fixed pixel** treat the value over time as a signal, **band-pass** it, amplify, add back — **no motion estimation at all**. For small variations this is dramatically simpler and more robust, and it amplifies **color** changes (a pulse) just as naturally as motion. The recurring moral: *picking the right frame of reference (fixed grid vs moving point) can turn a hard estimation problem into a trivial filtering one.*

The previous chapters made motion an *enemy to estimate* (optical flow) or a *cue to track* (KLT). Here motion is the thing we want to **edit**: handheld footage carries an involuntary high-frequency camera path on top of the intended one. Every method in this chapter is one pipeline — **measure the real path, design a better path, warp the frames onto it** — and the only thing you give up is the **border pixels** that the better path no longer sees.

💡 **Big lesson (recurrence of L16 · prefilter before you downsample / temporal aliasing):** stabilization is *temporal* signal processing on the camera-path signal — and the same Nyquist intuition applies. We **low-pass the trajectory** to remove jitter, and rolling-shutter "jello" is precisely a **temporal aliasing** artifact (rows sampled at different times under fast motion, the video cousin of the wagon-wheel effect). (→ see Big lesson **L16**, first placed in BASIC → Resampling; here it recurs as *smooth the path in time, and beware time-varying sampling within a frame*.)

A normal camera shoots, say, 30 fps; to play a moment 8× slower and smooth you'd need ~240 distinct instants per second that were never recorded. You can either **capture** them (a high-speed camera — true, but expensive and light-hungry) or **synthesize** them. Frame interpolation synthesizes: given two real frames, invent the ones that belong **between** them. The whole problem reduces to a question you already met in [[Morphing]] — *what moved where?* — answered here by **optical flow** or by a **learned** model.

💡 **Big lesson (recurrence of L14 · capture the full set, decide later — and its limits):** high-speed photography is the honest way to own every instant (capture the full temporal set, pick the moment/shutter later). Interpolation is the **budget substitute**: when you *didn't* capture the full set, a **prior** invents the missing instants. So this chapter sits at the seam between **L14** (capture everything) and **L10** (the prior is not optional) — the in-between frames are partly **reconstructed** (where flow is reliable) and partly **hallucinated** (across occlusions). (→ see Big lessons **L14**, first in MULTIPLE EXPOSURE, and **L10**, first in Super-resolution.)

This closing chapter spends the part's machinery. Editing is where alignment, flow, warping, and robust per-pixel statistics stop being algorithms and become **tools an editor reaches for**. It's deliberately lighter and demo-driven — and honest where a specific product or paper is uncertain (marked *queue*). The through-line: almost every "video effect" is **align the frames, then reduce or select across time**, or **re-time the timeline**.