Symbols, terms and frames

Accumulated linear attenuation along the finite path for detector pixel $p$ at energy $E$.

Manuscript §1.1. $L$ without a pixel subscript denotes the homogeneous-path optical depth. $L_p$ is also used when energy is fixed.

1 (dimensionless)

• Introduction: Before an image becomes a tensor

Probability of traversing the object without an interaction along pixel $p$'s primary path, equal to $\exp(-L_p(E))$.

Manuscript §1.1. $T_p$ or $T$ suppresses energy and pixel labels when fixed. This is uncollided survival, not total detector signal including scatter.

1 (dimensionless)

• Introduction: Before an image becomes a tensor

Expected detector output at pixel $p$ in the declared measurement domain.

Manuscript §1.1. The response model determines the output units. Chapter 10 uses unbarred $y_p$ for an expectation.

detector output units

• Introduction: Before an image becomes a tensor

One realised detector measurement including acquisition noise.

Manuscript §1.1. Chapter 5 specialises $y_p$ to observed photon counts. Chapter 10 instead uses $y_p$ for an expected signal.

detector output units

• Introduction: Before an image becomes a tensor

Pixel value after measurement processing or display mapping.

Manuscript §1.1. Chapter 1 notation. Chapter 3 reuses $d_p$ for physical source-to-pixel distance in mm.

processed image units

• Introduction: Before an image becomes a tensor

Parameters varied by the forward or inverse problem, initially the object pose.

Manuscript §1.2. Scalar $\theta$ later denotes one active parameter. Its units must be declared. Chapter 3 $\theta$ is specifically a rotation angle.

parameter-dependent

• Introduction: A model of the measurement

Remaining model inputs held fixed while the active parameters vary.

Manuscript §1.2. Chapter 1 convention. Chapter 3 bold $\boldsymbol\phi$ instead denotes a local rotation vector.

input-dependent

• Introduction: A model of the measurement

Deterministic map from active parameters and fixed model inputs to a prediction in the chosen comparison domain.

Manuscript §1.2. The output can be expected counts or another explicitly specified domain. Processing a mean need not equal the mean processed measurement.

declared prediction units

• Introduction: A model of the measurement

Expected open-beam photon population per unit energy associated with pixel $p$ over the exposure, before detection losses.

Manuscript §1.2. Distinct from energy-integrated $n_{0,p}$. Detector geometry is already included, so a second inverse-square factor is inappropriate.

photons $\mathrm{keV}^{-1}$

• Introduction: A model of the measurement

Expected detector output per incident photon at energy $E$, including detection efficiency.

Manuscript §1.2. Under the spatially local linear response assumption. Do not confuse $R_p$ with a rotation matrix or regularisation.

detector output units $\mathrm{photon}^{-1}$

• Introduction: A model of the measurement

Local coefficient for removal from the uncollided primary beam by absorption or scattering.

Manuscript §1.2. Energy is in keV and physical path distance is in mm. Chapter 2 makes these units explicit. Not an energy-absorption coefficient.

$\mathrm{mm}^{-1}$

• Introduction: A model of the measurement

Finite source-to-pixel path expressed in attenuation-field coordinates.

Manuscript §1.2. Here $\ell_p$ names a curve, not its scalar length. Chapter 4 reuses $\ell_p$ for the length of the clipped segment.

geometric path

• Introduction: A model of the measurement

Conditional probability law of the detector measurement given active and fixed model inputs.

Manuscript §1.2. Density units depend on the declared discrete or continuous measurement measure.

No single unit applies

• Introduction: A model of the measurement

Map from detector measurements to processed image values.

Manuscript §1.2. Nonlinear processing does not generally commute with expectation. Distinct from the grid orientation matrix bold $\mathbf Q$ in Chapter 4 and a single-photon random output $Q_p$ in Chapter 8.

No single unit applies

• Introduction: A model of the measurement

Derivative of predicted pixel $p$ with respect to local parameter $k$.

Manuscript §1.3. Each Jacobian column is a sensitivity image. Units vary across translation, rotation and material columns.

image units per parameter unit

• Introduction: What a sensitivity image tells us

Predicted image minus observed image in the chosen comparison domain.

Manuscript §1.3. Chapter 6 uses $\mathbf r$ for a whitened residual, which is dimensionless. The domain is part of the definition.

image units

• Introduction: What a sensitivity image tells us

Scalar objective comparing the forward prediction with observed data, optionally including priors.

Manuscript §1.3. For the illustrative unweighted squared-error loss in §1.3 its units are squared image units. Later likelihood objectives have different scaling.

objective-dependent

• Introduction: What a sensitivity image tells us

Expected primary photon count at detector pixel $p$, $n_{0,p} T_p$.

Manuscript §2.1. The Poisson model holds for the stated ideal primary count process. Chapter 7 $\lambda$ includes an explicitly added scatter mean.

photons

• Transmission: From incident photons to a transmission image

Expected photon population at pixel $p$ in the absence of the object, over the exposure.

Manuscript §2.1. Energy-integrated quantity. $n_{0,p}(E)$ denotes the spectral density.

photons

• Transmission: From incident photons to a transmission image

Realised primary photon count at pixel $p$.

Manuscript §2.1. A random count, distinct from its expectation $\lambda_p$.

photons

• Transmission: From incident photons to a transmission image

Realised incident photon count for the exposure before object interactions.

Manuscript §2.1. Conditioning on a fixed incident count gives binomial thinning. A Poisson incident population gives a Poisson surviving population.

photons

• Transmission: From incident photons to a transmission image

Physical distance measured along a path.

Manuscript §2.2. In Chapter 3 it is measured from the source. Chapter 7 reuses $s$ for expected scatter counts.

mm

• Transmission: Attenuation coefficients and units

Photon energy at the state being modelled.

Manuscript §2.2. Must match the coefficient and detector-response energy units.

keV

• Transmission: Attenuation coefficients and units

Material mass density used with tabulated mass attenuation coefficients.

Manuscript §2.2. Chapter 8 converts to $\mathrm{g}\,\mathrm{mm}^{-3}$ when using coefficients in $\mathrm{mm}^{2}\,\mathrm{g}^{-1}$. Bold $\boldsymbol\rho$ in Chapter 3 denotes pose translation coordinates.

$\mathrm{g}\,\mathrm{cm}^{-3}$

• Transmission: Attenuation coefficients and units

Linear attenuation coefficient divided by mass density.

Manuscript §2.2. Tabulated convention in Chapter 2. Multiply by density in $\mathrm{g}\,\mathrm{cm}^{-3}$, then divide $\mathrm{cm}^{-1}$ by ten to obtain $\mathrm{mm}^{-1}$.

$\mathrm{cm}^{2}\,\mathrm{g}^{-1}$

• Transmission: Attenuation coefficients and units

Physical distance travelled through homogeneous material.

