The Central Idea
Spatial data is information tied to location.
A digital representation typically connects:
\[ \boxed{ \text{location} + \text{stored value} + \text{coordinate system} \longrightarrow \text{spatial data} }. \]In an image, the stored values may describe brightness or color. In a map, they may describe elevation, land use, boundaries, roads, or measured quantities.
Raster Data
A raster represents space using a regular grid of cells.
In an image, these cells are pixels.
A raster can be represented as a matrix:
\[ \boxed{ A= \begin{bmatrix} a_{11} & a_{12} & \cdots\\ a_{21} & a_{22} & \cdots\\ \vdots & \vdots & \ddots \end{bmatrix} }. \]Each entry stores a value associated with one cell location.
Pixels
A pixel is one sample location in a digital image.
A pixel may store:
- brightness;
- color channels;
- depth;
- temperature;
- classification labels;
- other measured or derived values.
Pixel Coordinates
A pixel location is often indexed by row and column:
\[ \boxed{(r,c)}. \]Some systems instead use image coordinates
\[ \boxed{(u,v)}. \]The location of the origin and the direction of the vertical axis depend on the software convention.
Image Coordinate Conventions
In ordinary analytic geometry, the vertical coordinate often increases upward.
In many image systems, row or vertical pixel coordinates increase downward from the top of the image.
Therefore, a coordinate transformation may be required when switching between mathematical and image coordinate systems.
Image Dimensions
A raster image may be described by width and height in pixels:
\[ \boxed{W\times H}. \]For example,
\[ 1920\times1080 \]means \(1920\) columns and \(1080\) rows.
Total Pixel Count
The total number of pixel positions is
\[ \boxed{N=WH}. \]For a
\[ 1920\times1080 \]image,
\[ N=1920(1080)=2{,}073{,}600. \]Therefore,
\[ \boxed{N=2{,}073{,}600\text{ pixels}}. \]Spatial Resolution
Spatial resolution describes how finely space is sampled.
If one pixel represents square side length \(s\), then smaller \(s\) means finer spatial resolution.
Higher resolution can preserve smaller details but requires more data.
Ground Sample Distance
In mapped or aerial imagery, a useful concept is the physical distance represented by one pixel.
If one pixel corresponds to
\[ g\text{ m}, \]then a horizontal image measurement of \(n\) pixels represents
\[ \boxed{d=ng}. \]Example: Pixel Distance to Physical Distance
Suppose one pixel represents
\[ 0.25\text{ m}. \]A feature measures
\[ 80\text{ pixels} \]across.
Then
\[ d=80(0.25)=20. \]Therefore,
\[ \boxed{d=20\text{ m}}. \]Pixel Area
If one square pixel represents side length \(g\), then each pixel represents physical area
\[ \boxed{A_{\text{pixel}}=g^2}. \]If \(n\) pixels represent a region, an approximate area is
\[ \boxed{A\approx ng^2}. \]Example: Area from Pixel Count
Suppose one pixel represents
\[ 0.5\text{ m}\times0.5\text{ m}. \]Its area is
\[ 0.25\text{ m}^2. \]If a region contains \(400\) classified pixels, its approximate area is
\[ 400(0.25)=100. \]Therefore,
\[ \boxed{100\text{ m}^2}. \]Resolution and Feature Size
A feature much smaller than one pixel cannot be represented directly by its full geometry.
Small features may:
- disappear;
- merge with neighboring features;
- occupy only part of a pixel;
- appear with distorted boundaries.
Spatial resolution therefore limits what can be measured reliably.
Sampling
A digital raster samples a continuous spatial scene at discrete locations.
If a continuous field is
\[ f(x,y), \]a grid stores values such as
\[ \boxed{ f(x_i,y_j) }. \]Sampling converts continuous space into a finite numerical representation.
Sampling Interval
If neighboring samples are separated by
\[ \Delta x \]horizontally and
\[ \Delta y \]vertically, these spacings define the grid resolution.
Smaller intervals provide finer sampling.
Aliasing
If spatial detail changes more rapidly than the grid can represent, aliasing can occur.
