For foreground set $A$ and structuring element $B$, morphological dilation is their Minkowski sum:Equivalently, place a translated copy of $B$ at each foreground position and take their union. For $A=\{0,2\}$ and $B=\{0,1\}$, the result is $\{0,1,2,3\}$, connecting the separated input points.

In image processing, morphological dilation can enlarge bright regions or bridge small gaps. The grayscale version for a flat footprint takes a local maximum, with the footprint reflected according to the stated offset convention.^[ref] Asymmetric footprints make reflection conventions visible; symmetric disks hide that distinction. Morphological dilation alone generally enlarges a region, whereas morphological opening combines it with morphological erosion.

For foreground set $A$ and structuring element $B$, morphological erosion isThus an output position is retained only when every offset in the translated structuring element lands inside the foreground. This can shrink regions and remove features narrower than the neighborhood.^[ref]

For $A=\{0,1,2,5\}$ and $B=\{0,1\}$, only starts zero and one have their successor in $A$, giving $A\ominus B=\{0,1\}$. The point five disappears. For grayscale data with a flat footprint, the analogous operation takes a local minimum. An image implementation additionally chooses border rules; the set formula assumes the ambient space outside $A$ is background.

A structuring element specifies the neighborhood shape and reference position used by an operation in mathematical morphology. In binary image processing, represent it as a finite set $B$ of offsets. Morphological erosion tests where a translated $B$ fits inside the foreground; morphological dilation adds its offsets to foreground positions.

A three-by-three square uses all nine offsets around its center, while a cross uses the center and its four horizontal or vertical neighbors. They treat diagonal features differently despite having the same width. Scikit-image's morphology demonstration uses a disk-shaped footprint whose radius controls the affected scale.^[ref] The reference position matters for asymmetric shapes, and finite-image border rules must specify how neighborhoods outside the image are treated.

Morphological opening applies morphological erosion and then morphological dilation with the same structuring element. For a foreground set $A$ and structuring element $B$,It keeps the union of all translated copies of $B$ contained in $A$. Consequently the result is contained in $A$, and repeating the operation changes nothing: morphological opening is idempotent.

For a one-dimensional example, take $A=\{0,1,2,5\}$ and $B=\{0,1\}$. Morphological erosion gives $\{0,1\}$ and the following morphological dilation restores $\{0,1,2\}$; the isolated point at five disappears. The surviving run is restored rather than shifted. In image processing, selecting the structuring element controls which bright features survive. Scikit-image demonstrates removal of bright regions too thin to contain the chosen footprint.^[ref]

Institutional tutorial issued by the School of Materials Science and Engineering, UNSW Sydney. Distinguishes continuous and discontinuous reinforcement and explains the directional strength of aligned fibres. No personal author or publication date is inferred.

Official Biomedical Engineering faculty profile. Research Interests describes biomimetics, the relation between mineralized-tissue organization and mechanical properties, and biomineralization-inspired materials. Research Areas includes biomechanics and biomaterials. The profile links the external portrait used in David H. Kohn.

Official Bioengineering profile. Education lists University of Michigan degrees: BS in Materials Science and Engineering (1994), MS and PhD in Biomedical Engineering (2003), followed by a University of Washington postdoctoral fellowship (2006). Research Interests includes biomaterials, cell/tissue mechanics and device design; the profile supplies the external portrait used in Rupak M. Rajachar.

Primary teaching resource: www.matter.org.uk/matscicdrom/manual/co.html . Materials Science on CD-ROM, version 2.1. The page credits Bill Clyne, University of Cambridge, and Boban Tanovic, MATTER. Locators: Load Transfer, What happens when a Composite is Stressed? (equal strain); What about the Transverse Stiffness? and Figure 2 (equal-stress slab model, Reuss estimate, and limitations for real fibre geometry).

Primary recommendation: doi.org/10.1351/PAC-REC-10-12-04 . Pure and Applied Chemistry 84(2), 377–410 (2012). Verified publisher byline: Michel Vert, Yoshiharu Doi, Karl-Heinz Hellwich, Michael Hess, Philip Hodge, Przemyslaw Kubisa, Marguerite Rinaudo and François Schué. Official PDF: publications.iupac.org/pac/pdf/2012/pdf/8402x0377.pdf . Entries 30 (biomaterial) and 106 (biocompatibility in biomedical therapy). Entry 30 is reproduced in the primary IUPAC Gold Book goldbook.iupac.org/terms/view/13881 .