Manuscript §2.3. Distinct from processed-image values in Chapter 1 and observed signal $d$ in Chapter 10.

mm

• Transmission: The homogeneous path

Physical distance travelled by ray $p$ through material segment $j$.

Manuscript §2.4. Segment lengths multiply coefficients in inverse mm.

mm

• Transmission: Heterogeneous paths and optical depth

Quadrature weight including the physical distance represented by sample $j$ on path $p$.

Manuscript §2.4. Chapters 4–5 use $w_{pr}$. This is not a dimensionless spectral mixture weight or a statistical fitting weight.

mm

• Transmission: Heterogeneous paths and optical depth

Endpoint interpolation parameter, zero at the source-side endpoint and one at the detector-side endpoint.

Manuscript §2.4. Physical integration requires $\mathrm{d}s$ = endpoint distance times $\mathrm{d}t$. Appendix A.15 uses $t$ for slab thickness instead.

1 (dimensionless)

• Transmission: Heterogeneous paths and optical depth

Non-negative normalised weight of contributing path $i$ in a finite pixel average.

Manuscript §2.4. Average transmitted contributions before taking logarithms. $\alpha$ has other local meanings in later chapters.

1 (dimensionless)

• Transmission: Heterogeneous paths and optical depth

Weighted mean of primary transmissions over the paths contributing to pixel $p$.

Manuscript §2.4. Distinct averaging variable from the energy-weighted transmission of Chapter 8.

1 (dimensionless)

• Transmission: Heterogeneous paths and optical depth

Mean detector output per arriving photon in the local affine response model.

Manuscript §2.5. Chapter 7 uses scalar $g$ for fixed gain. Chapter 9 uses $g$ for a proposal density.

detector output units $\mathrm{photon}^{-1}$

• Transmission: From transmission to log projections

Mean additive electronic offset at pixel $p$.

Manuscript §2.5. Not a photon population and not subject to attenuation by the object.

detector output units

• Transmission: From transmission to log projections

Expected detector output in the absence of the object.

Manuscript §2.5. The affine model includes $b_p$. Chapter 8 spectral $\bar y_0$ is defined before the offset, so offset subtraction must follow the local model.

detector output units

• Transmission: From transmission to log projections

Additional mean contribution expressed in count-equivalent units.

Manuscript §2.5. It enters before count-to-signal gain and can represent scattered signal in the stated approximation.

photons

• Transmission: From transmission to log projections

Negative logarithm of an observed positive count divided by its open-beam expectation.

Manuscript §2.5. Noise-dependent estimate. Chapter 4 reuses $\widehat L_p$ for deterministic quadrature approximation. A zero count has no finite logarithm.

1 (dimensionless)

• Transmission: From transmission to log projections

Per-pixel Poisson negative log-likelihood with terms independent of optical depth omitted.

Manuscript §2.5. For the stated ideal count model. Later $\Phi_p$ denotes the same type of objective.

1 (dimensionless)

• Transmission: From transmission to log projections

Fraction removed from the uncollided primary beam, $1 - \exp(-L)$.

Manuscript §2.6. Evaluate as $-\operatorname{expm1}(-L)$ when needed near zero. Later chapters reuse $\delta$ for a feature displacement.

1 (dimensionless)

• Transmission: Evaluating transmission numerically

Homogeneous rigid transform mapping point coordinates from frame $B$ into frame $A$.

Manuscript §3.1. Destination subscript first, source second. Use column vectors, with the rightmost transform acting first. $\mathbf G_{WO}$ is the object pose in the world.

mixed: rotation 1, translation mm

• Frames and motion: Naming and framing

Column vector of coordinates of a physical point in the named frame $A$.

Manuscript §3.1. Superscript is a frame label, not exponentiation. The homogeneous point coordinate is one.

mm

• Frames and motion: Naming and framing

Rotation whose columns are frame $B$ axes expressed in frame $A$.

Manuscript §3.2. Orthogonal with determinant $+1$. A direction transforms by rotation only.

1 (dimensionless)

• Frames and motion: Rigid transforms and their inverses

Position of the origin of frame $B$ expressed in frame $A$.

Manuscript §3.2. Inverse translation is $-\mathbf R_{AB}^{\mathsf T}\mathbf t_{AB}$, not merely $-\mathbf t_{AB}$.

mm

• Frames and motion: Rigid transforms and their inverses

Group of proper three-dimensional rigid homogeneous transforms under composition.

Manuscript §3.2. Rotation blocks preserve lengths, angles and handedness. Translations are in mm.

No single unit applies

• Frames and motion: Rigid transforms and their inverses

Zero-based detector column index.

Manuscript §3.3. The array stores pixel $(i,j)$ at [j,i]. Chapter 4 separately uses $i$ as the fastest volume index.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

Zero-based detector row index.

Manuscript §3.3. Increasing $j$ follows the detector's second axis. No anatomical orientation follows from the index alone.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

Number of detector columns.

Manuscript §3.3. The centred column offset is $(N_c-1)/2$, with no extra half-pixel shift.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

Number of detector rows.

Manuscript §3.3. The centred row offset is $(N_r-1)/2$.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

Positive separation between adjacent detector pixel centres along increasing column index.

Manuscript §3.3. DICOM spacing is ordered row then column. Select spacing together with its calibration plane.

mm

• Frames and motion: Detector coordinates and pixel centres

Positive separation between adjacent detector pixel centres along increasing row index.

Manuscript §3.3. The active rectangle has dimensions $N_c\Delta_c$ by $N_r\Delta_r$ under contiguous pixel cells.

mm

• Frames and motion: Detector coordinates and pixel centres

Physical centre of detector pixel $(i,j)$ in the detector-centred frame.

Manuscript §3.3. Its coordinates are $((i-(N_c-1)/2)\Delta_c, (j-(N_r-1)/2)\Delta_r, 0)$.

mm

• Frames and motion: Detector coordinates and pixel centres

World position of the first detector pixel centre.

Manuscript §3.3. Alternative calibration origin to the geometric centre, paired with increasing-column and increasing-row unit vectors.

mm

• Frames and motion: Detector coordinates and pixel centres

Continuous column coordinate of the perpendicular projection of the source onto the detector plane.

Manuscript §3.3. The principal point need not coincide with the detector geometric centre.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

Continuous row coordinate of the perpendicular projection of the source onto the detector plane.

Manuscript §3.3. Defined using the actual source position in detector coordinates.

1 (dimensionless)

• Frames and motion: Detector coordinates and pixel centres

World position of the ideal focal point, equal to $\mathbf t_{WS}$.

Manuscript §3.4. Transform as a point to obtain $\mathbf s^O$. Bold $\mathbf s$ is not the scalar distance parameter $s$.

mm

• Frames and motion: Constructing source-to-detector rays

Positive Euclidean distance from source to detector pixel centre $p$.