Fine patterns may appear as:
- false larger patterns;
- jagged boundaries;
- incorrect textures;
- missing detail.
Adequate sampling is therefore essential for faithful spatial representation.
Vector Data
Vector spatial data represents geometry using explicit geometric objects rather than a fixed cell grid.
Common objects include:
- points;
- polylines;
- polygons;
- curves;
- meshes.
Raster Versus Vector
Raster data stores values over a grid.
Vector data stores explicit geometric features.
A satellite image is naturally raster-like, while a road centerline or property boundary is naturally vector-like.
Advantages of Raster Representation
Raster data is well suited for:
- imagery;
- continuous fields;
- elevation grids;
- temperature maps;
- classification surfaces.
Grid-based calculations are also often straightforward computationally.
Advantages of Vector Representation
Vector data is well suited for:
- precise boundaries;
- roads;
- parcel outlines;
- networks;
- explicit geometric editing.
Geometric objects can often be scaled without the same pixelation associated with raster enlargement.
Georeferencing
An image becomes more useful spatially when its pixel positions are tied to real-world coordinates.
A simplified transformation might have the form
\[ \boxed{ x=x_0+s_xu, \qquad y=y_0+s_yv }. \]More general georeferencing can include rotation, shear, and nonlinear corrections.
Affine Image Transformations
An affine transformation in two dimensions can be written
\[ \boxed{ \begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} a & b\\ c & d \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} e\\ f \end{bmatrix} }. \]This can represent translation, rotation, scaling, shear, and combinations of these transformations.
Control Points
Georeferencing often uses known point correspondences.
A control point links an image position
\[ (u_i,v_i) \]with a known spatial coordinate
\[ (x_i,y_i). \]Several control points can be used to estimate the transformation.
Registration
Image registration aligns two images or spatial datasets so corresponding features occupy matching positions.
Registration may correct:
- translation;
- rotation;
- scale difference;
- perspective change;
- local distortion.
Image Resampling
After an image is transformed, new pixel values must be assigned on the destination grid.
This process is called resampling.
Common approaches include:
- nearest-neighbor assignment;
- bilinear interpolation;
- higher-order interpolation.
Nearest-Neighbor Resampling
Nearest-neighbor resampling assigns the value of the nearest source sample.
It is simple and preserves original category values well, but can produce blocky results when enlarging imagery.
Linear Interpolation
Between two values \(A\) and \(B\), linear interpolation is
\[ \boxed{ L(t) = (1-t)A+tB, \qquad 0\le t\le1 }. \]Bilinear interpolation extends this idea across a rectangular grid cell.
Bilinear Interpolation
Suppose four surrounding raster values are
\[ f_{00},\quad f_{10},\quad f_{01},\quad f_{11}. \]A point inside the cell can be estimated by interpolating in one direction and then in the other.
The resulting value is a weighted combination of all four neighbors.
Distance in Pixel Space
For image points
\[ P_1=(u_1,v_1), \qquad P_2=(u_2,v_2), \]Euclidean pixel distance is
\[ \boxed{ d_p = \sqrt{ (u_2-u_1)^2 +(v_2-v_1)^2 } }. \]Converting Pixel Distance to Spatial Distance
If one pixel represents physical length \(g\) equally in both directions, then
\[ \boxed{ d=g\,d_p }. \]If horizontal and vertical scales differ, each coordinate difference must be scaled separately before computing distance.
Example: Diagonal Image Measurement
Two points differ by
\[ 30\text{ pixels} \]horizontally and
\[ 40\text{ pixels} \]vertically.
Their pixel distance is
\[ d_p=\sqrt{30^2+40^2}=50. \]If each pixel represents
\[ 0.2\text{ m}, \]the physical distance is
\[ 50(0.2)=10. \]Therefore,
\[ \boxed{10\text{ m}}. \]Segmentation
Segmentation divides an image into regions or objects.
A segmentation might identify:
- buildings;
- roads;
- water;
- vegetation;
- cells;
- manufactured components.
Once a region is identified, geometric measurements can be extracted from it.