Additional precise locator: entry 70, erosion, printed p. 391, defines degradation beginning at the surface and progressing inward; its notes discuss reaction versus reagent diffusion and discourage the wording bulk erosion. The official Gold Book reproduces this definition at goldbook.iupac.org/terms/view/13906 . Both the indexed primary recommendation and Gold Book entry were verified; direct-reader access to the PDF/entry returned 403. A mass-loss curve alone does not establish the spatial mechanism.

Primary publisher abstract: doi.org/10.1002/jbm.820050611 . Journal of Biomedical Materials Research 5(6), 117–141 (November 1971). Verified byline: L. L. Hench, R. J. Splinter, W. C. Allen and T. K. Greenlee; initials retained. The abstract compares in-vitro glass-ceramic solubility and interfacial hydroxyapatite crystallization studies with in-vivo rat-femur implant histology and ultrastructure.

Primary guide: www.matter.org.uk/matscicdrom/manual/MatSciCDROMv21_UserGuideSections1and2.pdf . Version 2.1. Printed p. 4 lists the MATTER Design and Development Team, including Boban Tanovic, and Academic Contributors, including Bill Clyne of Cambridge. The guide describes the interactive teaching modules and their use alongside materials-science instruction.

Professional-society profile: live.fems.org/professor-tw-bill-clyne . Locators: biography paragraph identifying the 1999 Cambridge chair in Mechanics of Materials, research on thermo-mechanics of composites and coatings, educational software and DoITPoMS. The profile explicitly connects the published TW initials with Bill; no further expansion of those initials is made.

Official primary resource: www.fda.gov/medical-devices/biocompatibility-assessment-resource-center/basics-biocompatibility-information-needed-assessment-fda . Locators: How the FDA Assesses or Evaluates Biocompatibility, concerning final finished form, processing and residuals; Biocompatibility Factors of Interest to the FDA, concerning nature, type, frequency and duration of contact. Companion endpoint tables: www.fda.gov/medical-devices/biocompatibility-assessment-resource-center/biocompatibility-evaluation-endpoints-device-category . These are evaluation principles, not a universal requirement to perform every listed test.

Author account: www.technagroup.it/data/image/pdf/Forum_2004/009-023_The_Story_of_Bioglass.pdf . Byline Larry Hench, Imperial College. Printed pp. 9–23 in Lessons in Nanotechnology from Traditional and Advanced Ceramics; first-page copyright 2005. Locators pp. 10–12, implant observations and original collaborators; pp. 13–14, proposed interfacial reaction sequence. The path contains Forum_2004; it is not taken as the publication year.

Primary collaborative NIST/NASA benchmark dataset README. Challenge Summary: laser-powder-bed-fusion titanium specimens underwent two heat-treatment conditions, machining/polishing and fully reversed four-point rotating-bending testing. Fatigue Testing specifies a 100 Hz ADMET eXpert 9313 apparatus and stress ratio minus one. Used for experimental arrangement, not to infer an allowable shaft stress or a lifetime for other materials.

Primary source: www.acs.org/molecule-of-the-week/archive/p/poly-ethylene-terephthalate.html

Verified locator: Molecule of the Week, 17 October 2022. Official ACS structural account identifies condensation of ethylene glycol and terephthalic acid. Repeat atom counts and finite-chain examples are explicitly derived in the article.

Total ion current is a sum of ion currents in mass spectrometry, with an important distinction about where it is measured. After mass analysis, it is the sum of separate ion currents contributing to the mass spectrum, sometimes called reconstructed ion current. Before mass analysis, it can mean the sum of all same-sign ion currents. A reconstructed total over a selected analyzer range need not include every ion initially produced.^[ref]

For discretely recorded spectral intensities $I_k$, a corresponding reconstructed signal is $S={\sum }_{k\in K}{I}_k$, where $K$ specifies the included mass-to-charge ratio bins. Instruments may store intensities in arbitrary response units or counts rather than amperes, so the reported total needs its acquisition convention. Equal-bin sums and integrals of intensity density also require different bin-width bookkeeping.

For an invented three-bin example, spectra $(4,6,0)$ and $(1,1,8)$ both have total $10$. The total alone cannot distinguish their different ionic compositions. If acquisition excludes the third bin, the two reconstructed totals become $10$ and $2$ without any change in the stipulated source outputs. A changing measurement range can thus change the reported total.