Manuscript §3.4. Rigid coordinate changes preserve it. Chapter 1 $d_p$ instead denotes a processed image value.

mm

• Frames and motion: Constructing source-to-detector rays

Unit vector from the source towards detector pixel centre $p$.

Manuscript §3.4. Computed by dividing the endpoint displacement by $d_p$. Translation must not act on it.

1 (dimensionless)

• Frames and motion: Constructing source-to-detector rays

Linear attenuation field defined in physical object coordinates and extended by zero outside the modelled object.

Manuscript §3.4. A posed field is queried by mapping world points through the inverse object pose.

$\mathrm{mm}^{-1}$

• Frames and motion: Constructing source-to-detector rays

Source pose relative to the object, $\mathbf G_{WO}^{-1}\mathbf G_{WS}$.

Manuscript §3.5. Unchanged by a common rigid change of all world poses.

mixed: rotation 1, translation mm

• Frames and motion: Move the object or move the acquisition

Detector pose relative to the object, $\mathbf G_{WO}^{-1}\mathbf G_{WD}$.

Manuscript §3.5. These relative transforms determine the primary paths through the fixed object field.

mixed: rotation 1, translation mm

• Frames and motion: Move the object or move the acquisition

Six local exponential coordinates ordered as translation $\boldsymbol\rho$ followed by rotation $\boldsymbol\phi$.

Manuscript §3.6. Left composition expresses increments in world axes. Right composition expresses them in current object axes. Mixed units require declared scaling.

translation mm, rotation rad

• Frames and motion: Pose parameters and local updates

Translation coordinates in the rigid-motion Lie algebra.

Manuscript §3.6. Finite translation is $\mathbf J(\boldsymbol\phi)\boldsymbol\rho$. $\boldsymbol\rho$ equals it only in special cases such as pure translation. Scalar $\rho$ elsewhere is mass density.

mm

• Frames and motion: Pose parameters and local updates

Local rotation vector with axis given by direction and angle by norm.

Manuscript §3.6. Chapter 1 bold $\boldsymbol\phi$ instead denotes fixed model inputs.

rad

• Frames and motion: Pose parameters and local updates

Norm of the local rotation vector in the rigid exponential.

Manuscript §3.6. Chapter 3 meaning. Later scalar $\theta$ is a general active parameter with declared units.

rad

• Frames and motion: Pose parameters and local updates

Matrix mapping translation exponential coordinates to the finite translation block.

Manuscript §3.6. Its product with $\boldsymbol\rho$ is the finite translation block. This matrix is distinct from the image and residual Jacobians.

1 (dimensionless)

• Frames and motion: Pose parameters and local updates

Translation-first local increment expressed along world axes and composed on the left.

Manuscript §3.6. A pure rotational increment rotates about the world origin.

translation mm, rotation rad

• Frames and motion: Pose parameters and local updates

Translation-first local increment expressed along current object axes and composed on the right.

Manuscript §3.6. A pure rotational increment leaves the object origin's world position fixed.

translation mm, rotation rad

• Frames and motion: Pose parameters and local updates

Chosen world point held fixed by a rotation-about-a-centre update.

Manuscript §3.6. The finite update translation block is $(\mathbf I-\mathbf R_\delta)\mathbf c$. Omitting it changes the pivot.

mm

• Frames and motion: Pose parameters and local updates

Dimensionless perturbation multiplying a pose-check direction whose components already carry mm and radians.

Manuscript §3.7. Chapter 10 can instead use $h$ in physical parameter units.

1 (dimensionless)

• Frames and motion: Geometry checks that catch plausible mistakes

Stored attenuation coefficient assigned to the declared physical sample centre.

Manuscript §4.1. Array order is k,j,i. The physical coefficient interpretation and boundary extension are modelling choices.

$\mathrm{mm}^{-1}$

• Volumes and rays: What a voxel represents

Physical location of the first stored volume sample centre.

Manuscript §4.1. It is a sample-centre anchor, not a support corner.

mm

• Volumes and rays: What a voxel represents

Orthonormal grid-axis directions expressed as columns in object coordinates.

Manuscript §4.1. The grid map also contains spacing. Dimensionless grid coordinates are not a rigid physical frame.

1 (dimensionless)

• Volumes and rays: What a voxel represents

Diagonal matrix of positive physical voxel spacings $\Delta_x$, $\Delta_y$ and $\Delta_z$.

Manuscript §4.1. Not the primary/scatter signal vectors used in Chapter 7.

mm

• Volumes and rays: What a voxel represents

Number of stored samples along volume grid axis $a$.

Manuscript §4.1. Cell bounds run from $-1/2$ to $N_a-1/2$ in grid coordinates. Interpolation support depends separately on the field extension.

1 (dimensionless)

• Volumes and rays: What a voxel represents

Continuous sample-lattice coordinates $\mathbf S^{-1}\mathbf Q^{\mathsf T}(\mathbf x^O-\mathbf o^O)$.

Manuscript §4.2. Integer components select sample centres. Later bold $\mathbf u$ denotes scaled pose parameters rather than spatial indices.

1 (dimensionless)

• Volumes and rays: Physical coordinates to array indices

One-dimensional hat function $\max(1-|u|,0)$ defining the trilinear basis.

Manuscript §4.3. Chapter 6 $\beta$ is an optimisation damping coefficient.

1 (dimensionless)

• Volumes and rays: Interpolation and the field between samples

Continuous field defined by volume coefficients and their product hat basis with the declared boundary extension.

Manuscript §4.3. Different from the piecewise-constant cell field even at identical stored values.

$\mathrm{mm}^{-1}$

• Volumes and rays: Interpolation and the field between samples

Fractional position within the current interpolation cell along its first grid axis.

Manuscript §4.3. $\alpha_y$ and $\alpha_z$ are the analogous fractions on the other axes. These fractions are distinct from spectral or path mixture weights.

1 (dimensionless)

• Volumes and rays: Interpolation and the field between samples

Spatial derivative of the interpolated attenuation field in physical object coordinates.

Manuscript §4.3. Transform the grid derivative with $\mathbf Q\mathbf S^{-\mathsf T}$. Anisotropic spacings matter.

$\mathrm{mm}^{-2}$

• Volumes and rays: Interpolation and the field between samples

Source position transformed into continuous grid coordinates.

Manuscript §4.4. Used in $\mathbf u_p(t)=\mathbf a_p+t\mathbf b_p$. This is not an attenuation coefficient.

1 (dimensionless)

• Volumes and rays: Intersecting a ray with the volume

Source-to-detector endpoint displacement in continuous grid coordinates.

Manuscript §4.4. Physical length must still be obtained from the physical endpoint distance.