Binary Masks
A simple segmentation can be represented by a binary mask:
\[ \boxed{ M(i,j) = \begin{cases} 1, & \text{pixel belongs to the region},\\ 0, & \text{otherwise}. \end{cases} }. \]Counting the \(1\)-valued pixels gives the region's raster area in pixel units.
Area from a Binary Mask
If a segmented region contains \(N\) pixels and each pixel represents area \(a\), then
\[ \boxed{ A=Na }. \]Centroid of a Pixel Region
Suppose a region contains pixel coordinates
\[ (u_i,v_i), \qquad i=1,\ldots,N. \]Its coordinate centroid is
\[ \boxed{ \bar{u} = \frac1N\sum_i u_i, \qquad \bar{v} = \frac1N\sum_i v_i }. \]Bounding Boxes
A simple way to summarize a detected object is with an axis-aligned bounding rectangle.
If its coordinate extremes are
\[ u_{\min},u_{\max},v_{\min},v_{\max}, \]then width and height are
\[ \boxed{ W=u_{\max}-u_{\min} } \]and
\[ \boxed{ H=v_{\max}-v_{\min} }. \]Aspect Ratio of a Detected Region
If a bounding box has width \(W\) and height \(H\), its aspect ratio is
\[ \boxed{ \frac{W}{H} }. \]Aspect ratio can help distinguish long narrow objects from compact ones.
Perimeter in Raster Data
Estimating perimeter from pixels is more difficult than estimating area.
Pixel boundaries create stair-step approximations to smooth curves.
The measured perimeter can therefore depend strongly on resolution and the method used.
Connectivity
Raster regions can be analyzed according to which neighboring pixels count as connected.
Common choices include:
- edge-sharing neighbors;
- edge- or corner-sharing neighbors.
The connectivity rule affects how regions are classified as separate or connected.
Four-Neighbor Connectivity
Under a four-neighbor rule, a pixel is adjacent to the cells directly:
- above;
- below;
- left;
- right.
Diagonal contact alone does not create connectivity.
Eight-Neighbor Connectivity
Under an eight-neighbor rule, diagonal neighbors are also included.
Two regions touching only at one corner may therefore be separate under one convention and connected under another.
Edges
Image edges occur where pixel values change rapidly across space.
If brightness is represented by
\[ I(x,y), \]large values of
\[ \|\nabla I\| \]can indicate strong local transitions.
Image Gradient
The image gradient is
\[ \boxed{ \nabla I = \left\langle \frac{\partial I}{\partial x}, \frac{\partial I}{\partial y} \right\rangle }. \]It points toward the strongest local increase in image intensity.
Discrete Image Gradients
Digital images do not provide continuous derivatives directly.
A simple finite-difference approximation is
\[ \boxed{ I_x \approx I(i,j+1)-I(i,j) }. \]Similar differences estimate vertical change.
Contours
A detected boundary can be represented as an ordered sequence of points:
\[ \boxed{ P_1,P_2,\ldots,P_n }. \]This converts a raster boundary into a polygonal or polyline-like geometric representation.
Polygonal Approximation
A complex contour may contain many points.
A simplified polygon can approximate the same boundary with fewer vertices.
This trades geometric detail for compact representation.
Convex Hull
The convex hull of a set of points is the smallest convex region containing them.
It provides a useful outer envelope for an irregular detected shape.
Area Ratio to Convex Hull
If a region has area \(A\) and its convex hull has area \(A_h\), then
\[ \boxed{ \frac{A}{A_h} }. \]can be used as a simple measure of how much the shape fills its convex envelope.
Shape Measurements
Image-derived objects can be characterized by:
- area;
- perimeter;
- centroid;
- bounding box;
- aspect ratio;
- orientation;
- convex hull;
- compactness;
- distance to neighboring objects.
Orientation
An elongated detected region may have a dominant direction.
This orientation can be represented by an angle
\[ \boxed{\theta}. \]Orientation allows shape alignment, comparison, and rotated bounding geometry.
Image Moments
Image moments summarize how pixel values or region mass are distributed.