Plotting reconstructed totals from successive spectra against time gives a total ion current profile; in coupled liquid chromatography, this can display when substantial summed signal reaches the detector.^[ref] Individual chemical signals can overlap or move in opposite directions while their sum remains unchanged. Total ion current normalization uses such a sum to rescale a spectrum, but the arithmetic does not independently establish chemical identity, detector linearity or equal response per molecule.

An extracted ion profile plots selected mass-to-charge ratio signal across a time-ordered series of mass spectra. The selection can be one value or a set of values. When time follows a chromatographic separation, the result is an extracted ion chromatogram; the broader profile term also covers other time-dependent sampling.^[ref]

For recorded spectra $I_k(t_j)$, an explicit extracted ion mass window $W$ gives the reconstructed traceThis selects information already present in the spectra. It does not create missing mass measurements or turn the acquisition into a different instrumental measurement. Unlike a total ion current profile, it excludes signal outside the stated window.

An original three-scan example shows how the window changes the result. At mass coordinates $499.98$, $500.00$ and $500.03$, let the intensity vectors be $(0,1,0)$, $(3,8,5)$ and $(0,2,10)$ at successive times. A window $500.00\pm 0.01$ selects only the middle coordinate and gives $(1,8,2)$. Widening to $500.00\pm 0.04$ includes all three coordinates and gives $(1,16,12)$. The larger late signal in the broad trace comes from added spectral content; it is not evidence that the middle-coordinate signal remained high.

Choosing a window requires considering mass resolving power, mass-coordinate accuracy and what the instrument actually recorded. A narrow window can exclude a shifted target peak; a wide window can include neighbors. Even an isolated mass coordinate need not identify a unique chemical species, because different ions can share it. Preserve the raw spectra and state the window, summation rule and time axis so the profile can be reproduced and interpreted.

A mass spectrometry matrix factor compares analyte response in a sample matrix with response from an equal amount in a specified reference solution. For matched acquisition and blank-corrected peak areas, a common post-preparation comparison isThe post-preparation sample is a prepared blank matrix to which the analyte is added after extraction. This separates response from extraction recovery under the matching assumptions. MF below 1 indicates ion suppression relative to the reference; MF above 1 indicates enhanced response. The number is not an absolute ionization efficiency.

The original method study distinguishes matrix effects, recovery and variation across sample sources.^[ref] For an invented example, take neat response $1000$, post-preparation response $750$ and response $450$ when the same nominal amount is added before preparation. Then MF is $0.75$, the estimated recovery is $450/750=0.60$, and mass spectrometry process efficiency is $450/1000=0.45$. The product $0.75\times 0.60=0.45$ identifies which two effects this model combines.

An internal-standard-normalized matrix factor divides analyte MF by the corresponding MF for an internal standard. If both are $0.50$, the normalized value is 1 even though both raw responses are suppressed. If the analyte MF is $0.80$ and the standard MF is $0.40$, the normalized value is 2. Normalization can therefore fail when the two responses change differently.

Magnitude and variability are separate questions. Three matrices giving factors $(0.50,0.50,0.50)$ have a large common suppression but no between-matrix variation in this invented set. Factors $(0.50,0.70,0.90)$ differ between matrices despite their less suppressed mean. Report the reference, analyte level, sample preparation and response calculation; near-zero reference signals make a ratio unstable, and unmatched volumes or uncorrected offsets can invalidate its interpretation.

Mass spectrometry image normalization rescales the spectra associated with individual image pixels to make a specified comparison more meaningful. In mass spectrometry imaging, raw intensity depends on both the sample and measurement conditions; MALDI matrix distribution can change the recorded intensity. A normalization rule therefore embodies an assumption about which variation should be removed.^[ref]

For example, total ion current normalization divides each nonzero spectrum by the sum of its included intensities. Write pixel $p$ as values $I_{p,k}$ and setIn an invented two-feature example, spectra $(2,6)$ and $(4,12)$ both become $(1/4,3/4)$. This removes a common multiplicative change. But $(2,14)$ becomes $(1/8,7/8)$: the first feature now appears half as strong as in $(2,6)$ even though its raw intensity is unchanged. The normalized image represents a relative share of the included signal. It answers a different question from an image of raw intensity or calibrated molecular abundance.

In actual MALDI datasets, Deininger and colleagues found that locally dominant signals could introduce artifacts under total-signal normalization; they compared median and noise-based alternatives. Their examples motivate inspecting the denominator as well as the selected ion image.^[ref] Specify the mass range, excluded peaks and zero-signal handling. Neither total ion current normalization nor another single scalar correction can, in general, undo selective suppression of one molecular signal.