1 (dimensionless)

• Volumes and rays: Intersecting a ray with the volume

Entry parameter of the accepted finite ray interval after volume and endpoint clipping.

Manuscript §4.4. No contribution when exit is not greater than entry. Parallel slabs require explicit handling.

1 (dimensionless)

• Volumes and rays: Intersecting a ray with the volume

Exit parameter of the accepted finite ray interval after volume and endpoint clipping.

Manuscript §4.4. The finite ray is restricted to $t$ between zero and one.

1 (dimensionless)

• Volumes and rays: Intersecting a ray with the volume

Physical length $d_p(t_p^+-t_p^-)$ of an accepted clipped ray segment.

Manuscript §4.4. Chapter 4 meaning. Earlier $\ell_p$ names the source-to-detector path itself.

mm

• Volumes and rays: Intersecting a ray with the volume

Positive integer number of midpoint samples over the accepted ray interval.

Manuscript §4.5. A pose-dependent change in this count changes the discrete calculation being differentiated.

1 (dimensionless)

• Volumes and rays: Quadrature along a physical path

Dimensionless ray parameter locating midpoint sample $r$ inside the clipped interval for pixel $p$.

Manuscript §4.5. Sample physical position follows from the object-space ray endpoints.

1 (dimensionless)

• Volumes and rays: Quadrature along a physical path

Optical depth approximated by the declared physical-path quadrature.

Manuscript §4.5. Chapter 4 meaning. Chapter 2 $\widehat L_p$ is a noisy logarithmic estimate from counts.

1 (dimensionless)

• Volumes and rays: Quadrature along a physical path

Combined physical path and interpolation weight linking volume coefficient $v$ to ray $p$ at fixed geometry.

Manuscript §4.6. Represents the linear map $L_p=\sum_v W_{pv}a_v$. No dense matrix storage is implied.

mm

• Volumes and rays: A GPU projection operator

Volume attenuation coefficient indexed by flattened sample $v$.

Manuscript §4.6. The spectral material field $a_m(\mathbf x)$ in Chapter 8 is dimensionless instead.

$\mathrm{mm}^{-1}$

• Volumes and rays: A GPU projection operator

Exact line integral of the chosen interpolated field.

Manuscript §4.7. Separates field representation error from quadrature and floating-point errors.

1 (dimensionless)

• Volumes and rays: Resolution, truncation and convergence

Optical depth returned by floating-point evaluation of the chosen numerical operator.

Manuscript §4.7. A convergence study must distinguish numerical quadrature error from arithmetic and input-field error.

1 (dimensionless)

• Volumes and rays: Resolution, truncation and convergence

Direction in the active parameter coordinates used for a Jacobian-vector product.

Manuscript §5.1. Its components carry the units of the corresponding parameters. Chapter 7 $\boldsymbol\eta$ instead collects nuisance parameters.

parameter-dependent

• Projection gradients: Which projection derivative do we need?

Detector-space incoming derivative with respect to expected primary counts in reverse differentiation.

Manuscript §5.1. An overbar here denotes a cotangent, unlike bars denoting detector expectations. It is not a mean count.

objective units $\mathrm{photon}^{-1}$

• Projection gradients: Which projection derivative do we need?

Diagonal map from dimensionless local increments into physical parameter increments.

Manuscript §5.1. Chapter 6 specialises it to translation and rotation scales. It is not the detector frame $D$.

parameter units

• Projection gradients: Which projection derivative do we need?

Incoming objective derivative with respect to dimensionless optical depth.

Manuscript §5.2. Reverse result is $-\lambda_p\overline\lambda_p$. Chapter 8 $\bar L$ instead denotes apparent spectral optical depth.

objective units

• Projection gradients: Attenuation and transmission derivatives

Objective derivative accumulated for stored attenuation coefficient $v$.

Manuscript §5.2. Requires accumulation over every contributing ray/sample.

objective units mm

• Projection gradients: Attenuation and transmission derivatives

Objective derivative with respect to expected open-beam count.

Manuscript §5.2. Can be evaluated directly through transmission, without dividing by $n_{0,p}$.

objective units $\mathrm{photon}^{-1}$

• Projection gradients: Attenuation and transmission derivatives

Product interpolation basis associated with stored coefficient $v$.

Manuscript §5.5. Its physical spatial derivative contributes to the sample-position cotangent.

1 (dimensionless)

• Projection gradients: Reverse-mode differentiation in Warp

Objective derivative with respect to the physical object-space quadrature sample position.

Manuscript §5.5. A cotangent, not an averaged point location.

objective units $\mathrm{mm}^{-1}$

• Projection gradients: Reverse-mode differentiation in Warp

Objective derivative with respect to the physical quadrature weight.

Manuscript §5.5. Geometry derivatives can require both sample-position and weight derivatives.

objective units $\mathrm{mm}^{-1}$

• Projection gradients: Reverse-mode differentiation in Warp

Relative discrepancy between a directional derivative and its central finite difference, with a declared absolute reference scale.

Manuscript §5.6. Check a sequence of step sizes, including the transition from truncation to round-off error.

1 (dimensionless)

• Projection gradients: Independent directional checks

Absolute remainder after subtracting the predicted first-order objective change.

Manuscript §5.6. Not a rotation matrix or a detector response.

objective units

• Projection gradients: Independent directional checks

Inverse of the positive definite matrix defining the parameter-space inner product for an adjoint.

Manuscript §5.8. A Euclidean transpose becomes an adjoint only after the inner products are declared.

metric-dependent

• Projection gradients: Gradient validation as an executable contract

Positive definite matrix defining the detector-space inner product.

Manuscript §5.8. The corresponding adjoint is $\mathbf M_\theta^{-1}\mathbf J^{\mathsf T}\mathbf M_y$.

metric-dependent

• Projection gradients: Gradient validation as an executable contract

Estimated rigid transform placing the object in world coordinates.

Manuscript §6.1. A pose estimate depends on the discrepancy, measurement model, prior and fixed acquisition inputs.

mixed: rotation 1, translation mm

• Pose recovery: Pose recovery as an inverse problem

Scalar measure of disagreement between simulated and observed images in a declared measurement domain.

Manuscript §6.1. Chapter 5 uses this symbol for the differentiable discrepancy as well.

discrepancy-dependent

• Pose recovery: Pose recovery as an inverse problem

Pose prior or regularisation term in the inverse objective.

Manuscript §6.1. Regularisation can constrain directions that the image data do not identify. Distinguish prior from data information.

objective units

• Pose recovery: Pose recovery as an inverse problem

Fixed set of detector pixels included in the pose fit.

Manuscript §6.2. Data-dependent masks change the problem and must not be silently treated as fixed.

No single unit applies

• Pose recovery: What should the images agree on?