For a binary region, a basic moment is
\[ \boxed{ m_{pq} = \sum_x\sum_y x^py^qI(x,y) }. \]where \(I(x,y)\) is \(1\) inside the region and \(0\) outside.
Centroid from Moments
For a nonempty binary region,
\[ \boxed{ \bar{x} = \frac{m_{10}}{m_{00}}, \qquad \bar{y} = \frac{m_{01}}{m_{00}} }. \]The quantity \(m_{00}\) corresponds to the region's pixel area.
Intensity Profiles
A line through an image can be sampled to produce a one-dimensional profile.
If a path is
\[ \mathbf{r}(t), \]then the sampled image intensity is
\[ \boxed{ I(\mathbf{r}(t)) }. \]Peaks, valleys, and transitions can reveal spatial structure along that path.
Digital Elevation Models
A digital elevation model stores height values over a spatial grid.
It can be represented as
\[ \boxed{z=f(x,y)}. \]From this raster, we can derive slope, aspect, contours, drainage direction, and other geometric quantities.
Slope from Elevation Data
If elevation is
\[ z=f(x,y), \]the gradient
\[ \nabla f \]describes local horizontal change in elevation.
Larger gradient magnitude corresponds to steeper terrain.
Contours from Elevation Data
A contour line is a level curve:
\[ \boxed{ f(x,y)=c }. \]Every point on the contour has the same elevation \(c\).
Point Clouds
A point cloud stores many sampled spatial positions:
\[ \boxed{ P_i=(x_i,y_i,z_i) }. \]Points may also include attributes such as color, intensity, classification, or time.
From Point Clouds to Surfaces
A point cloud does not automatically define which points should be connected.
Surface reconstruction may create:
- triangular meshes;
- grids;
- implicit surfaces;
- other geometric approximations.
Nearest-Neighbor Queries
Given query point \(Q\), a nearest-neighbor problem seeks point \(P_k\) minimizing
\[ \boxed{ d(Q,P_k) }. \]This supports matching, classification, reconstruction, and spatial indexing.
Spatial Indexing
Large datasets benefit from structures that organize points or regions by location.
Spatial indexes help answer queries such as:
- what is nearby?;
- what intersects this region?;
- what lies inside this box?;
- which object is nearest?
The mathematical goal is to avoid checking every object unnecessarily.
Image Histograms
An image histogram counts how frequently intensity or color values occur.
A histogram summarizes value distribution but does not preserve the original spatial arrangement.
Two images can have similar histograms while having very different geometry.
Spatial Information Versus Value Information
Knowing which values occur is different from knowing where they occur.
Spatial analysis depends on location relationships such as:
- adjacency;
- distance;
- containment;
- orientation;
- connectivity;
- overlap.
Multiband Images
A pixel may store more than one value.
A multiband image can be modeled as
\[ \boxed{ \mathbf{I}(x,y) = \langle I_1(x,y), I_2(x,y), \ldots, I_n(x,y) \rangle }. \]Each band represents a different measured or derived quantity.
Three-Dimensional Imaging
Some imaging systems produce volumetric data rather than a single planar image.
A volume can be sampled on a three-dimensional grid:
\[ \boxed{ I(i,j,k) }. \]Each sample is often called a voxel.
Voxels
A voxel is the three-dimensional analogue of a pixel.
If voxel side lengths are
\[ \Delta x,\Delta y,\Delta z, \]then the represented volume per voxel is
\[ \boxed{ V_{\text{voxel}} = \Delta x\Delta y\Delta z }. \]Volume from Voxel Count
If a segmented object contains \(N\) voxels, then its approximate volume is
\[ \boxed{ V = N\Delta x\Delta y\Delta z }. \]Example: Voxel Volume
Suppose each voxel represents
\[ 0.5\text{ mm} \times 0.5\text{ mm} \times 1.0\text{ mm}. \]One voxel therefore represents
\[ 0.25\text{ mm}^3. \]If a segmented object contains \(4000\) voxels, its approximate volume is
\[ 4000(0.25)=1000. \]Therefore,
\[ \boxed{1000\text{ mm}^3}. \]Projection Imaging
Some imaging systems record a two-dimensional projection of three-dimensional structure.
