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Image resampling evaluates image data on a new sampling grid, often while resizing, rotating or rectifying an image. An output pixel is mapped to a source location, which may lie between stored source pixels. An interpolation rule estimates its value; a boundary rule handles locations outside the recorded image.^[ref]

For example, bilinear interpolation at the center of four source samples with values $0,0,0,100$ gives their equally weighted mean, 25. Nearest-neighbor interpolation instead selects one sample and retains its value. These methods trade smoothness against preservation of discrete labels. Reducing image size also changes the sampling rate: a low-pass filter before subsampling can suppress aliasing, whereas interpolation alone cannot guarantee its removal. Stereo rectification uses image resampling to construct aligned virtual views without physically moving either camera.

Image stack axis calibration assigns physical coordinates to the indices of an image stack. It states what each axis means, its spacing, origin and orientation. Pixel indices, section indices and frame indices are distinct from positions and times. In ImageJ, image properties include pixel width, pixel height, voxel depth and frame interval; its Origin fields are expressed in pixels.^[ref]

For an original general spatial model, let the index vector be $\mathbf{n}=(i,j,k{)}^T$ and definewhere ${\mathbf{r}}_0$ is a physical origin and $R$ is a rotation matrix. This is an affine map; the rotation represents a chosen acquisition-to-world orientation rather than a claim that ImageJ's Properties dialog implements an arbitrary rotation. Convert any pixel-unit origin into the chosen physical convention before using this equation.

Take spacings $(s_x,s_y,s_z)=(0.2,0.3,1.0)\mu \mathrm{m}$, origin ${\mathbf{r}}_0=(10,-4,2)\mu \mathrm{m}$ and a 90° rotation about the positive third axis:Index $(3,2,1)$ first maps to local displacement $(0.6,0.6,1.0)\mu \mathrm{m}$ and then to world position $(9.4,-3.4,3.0)\mu \mathrm{m}$. Its distance from the physical origin is $\sqrt{0.6^2+0.6^2+1^2}\simeq 1.31\mu \mathrm{m}$, unchanged by the rotation.

Cropping and axis permutation require corresponding metadata changes. If a crop begins at old index $\mathbf{c}$, its new physical origin is ${\mathbf{r}}_0^{\prime }={\mathbf{r}}_0+R\mathrm{diag}(s_x,s_y,s_z)\mathbf{c}$. For irregular section positions, store those positions explicitly instead of inventing one uniform depth increment. Image resampling may alter the grid, but it does not create independently acquired depth information.

Section-thickness anisotropy describes unequal spatial sampling along and across a serial-section image stack. For uniform section thickness $\Delta z$ and square lateral pixel pitch $p$, a useful dimensionless ratio isIt compares assigned sampling scales, rather than automatically specifying an achieved spatial resolution. Section collection followed by image alignment and stacking produces a physical cutting coordinate that must be calibrated separately from lateral image dimensions.^[ref]

For an original example, let $p=4\mathrm{nm}$ and $\Delta z=50\mathrm{nm}$. Then $r=12.5$ and a rectangular voxel has volume $p^2\Delta z=800\mathrm{n}{\mathrm{m}}^3$. A displacement of three pixels laterally and two section intervals axially has physical lengthTreating all three index coordinates as though they had spacing $p$ would instead give $p\sqrt{3^2+2^2}\simeq 14.42\mathrm{nm}$, a serious geometrical error.

Image resampling to $4\mathrm{nm}$ cubic voxels can regularize display coordinates, but the inserted axial samples are interpolation, not new independent observations. Likewise, 200 uniform sections represent $200\Delta z=10\mu \mathrm{m}$ of thickness while the first and last centers are only $199\Delta z=9.95\mu \mathrm{m}$ apart. For variable or missing sections, retain the actual section coordinates rather than assuming a single constant $\Delta z$.

Serial-section electron microscopy reconstructs a volume from an ordered image stack of thin physical slices. In the transmission electron microscopy implementation, an ultramicrotome cuts successive slices, which are collected on a support and imaged through the specimen. Image registration aligns corresponding features; image segmentation identifies structures to follow between slices. JEOL's glossary describes sections of 40–100 nm and illustrates the separate registration and segmentation stages.^[ref]

The motivation is geometrical: one isolated slice cannot show whether a feature continues, divides, or ends outside that slice. An ordered image stack provides observations along the third coordinate, although finite section-thickness anisotropy and missing material limit what can be inferred. A change between neighboring images may come from specimen structure, a slice deformation, or imperfect image registration.

For a chosen numerical example, suppose each of $N=200$ consecutive sections has uniform thickness $\Delta z=50\mathrm{nm}$, with lateral pixel pitch $p=4\mathrm{nm}$. Ignoring lost material and thickness variation, the sections coverThe voxel sampling is 12.5 times coarser along the cutting direction. These are chosen dimensions, not measurements from either cited experiment. The first and last section centers are only $(N-1)\Delta z$ apart: center-to-center extent differs from total represented thickness. Neither spacing is automatically the achieved spatial resolution.

Image resampling onto a cubic grid can make an image stack easier to display, but interpolation supplies no additional independent observations. Likewise, smooth image registration cannot resolve missing-section ambiguity in volume reconstruction: a small connecting feature could lie wholly inside lost material. Checking a reconstructed connection therefore needs evidence in the acquired slices, not just a visually smooth rendered surface.

A historically important automation change illustrates the distinction between section and surface imaging. In 2004, Winfried Denk and Heinz Horstmann placed an ultramicrotome with a diamond knife inside a scanning electron microscope. Their serial block-face scanning electron microscopy imaged the exposed block surface between cuts instead of collecting the removed sections. They reported stacks several hundred layers deep, 50–70 nm sectioning, and typically less than 10 nm lateral position jitter. These observations tested an apparatus for acquiring closely aligned serial images; they did not remove the finite spacing between observations.^[ref]

A mechanical detail makes the measurement chain concrete. Their specimen-advance lever reduced the motorized micrometer's motion by 0.11. Thus its stated position measurement uncertainty of 50 nm contributed $0.11\times 50=5.5\mathrm{nm}$ after that reduction; other sources of section-thickness error still required assessment.^[ref]

Stereo rectification transforms a pair of images so that corresponding epipolar lines become aligned scan lines. In a horizontal stereo arrangement, a scene point visible in both rectified views should have the same vertical image coordinate. This turns a two-dimensional matching search into a one-dimensional search along a row.^[ref]

For example, a feature initially at left row 100 and right row 104 can be mapped to the same row in virtual aligned views. Camera calibration supplies geometry for the transforms, and image resampling evaluates the transformed images on a new pixel grid. The new images need not retain every border region. Stereo rectification aligns the geometric search; it does not identify the correct partner among repeated patterns or restore a feature hidden in one view. The rectified focal length and coordinate offsets must also be used when converting stereo disparity to depth.