Specified covariance of the measurement errors used for whitening.

Manuscript §6.2. $\mathbf C^{-1/2}$ maps residuals to noise-normalised coordinates. Filtering can create off-diagonal covariance.

image units squared

• Pose recovery: What should the images agree on?

Prediction-minus-observation residual after the specified measurement whitening.

Manuscript §6.2. Chapter 1 $\mathbf r$ is the unwhitened image residual. Units and statistical assumptions follow the local definition.

1 (dimensionless)

• Pose recovery: What should the images agree on?

Correlation between centred image patterns at a candidate pose.

Manuscript §6.2. Correlation discards scale and offset information. Its geometry sensitivity differs from a count likelihood.

1 (dimensionless)

• Pose recovery: What should the images agree on?

Dimensionless local pose coordinate mapped to translation-first physical increments by $\mathbf D$.

Manuscript §6.3. Distinct from the continuous spatial grid coordinate $\mathbf u$ in Chapter 4.

1 (dimensionless)

• Pose recovery: Parameter scales and pose updates

Reference physical displacement used to set local parameter scales.

Manuscript §6.3. Paired with characteristic radius $r_*$ to set a comparable rotational scale $\ell/r_*$.

mm

• Pose recovery: Parameter scales and pose updates

Declared characteristic distance from the update origin used to scale rotations.

Manuscript §6.3. A modelling choice. Image sensitivity still depends on actual anatomy and geometry.

mm

• Pose recovery: Parameter scales and pose updates

Jacobian of the whitened residual with respect to physical local pose coordinates.

Manuscript §6.3. Composition side and update origin are part of the derivative definition.

$\mathrm{mm}^{-1}$ for translation, $\mathrm{rad}^{-1}$ for rotation

• Pose recovery: Parameter scales and pose updates

Jacobian of the whitened residual with respect to dimensionless local pose coordinates, $\mathbf J_\xi\mathbf D$.

Manuscript §6.3. Column scales affect conditioning and local information interpretation.

1 (dimensionless)

• Pose recovery: Parameter scales and pose updates

Positive damping coefficient added to the scaled local least-squares normal system.

Manuscript §6.3. Not the trilinear hat basis $\beta$ in Chapter 4 or transmitted spectral weighting $\beta_d$ in Chapter 8.

1 (dimensionless)

• Pose recovery: Parameter scales and pose updates

Local data information $\mathbf J_u^{\mathsf T}\mathbf J_u$ in scaled pose coordinates.

Manuscript §6.5. Small eigenvalues identify weak local directions. This does not establish global uniqueness.

1 (dimensionless)

• Pose recovery: Single-view ambiguity and identifiability

Singular values of the whitened, scaled pose residual Jacobian.

Manuscript §6.5. Their meaning depends on the declared parameter scales, measurement covariance and included pixels.

1 (dimensionless)

• Pose recovery: Single-view ambiguity and identifiability

Point depth along the source coordinate axis in the centred projection example.

Manuscript §6.5. Appendix A uses $Z$ for atomic number, a dimensionless quantity.

mm

• Pose recovery: Single-view ambiguity and identifiability

Euclidean translation error at the declared object origin.

Manuscript §6.7. Translation error depends on the origin. Use the same origin for estimated and reference poses.

mm

• Pose recovery: Evaluate recovery independently

Relative rotation angle between estimated and reference rotations.

Manuscript §6.7. Separate angular error from target displacement. Do not combine mm and radians without a declared scale.

rad

• Pose recovery: Evaluate recovery independently

Root-mean-square displacement of selected object-space targets under estimated and reference poses.

Manuscript §6.7. Its value depends on the chosen targets and their distribution, which must be reported.

mm

• Pose recovery: Evaluate recovery independently

Mean image-model mismatch after measurement whitening.

Manuscript §6.8. Different from the processed image vector $\mathbf d$ in Chapter 1 or fixed observed scalar $d$ in Chapter 10.

1 (dimensionless)

• Pose recovery: When a lower loss is the wrong answer

First-order displacement of scaled pose coordinates induced by the whitened model mismatch.

Manuscript §6.8. The local calculation explains how a better image fit can bias physical pose.

1 (dimensionless)

• Pose recovery: When a lower loss is the wrong answer

Orthogonal projector onto the column space of the pose image Jacobian.

Manuscript §6.8. Only the mismatch component in that space can be explained by a local pose change.

1 (dimensionless)

• Pose recovery: When a lower loss is the wrong answer

Expected primary counts at unit reference exposure.

Manuscript §7.1. Physical primary counts are scaled by the relative exposure $a$ before the fixed detector gain.

photons

• Acquisition mismatch: Separate motion from acquisition changes

Expected scattered counts at unit reference exposure.

Manuscript §7.1. Distinct from the voxel-spacing matrix $\mathbf S$ and the single-history detector score $S_p$.

photons

• Acquisition mismatch: Separate motion from acquisition changes

Acquisition or nuisance parameters estimated or varied alongside pose.

Manuscript §7.1. Distinct from $\boldsymbol\eta$ as a directional-check vector in Chapters 3 and 5.

parameter-dependent

• Acquisition mismatch: Separate motion from acquisition changes

Image Jacobian with respect to acquisition nuisance parameters.

Manuscript §7.1. Comparing its image directions with pose directions exposes local confounding.

image units per nuisance-parameter unit

• Acquisition mismatch: Separate motion from acquisition changes

Positive exposure relative to the reference exposure.

Manuscript §7.2. Scaling exposure changes photon statistics. Detector gain $g$ instead changes signal per count.

1 (dimensionless)

• Acquisition mismatch: Exposure, gain and offsets

Variance of additive electronic noise in the signal-domain model.

Manuscript §7.2. Distinguish the electronic contribution from photon-count variance multiplied by gain squared.

detector output units squared

• Acquisition mismatch: Exposure, gain and offsets

Positive multiplicative image-domain coefficient in the affine intensity fit.

Manuscript §7.2. Its unconstrained least-squares estimate must respect the chosen positivity constraint.

1 (dimensionless)

• Acquisition mismatch: Exposure, gain and offsets

Fixed positive statistical or fitting weight assigned to pixel $p$.

Manuscript §7.2. For inverse-variance weighting the units are inverse squared signal units. Not a path-length quadrature weight.

weight-dependent

• Acquisition mismatch: Exposure, gain and offsets

Normalised non-negative incident weight of spectral component $k$.

Manuscript §7.3. Chapter 7 mixture weights sum to one. Chapter 8 $w_k$ instead carries keV as an energy-quadrature weight.

1 (dimensionless)

• Acquisition mismatch: Spectrum, filtration and beam hardening

Fraction of the transmitted signal contributed by spectral component $k$ after thickness $\ell$.