In such cases, depth information may be compressed or superimposed.
Recovering three-dimensional geometry can require multiple viewpoints, calibration, or additional physical modeling.
Stereo Geometry
Two images from different viewpoints can provide depth information through parallax.
A point appears at different horizontal image positions in the two views.
The difference is called disparity.
Simple Stereo Depth Relationship
In an idealized parallel-camera model,
\[ \boxed{ Z=\frac{fB}{d} }, \]where:
- \(Z\) is depth;
- \(f\) is focal length in compatible image units;
- \(B\) is camera baseline;
- \(d\) is image disparity.
Larger disparity corresponds to smaller depth in this simplified model.
Perspective Measurement
Image length is not automatically proportional to physical length when depth varies.
Under perspective projection, an object farther from the camera appears smaller.
Reliable physical measurement may therefore require camera calibration or a known reference plane.
Camera Calibration
Camera calibration estimates the parameters connecting three-dimensional rays with image coordinates.
Parameters can include:
- focal length;
- principal point;
- pixel scaling;
- camera orientation;
- camera position;
- lens distortion.
Perspective Projection Model
A simplified pinhole-camera model is
\[ \boxed{ u=\frac{fx}{z}, \qquad v=\frac{fy}{z} }. \]The model demonstrates why image size depends on depth.
Lens Distortion
Real lenses can distort ideal perspective geometry.
Straight lines may appear curved, especially near image edges.
Calibration can estimate distortion so measurements can be corrected.
Image Mosaics
Multiple overlapping images can be aligned and combined into a larger composite image.
The process requires:
- feature correspondence;
- registration;
- geometric transformation;
- resampling;
- overlap handling.
Change Detection
Registered images from different times can be compared to detect spatial change.
If images are not aligned correctly, apparent changes may be caused by registration error rather than real physical change.
Spatial Uncertainty
Image-derived coordinates are rarely infinitely exact.
Uncertainty can arise from:
- pixel resolution;
- blur;
- segmentation error;
- calibration error;
- registration error;
- projection distortion;
- measurement noise.
Coordinate Uncertainty
A detected point may be represented as
\[ (x,y) \]with uncertainty
\[ \pm\Delta x, \qquad \pm\Delta y. \]Geometric calculations based on that point inherit some of this uncertainty.
Resolution-Limited Measurement
Suppose a boundary can only be located to within approximately one pixel and one pixel represents
\[ 0.4\text{ m}. \]Then the boundary-location uncertainty is already on the order of
\[ \boxed{0.4\text{ m}} \]before other error sources are considered.
Scale Bars and Image Measurement
Images may include a known scale bar.
If a scale bar of physical length \(L\) measures \(p\) pixels, then the physical length represented by one pixel is
\[ \boxed{ g=\frac{L}{p} }. \]This can be used to measure other features in the same image when the scale is spatially valid.
Example: Scale-Bar Calibration
A \(10\)-millimeter scale bar measures
\[ 200\text{ pixels}. \]Then
\[ g=\frac{10}{200}=0.05\text{ mm/pixel}. \]A feature measuring \(120\) pixels corresponds to
\[ 120(0.05)=6. \]Therefore,
\[ \boxed{6\text{ mm}}. \]Common Error: Treating Pixels as Physical Units
A measurement of
\[ 100\text{ pixels} \]is not automatically \(100\) millimeters, meters, or any other physical length.
A scale relationship is required.
Common Error: Ignoring Coordinate Conventions
Image coordinates may use a top-left origin and downward-positive vertical direction.
Analytic geometry may use a lower-left or centered origin and upward-positive vertical direction.
Coordinate systems must be reconciled before combining measurements.
Common Error: Measuring a Resized Image with an Old Scale
If an image is enlarged or reduced, the original pixel-to-distance relationship changes unless the scale information is transformed with it.
Always calibrate the current image representation.
Common Error: Ignoring Perspective
Objects at different depths can have different pixel scales in perspective images.
A single global pixels-per-meter value may be invalid across the entire scene.
Common Error: Assuming More Pixels Always Mean More Information
Enlarging an existing low-resolution image creates more stored pixels but does not recreate missing physical detail automatically.
Interpolation can smooth or estimate intermediate values, but it does not recover information that was never sampled.
Common Error: Confusing Raster and Vector Precision
A raster boundary is limited by its grid representation.
A vector line may store coordinates more directly.
Converting between the two can change geometric precision and topology.
Common Error: Ignoring Connectivity Rules
Whether diagonal pixel contact counts as connected depends on the chosen neighborhood convention.
Region counts can change when switching between four-neighbor and eight-neighbor connectivity.
Common Error: Treating Segmentation as Ground Truth
A segmented region is an interpreted representation.
Classification errors can affect area, perimeter, centroid, and shape measurements.
Image-derived geometry should be evaluated together with segmentation quality.
A Reliable Procedure for Image Measurement
- Identify the image coordinate system.
- Determine the current pixel dimensions and spatial scale.
- Verify whether the image has been resized or warped.
- Identify the feature to measure.
- Measure its pixel geometry.
- Convert pixel lengths or areas using the correct spatial scale.
- Account for perspective or distortion when relevant.
- Report uncertainty consistent with image resolution and calibration.
A Reliable Procedure for Raster Analysis
- Identify the raster dimensions and cell resolution.
- Identify what each cell value represents.
- Choose the appropriate neighborhood or connectivity rule.
- Segment or classify the region if necessary.
- Count or summarize relevant cells.
- Convert cell counts into physical area or volume.
- Use interpolation or derivatives only at a resolution justified by the data.
- Check sensitivity to sampling and classification error.
A Reliable Procedure for Spatial Registration
- Identify corresponding control points or features.
- Choose an appropriate transformation model.
- Estimate transformation parameters.
- Transform one dataset into the coordinate system of the other.
- Resample the transformed raster if necessary.
- Measure residual mismatch at control or check points.
- Verify that the alignment is accurate enough for the intended analysis.
Example: Full Imaging Workflow
A segmented object occupies
\[ 2500 \]pixels in an image.
A scale bar shows that
\[ 100\text{ pixels}=20\text{ mm}. \]Therefore, one pixel represents
\[ \frac{20}{100}=0.2\text{ mm}. \]One square pixel represents
\[ (0.2)^2=0.04\text{ mm}^2. \]The estimated object area is
\[ 2500(0.04)=100\text{ mm}^2. \]Therefore,
\[ \boxed{A\approx100\text{ mm}^2}. \]The result remains an image-derived estimate and should be interpreted in light of segmentation accuracy, boundary resolution, and scale calibration.
Practice 1: Pixel Count
How many pixels are in an image measuring
\[ 800\times600? \]Practice 2: Physical Distance from Pixels
One pixel represents
\[ 0.4\text{ m}. \]A line measures
\[ 75\text{ pixels}. \]Find its physical length.
Practice 3: Pixel Area
One square pixel represents
\[ 0.2\text{ m}\times0.2\text{ m}. \]Find the area represented by one pixel.
Practice 4: Region Area
A segmented region contains
\[ 1500 \]pixels. Each pixel represents
\[ 0.04\text{ m}^2. \]Find the region area.
Practice 5: Pixel Distance
Two image points differ by
\[ 9 \]pixels horizontally and
\[ 12 \]pixels vertically.
Find their Euclidean pixel distance.
Practice 6: Scale-Bar Calibration
A scale bar representing
\[ 5\text{ cm} \]measures
\[ 250\text{ pixels}. \]Find the physical length per pixel in centimeters.
Practice 7: Bounding Box
A detected region has
\[ u_{\min}=20, \quad u_{\max}=140, \quad v_{\min}=30, \quad v_{\max}=90. \]Find its bounding-box width and height.
Practice 8: Voxel Volume
A voxel measures
\[ 1\text{ mm}\times1\text{ mm}\times2\text{ mm}. \]Find the volume represented by \(300\) voxels.
Practice 9: Simple Stereo Depth
In the model
\[ Z=\frac{fB}{d}, \]let
\[ f=800, \qquad B=0.10\text{ m}, \qquad d=40. \]Find \(Z\).