Manuscript §7.3. Weights depend on attenuation and thickness even when the incident spectrum is fixed.

1 (dimensionless)

• Acquisition mismatch: Spectrum, filtration and beam hardening

Physical length of the path through material $m$.

Manuscript §7.3. Multiple material lengths can require distinct spectral responses. No universal effective attenuation coefficient follows.

mm

• Acquisition mismatch: Spectrum, filtration and beam hardening

Local expected primary contribution in the same domain as the scatter contribution $S$.

Manuscript §7.4. Scalar signal notation, not the number of predicted pixels in Chapter 1.

detector output units

• Acquisition mismatch: Scatter and background structure

Local expected scattered contribution in the same domain as primary signal $P$.

Manuscript §7.4. Chapter 9 $S$ instead denotes a random free-flight distance.

detector output units

• Acquisition mismatch: Scatter and background structure

Additional expected scatter count in the count-domain likelihood.

Manuscript §7.4. Distinct from physical path coordinate $s$ in the transport integral.

photons

• Acquisition mismatch: Scatter and background structure

Background basis function $q$ evaluated at pixel $p$.

Manuscript §7.4. Coefficients $c_q$ and basis units must combine into the stated background signal. Flexible background fitting can remove pose information.

basis-dependent

• Acquisition mismatch: Scatter and background structure

Deterministic linear filter applied to observed photon counts.

Manuscript §7.5. Filtered covariance is $\mathbf H\operatorname{diag}(\boldsymbol\lambda)\mathbf H^{\mathsf T}$ plus post-filter electronic covariance. This operator is not the rigid frame-change $\mathbf H$ of Chapter 3.

filter-dependent

• Acquisition mismatch: Detector blur, noise and processing

Covariance of electronic noise added after filtering.

Manuscript §7.5. Off-diagonal entries and the filter domain must be retained when relevant.

detector output units squared

• Acquisition mismatch: Detector blur, noise and processing

Centroid displacement of the blur kernel along coordinate $x$.

Manuscript §7.5. Not a mass attenuation coefficient $\kappa_m(E)$ in Chapter 8. An asymmetric kernel can imitate feature motion.

coordinate units

• Acquisition mismatch: Detector blur, noise and processing

Whitened image Jacobian for dimensionless scaled pose coordinates in the joint local fit.

Manuscript §7.6. Not the stored attenuation array $A$ of Chapter 4 or material-path matrix $A_{pm}$ of Chapter 8.

1 (dimensionless)

• Acquisition mismatch: Jointly estimate pose and nuisance parameters

Whitened image Jacobian with respect to nuisance parameters.

Manuscript §7.6. The orthogonal projector onto its column space identifies image changes that nuisance fitting can explain.

inverse nuisance-parameter units

• Acquisition mismatch: Jointly estimate pose and nuisance parameters

Orthogonal projector onto the nuisance Jacobian column space.

Manuscript §7.6. Pose information remaining after unconstrained nuisance fitting is $\mathbf A^{\mathsf T}(\mathbf I-\mathbf P_B)\mathbf A$.

1 (dimensionless)

• Acquisition mismatch: Jointly estimate pose and nuisance parameters

Local pose information after fitting unconstrained nuisance increments.

Manuscript §7.6. A Schur-complement calculation under the specified whitening and local linear model.

1 (dimensionless)

• Acquisition mismatch: Jointly estimate pose and nuisance parameters

Local precision supplied by calibration or a prior on nuisance parameters.

Manuscript §7.6. Adding prior precision must be distinguished from image-derived information.

inverse nuisance-parameter units squared

• Acquisition mismatch: Jointly estimate pose and nuisance parameters

Translation-first left Lie-algebra error for repetition $r$ relative to the reference pose.

Manuscript §7.7. A local error convention. Report frame, composition side and origin with the statistic.

translation mm, rotation rad

• Acquisition mismatch: Controlled mismatch experiments

Sample covariance of repeated local pose-error vectors.

Manuscript §7.7. Sampling variation and mean systematic bias are separate outputs. This is not evidence of executed experiments.

mixed: $\mathrm{mm}^{2}$, $\mathrm{mm}\,\mathrm{rad}$, $\mathrm{rad}^{2}$

• Acquisition mismatch: Controlled mismatch experiments

Normalised open-beam spectral density, $n_{0,p}(E)/n_{0,p}$, for a positive total population.

Manuscript §8.1. Its energy integral is one. Line spectra need a compatible discrete or mixed measure.

$\mathrm{keV}^{-1}$

• Spectra and detectors: The source spectrum and its units

Non-negative local mass fraction of constituent $m$ in a mixture.

Manuscript §8.2. Fractions sum to one. These are neither energy weights nor physical path lengths.

1 (dimensionless)

• Spectra and detectors: Energy-dependent material attenuation

Mass attenuation coefficient of constituent $m$ in the all-mm mixture formula.

Manuscript §8.2. Multiply by density in $\mathrm{g}\,\mathrm{mm}^{-3}$ to obtain $\mathrm{mm}^{-1}$. Chapter 2 tabulates $\mu/\rho$ in $\mathrm{cm}^{2}\,\mathrm{g}^{-1}$.

$\mathrm{mm}^{2}\,\mathrm{g}^{-1}$

• Spectra and detectors: Energy-dependent material attenuation

Non-negative dimensionless spatial material-basis coefficient at the declared reference density.

Manuscript §8.2. Can be an indicator or relative concentration. It differs from stored monochromatic attenuation coefficient $a_v$ in $\mathrm{mm}^{-1}$.

1 (dimensionless)

• Spectra and detectors: Energy-dependent material attenuation

Linear attenuation coefficient of basis material $m$ at its declared reference density.

Manuscript §8.2. Its energy dependence multiplies dimensionless spatial material coefficients.

$\mathrm{mm}^{-1}$

• Spectra and detectors: Energy-dependent material attenuation

Integral of material-basis field $a_m$ along the finite path for pixel $p$.

Manuscript §8.2. A literal material length only when the basis field is an indicator. Its definition governs interpretation.

mm

• Spectra and detectors: Energy-dependent material attenuation

Normalised incident spectrum weighted by non-negative detector response and divided by positive open-beam output.

Manuscript §8.3. The averaging weights include detector response, not only photon-number spectrum.

$\mathrm{keV}^{-1}$

• Spectra and detectors: Polychromatic transmission

Ratio of expected primary output to open-beam output in the declared detector domain, before additive offsets.

Manuscript §8.3. An energy-weighted mean of transmissions. The spatial average in Chapter 2 has a different averaging variable.

1 (dimensionless)

• Spectra and detectors: Polychromatic transmission

Negative logarithm of detector-domain spectral primary transmission.

Manuscript §8.3. Generally differs from an average optical depth and from a single monochromatic line integral. In Chapter 5 an overbar denotes a cotangent.