Practice 10: Identify the Error
A student enlarges an image to twice its displayed width and then continues using the original millimeters-per-pixel scale.
Explain the error.
Practice Answers
Practice 1
\[ 800(600)=480{,}000. \]Therefore,
\[ \boxed{480{,}000\text{ pixels}}. \]Practice 2
\[ d=75(0.4)=30. \]Therefore,
\[ \boxed{30\text{ m}}. \]Practice 3
\[ A=(0.2)(0.2)=0.04. \]Therefore,
\[ \boxed{0.04\text{ m}^2}. \]Practice 4
\[ A=1500(0.04)=60. \]Therefore,
\[ \boxed{60\text{ m}^2}. \]Practice 5
\[ d_p=\sqrt{9^2+12^2} = \sqrt{225} = 15. \]Therefore,
\[ \boxed{15\text{ pixels}}. \]Practice 6
\[ g=\frac{5}{250}=0.02. \]Therefore,
\[ \boxed{0.02\text{ cm/pixel}}. \]Practice 7
\[ W=140-20=120, \] \[ H=90-30=60. \]Therefore,
\[ \boxed{120\times60\text{ pixels}}. \]Practice 8
One voxel represents
\[ 1(1)(2)=2\text{ mm}^3. \]Therefore,
\[ 300(2)=600\text{ mm}^3. \]Thus,
\[ \boxed{600\text{ mm}^3}. \]Practice 9
\[ Z=\frac{800(0.10)}{40} = \frac{80}{40} = 2. \]Therefore,
\[ \boxed{Z=2\text{ m}} \]under the stated compatible-unit model.
Practice 10
Doubling the displayed image width changes the number of displayed pixels spanning the same physical feature.
The original physical-length-per-displayed-pixel scale is therefore no longer valid.
The current image must be recalibrated, or the scale must be adjusted by the same resize factor.
What to Remember
- Spatial data combines stored values with positions in a coordinate system.
- Raster data uses regular grids, while vector data stores explicit geometric objects.
- Pixel and voxel dimensions determine spatial resolution.
- Physical lengths, areas, and volumes can be estimated from pixel or voxel counts only when a valid spatial scale is known.
- Sampling converts continuous spatial structure into discrete data and can lose detail.
- Aliasing occurs when spatial variation is too fine for the sampling grid.
- Georeferencing ties image coordinates to real-world coordinates.
- Registration aligns datasets into a common coordinate system.
- Resampling estimates destination-grid values after geometric transformation.
- Segmentation identifies regions from which geometric measurements can be extracted.
- Centroids, bounding boxes, contours, convex hulls, and moments summarize image-derived shape.
- Digital elevation models represent terrain as scalar fields over a grid.
- Point clouds represent sampled three-dimensional locations and can be reconstructed into surfaces.
- Perspective and lens distortion complicate direct measurement from ordinary photographs.
- Stereo imagery can recover depth from geometric disparity.
- Spatial uncertainty should be reported at a level consistent with image resolution, calibration, and segmentation quality.
The central relationship is:
\[ \boxed{ \text{sampled data} + \text{coordinates} + \text{geometric interpretation} \longrightarrow \text{spatial information} }. \]Bringing Applied Spatial Problem Families Together
The lessons in this section show that the same spatial ideas reappear across many practical domains.
Measurement connects unknown quantities to known geometry. Scale drawings connect representations to physical size. Navigation and surveying connect distance and direction to position. Architecture and engineering turn geometry into buildable and manufacturable systems. Motion and robotics add time. Graphics and imaging convert space into digital representations. Geographic and astronomical positioning extend coordinate reasoning to global and celestial scales. Waves and fields distribute behavior through space. Packing, covering, routing, and optimization turn geometry into decisions.
Across all of them, the recurring structure is:
\[ \boxed{ \text{represent} \longrightarrow \text{measure} \longrightarrow \text{relate} \longrightarrow \text{model} \longrightarrow \text{solve} \longrightarrow \text{verify} }. \]That pattern is one of the central practical powers of spatial mathematics.