1 (dimensionless)

• Spectra and detectors: Polychromatic transmission

Normalised transmitted spectral-response weighting after homogeneous thickness $d$.

Manuscript §8.3. For fixed incident weights, the derivative of apparent optical depth is its weighted mean attenuation.

$\mathrm{keV}^{-1}$

• Spectra and detectors: Polychromatic transmission

Photon energy of node $k$ in the declared spectral quadrature.

Manuscript §8.4. Coefficient interpolation must retain the two sides of absorption edges.

keV

• Spectra and detectors: Energy quadrature and material paths

Non-negative integration weight for energy node $k$ in a continuous-spectrum quadrature.

Manuscript §8.4. Unlike Chapter 7's normalised dimensionless mixture weights, this includes an energy interval.

keV

• Spectra and detectors: Energy quadrature and material paths

Open-beam response contribution $w_k n_{0,p}(E_k) R_p(E_k)$ at node $k$.

Manuscript §8.4. The discrete expected primary output sums $c_{pk}\exp(-L_{pk})$.

detector output units

• Spectra and detectors: Energy quadrature and material paths

Optical depth of ray $p$ evaluated at energy node $k$.

Manuscript §8.4. Equal to the sum over material basis coefficients $\mu_m(E_k)A_{pm}$.

1 (dimensionless)

• Spectra and detectors: Energy quadrature and material paths

Probability that one incident photon of energy $E$ gives an accepted count in detector pixel $p$ and energy bin $b$.

Manuscript §8.5. A mean response probability. Real pile-up or correlated multi-count responses require the corresponding measurement model.

1 (dimensionless)

• Spectra and detectors: What the detector measures

Expected accepted photon count in pixel $p$ and energy bin $b$.

Manuscript §8.5. Computed by weighting the transmitted spectral population with the bin acceptance.

photons

• Spectra and detectors: What the detector measures

Linear coefficient carrying input $r$ into readout pixel $p$.

Manuscript §8.6. Its units depend on the input/output domains. The noise law depends on whether it represents independent routing or deterministic sharing.

response-dependent

• Spectra and detectors: Spatial response and measurement noise

Random contribution of one incident photon to readout pixel $p$.

Manuscript §8.6. Its first and second moments determine compound-Poisson signal and covariance. This readout is not the processing map $Q$ or grid orientation $\mathbf Q$.

detector output units

• Spectra and detectors: Spatial response and measurement noise

Incoming objective derivative with respect to expected detector output at pixel $p$.

Manuscript §8.7. The spectral transpose-Jacobian calculation accumulates its contributions across energy and material paths.

objective units per detector output unit

• Spectra and detectors: Differentiate the spectral image

Unit photon propagation direction at its current state.

Manuscript §9.1. Direction integration is over solid angle. This direction is not a position vector or ray length.

1 (dimensionless)

• Monte Carlo transport: Beyond the uncollided photon

Exposure-integrated angular photon fluence as a function of position, direction and energy.

Manuscript §9.1. Distinct from radiance or detector signal. The transport equation is stationary and linear under the stated assumptions.

photons $\mathrm{mm}^{-2}\,\mathrm{sr}^{-1}\,\mathrm{keV}^{-1}$

• Monte Carlo transport: Beyond the uncollided photon

Sum of absorption and included scattering process coefficients at the local position and energy.

Manuscript §9.1. The process set must match event sampling and excludes the physical processes omitted by the declared reduced model.

$\mathrm{mm}^{-1}$

• Monte Carlo transport: Beyond the uncollided photon

Internal source density in the angular-energy transport equation.

Manuscript §9.1. An incoming boundary source is an alternative representation. Do not count the same source twice. Chapter 9.6 $q$ is instead roulette survival probability.

photons $\mathrm{mm}^{-3}\,\mathrm{sr}^{-1}\,\mathrm{keV}^{-1}$

• Monte Carlo transport: Beyond the uncollided photon

Normalised outgoing direction-and-energy probability kernel for scattering process $a$.

Manuscript §9.1. An elastic energy kernel can contain a delta distribution. Use a compatible probability measure rather than assume an ordinary density.

$\mathrm{sr}^{-1}\,\mathrm{keV}^{-1}$ for a continuous density

• Monte Carlo transport: Beyond the uncollided photon

Random distance to the next interaction in a homogeneous region with positive total coefficient.

Manuscript §9.2. Exponentially distributed, with mean free path $1/\mu_t$. Vacuum needs geometric traversal rather than division by zero.

mm

• Monte Carlo transport: Free flights and interaction sampling

Uniform random draw strictly between zero and one for inverse-survival sampling.

Manuscript §9.2. Chapter 10 uses $U$ more generally for parameter-independent base randomness.

1 (dimensionless)

• Monte Carlo transport: Free flights and interaction sampling

Macroscopic coefficient of interaction process $a$ at the local state.

Manuscript §9.2. At a real collision, choose the process with probability $\mu_a/\mu_t$ including absorption.

$\mathrm{mm}^{-1}$

• Monte Carlo transport: Free flights and interaction sampling

Unit-exponential optical-depth draw consumed along a heterogeneous photon flight.

Manuscript §9.2. Carry residual optical depth across boundaries consistently. Chapter 10 also uses $\tau$ for the deterministic accumulated depth of a fixed segment.

1 (dimensionless)

• Monte Carlo transport: Free flights and interaction sampling

Valid upper bound on the total interaction coefficient throughout the sampling region.

Manuscript §9.2. A majorant violation changes the null-collision sampling law. This overbar is not a cotangent or expectation.

$\mathrm{mm}^{-1}$

• Monte Carlo transport: Free flights and interaction sampling

Outgoing photon energy after a Compton event in the free-electron-at-rest model.

Manuscript §9.3. The same relation appears in Appendix A.4. Incident energies and electron rest energy must use the same units.

keV

• Monte Carlo transport: Scattering angle and energy change

Electron rest energy in the free-electron Compton relation.

Manuscript §9.3. No numerical value is required by the registry. This is not a statement about local energy deposition.

keV

• Monte Carlo transport: Scattering angle and energy change

Angle between the incoming and outgoing photon directions.

Manuscript §9.3. Distinct from the pose rotation vector and from a general active parameter $\theta$.

rad

• Monte Carlo transport: Scattering angle and energy change

Classical electron radius in a consistent length system for the Klein–Nishina cross section.

Manuscript §9.3. Its square supplies the area units. The manuscript does not specify a numeric radius.

mm

• Monte Carlo transport: Scattering angle and energy change

Ratio $E'/E$ of outgoing to incident photon energy in the free-electron model.

Manuscript §9.3. Not a volume storage index, energy node index or optimiser iteration counter.

1 (dimensionless)

• Monte Carlo transport: Scattering angle and energy change

Azimuth about the incoming photon direction in the constructed transverse basis.

Manuscript §9.3. Uniform for the stated unpolarised free-electron model. Future polarised models require different assumptions.

rad

• Monte Carlo transport: Scattering angle and energy change

First unit transverse axis perpendicular to the incoming photon direction.

Manuscript §9.3. Choose the second as $\mathbf e_2 = \boldsymbol\omega\times\mathbf e_1$ to fix the right-handed local direction basis.

1 (dimensionless)

• Monte Carlo transport: Scattering angle and energy change

Source emission state comprising position, direction and energy.

Manuscript §9.4. Different coordinates have different units and can have discrete or delta-supported components.

No single unit applies

• Monte Carlo transport: Sources, boundaries and detector scores

Non-negative physical source intensity density integrated over the exposure.

Manuscript §9.4. Its integral is $N_{\mathrm{src}}$. Source importance weighting must use the same reference measure as the proposal.

photons per source phase-space measure

• Monte Carlo transport: Sources, boundaries and detector scores

Normalised probability density used to sample source states.

Manuscript §9.4. Positive wherever the physical source contributes. $F/g$ is the source importance weight.

inverse source phase-space measure

• Monte Carlo transport: Sources, boundaries and detector scores

Expected number of photons emitted by the physical source over the exposure.

Manuscript §9.4. Distinct from the computational history count $N$. It remains an explicit prefactor in Chapter 10 gradients.

photons

• Monte Carlo transport: Sources, boundaries and detector scores

Contribution assigned to detector pixel $p$ by a history under the physical interaction law.

Manuscript §9.4. A count score can be one for an accepted arrival. This detector score differs from the likelihood score $\mathfrak s_\theta$.

detector output units per emitted-photon history

• Monte Carlo transport: Sources, boundaries and detector scores

Complete contribution of source history $h$ to pixel $p$, including source and other importance weights.

Manuscript §9.5. Aggregate all split descendants of a source history before estimating independent-history variance.

detector output units

• Monte Carlo transport: Estimators, variance and uncertainty

Number of independently launched source histories in the computational estimate.

Manuscript §9.5. Controls estimator precision, not exposure. Distinct from an observed physical count $N_p$.

1 (dimensionless)

• Monte Carlo transport: Estimators, variance and uncertainty

Sample-mean Monte Carlo estimate of the expected detector output.

Manuscript §9.5. Monte Carlo sampling uncertainty is distinct from physical acquisition noise.

detector output units

• Monte Carlo transport: Estimators, variance and uncertainty

Target probability density of physical paths used in importance sampling.

Manuscript §9.6. Must share a compatible reference measure with the proposal $g(H)$.

inverse history measure

• Monte Carlo transport: Variance reduction without changing the answer

Probability density used to propose full photon histories.

Manuscript §9.6. Requires support on all contributing target histories and the correct target-to-proposal weight.

inverse history measure

• Monte Carlo transport: Variance reduction without changing the answer

Positive probability, at most one, of retaining a history during Russian roulette.

Manuscript §9.6. A retained history's weight is divided by $q$ to preserve expectation. This probability is distinct from internal source density $q$.

1 (dimensionless)

• Monte Carlo transport: Variance reduction without changing the answer

Random post-roulette weight, zero on termination and $W/q$ on survival.

Manuscript §9.6. Roulette changes variance and work while preserving the conditional expected weight under the stated rule.

same as pre-roulette weight

• Monte Carlo transport: Variance reduction without changing the answer

Photon energy transferred to recoil electrons in Compton interactions.

Manuscript §9.8. Photon energy bookkeeping does not establish that the energy is deposited at the collision site.

keV

• Monte Carlo transport: Validate before comparing speed

Remaining photon energy assigned to unresolved low-energy cutoff termination.

Manuscript §9.8. Keep this separate from detector escape and explicitly modelled transfer when checking energy accounting.

keV

• Monte Carlo transport: Validate before comparing speed

Expected detector signal as a function of one active scalar physical parameter.

Manuscript §10.1. Chapter 10 drops the expectation overbar used earlier. This $y_p$ is not an observed sample or a Monte Carlo estimate.

detector output units

• Transport gradients: Differentiate an expected measurement

Complete physical history of one emitted photon, including discrete choices, continuous states and the terminal event.

Manuscript §10.1. Not the deterministic image filter bold $\mathbf H$ of Chapter 7 or common rigid transform bold $\mathbf H$ of Chapter 3.

No single unit applies

• Transport gradients: Differentiate an expected measurement

Normalised physical history density at the current parameter value.

Manuscript §10.1. History integration includes sums over discrete choices and lengths, as well as continuous integrals.

inverse history measure

• Transport gradients: Differentiate an expected measurement

Logarithmic derivative $\partial_\theta\log f_\theta$ of the physical history density where it is positive.

Manuscript §10.1. Distinct from detector score $S_p$. A simple fixed-domain score identity needs differentiability and an integrable dominating bound. Moving support requires further terms.

inverse parameter units

• Transport gradients: Differentiate an expected measurement

Derivative of the detector score with the history coordinates held fixed.

Manuscript §10.1. The full expected derivative also accounts for the change in history probability and any varying source prefactor.

detector score units per parameter unit

• Transport gradients: Differentiate an expected measurement

Parameter-independent random inputs from which a physical history is generated.

Manuscript §10.2. Can collect many continuous random variables. The uniform scalar in Chapter 9 is a special case.

base-variable-dependent

• Transport gradients: Pathwise and score-function estimators

Map from parameter-independent base randomness to a history sampled at parameter $\theta$.

Manuscript §10.2. Its dependence includes changing flight positions and energies under the stated construction.

map-dependent

• Transport gradients: Pathwise and score-function estimators

Detector score expressed through base randomness, $S_p(h_\theta(U),\theta)$.

Manuscript §10.2. The hybrid construction in §10.2 conditions this score on a discrete choice $D$ as $A_p(\theta,d,U)$.

detector score units

• Transport gradients: Pathwise and score-function estimators

Probability of the sampled discrete choice under the parameter-dependent law.

Manuscript §10.2. Its likelihood derivative handles that choice in the conditional hybrid estimator. Do not add two complete estimators of the same derivative.

1 (dimensionless)

• Transport gradients: Pathwise and score-function estimators

Indicator equal to one for a real collision before escape and zero for escape.

Manuscript §10.3. Not the covariance $C$ used later in §10.5.

1 (dimensionless)

• Transport gradients: Differentiate the sampling law

Distance actually travelled before collision or escape from the fixed slab.

Manuscript §10.3. Section 10.3 first uses $\ell$ for an uncensored exponential flight, then $\min(\text{flight},d)$, and later for a fixed segment length. The conditioning must be kept explicit.