Invariant Shape Research Articles

Issues of Lorentz-invariance in f(T) gravity and calculations for spherically symmetric solutions

The f(T) gravity is nowadays being widely used for cosmological model building, as well as for constructing spherically symmetric solutions. In its classical pure tetrad formulation it violates the local Lorentz symmetry in the space of tetrads. By using an appropriate spin connection it can be brought to a formally Lorentz invariant shape. However, despite some mathematical elegance obtained and new options of looking for non-standard couplings to matter probably made possible, it is fully equivalent in itself to the initial formulation. It seems that this fact, that the covariantisation does not in principle change anything, is not understood well by the community. Therefore we give a pedagogical introduction to these topics. And, on top of that, we show that obtaining the explicit equations for spherically symmetric solutions in f(T) gravity is not very difficult computationally and can be easily done even without any computer at hand.

Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence

Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images.

Roll-Over Shape–Based Design of Novel Biomimetic Ankle-Foot Prosthesis

ABSTRACT Introduction Past research suggests that able-bodied individuals actively control their gait kinematics to produce one invariant roll-over shape (ROS). The ROS is a spatial representation of the center of pressure in a shank-based coordinate system. Because individuals with an amputation do not have adaptive control over their center of pressure during gait, the ankle-foot prosthesis determines the resultant ROS. This study aimed to design the mechanical properties of ankle-foot prostheses to achieve specific ROSs during gait. Methods The authors developed a mathematical model that, when used in combination with ground reaction force and shank angle experimental data, predicts the ROS created by the novel Compliant and Articulating Prosthetic Ankle (CAPA) foot. The CAPA consists of four articulating components connected by torsion springs. Able-bodied individuals walked wearing a prosthetic simulator with the solid ankle cushioned heel foot, a dynamic response foot (the Renegade AT), and versions of the CAPA with distinctly different predicted ROSs. The experimental ROS of each trial was presented, and quantitative ROS characteristics were evaluated for statistical significance. Predicted and experimental shapes for each version of the CAPA were compared by computing the distance between the shapes and ROS characteristics. Finally, we evaluated whether our method predicts the same ROS regardless of the experimental data used for the prediction. Results Changes to the mechanical properties of the CAPA resulted in statistically significant differences in ROS characteristics, notably forward position, forward length, and orientation (P < 0.05). The mean distance between points on the predicted and experimental ROSs is 1.54% of the length of the CAPA. We demonstrated that experimental ROSs can be predicted using gait data from a different version of the CAPA and a different individual. Conclusions This study presents and experimentally validates a method to predict the ROSs created by eight versions of the CAPA during gait. The method enables prosthetists and researchers to design ankle-foot prostheses based on the ROS created during gait.

Rigid motion invariant statistical shape modeling based on discrete fundamental forms: Data from the osteoarthritis initiative and the Alzheimer’s disease neuroimaging initiative

We present a novel approach for nonlinear statistical shape modeling that is invariant under Euclidean motion and thus alignment-free. By analyzing metric distortion and curvature of shapes as elements of Lie groups in a consistent Riemannian setting, we construct a framework that reliably handles large deformations. Due to the explicit character of Lie group operations, our non-Euclidean method is very efficient allowing for fast and numerically robust processing. This facilitates Riemannian analysis of large shape populations accessible through longitudinal and multi-site imaging studies providing increased statistical power. Additionally, as planar configurations form a submanifold in shape space, our representation allows for effective estimation of quasi-isometric surfaces flattenings. We evaluate the performance of our model w.r.t. shape-based classification of hippocampus and femur malformations due to Alzheimer’s disease and osteoarthritis, respectively. In particular, we outperform state-of-the-art classifiers based on geometric deep learning as well as statistical shape modeling especially in presence of sparse training data. To provide insight into the model’s ability of capturing biological shape variability, we carry out an analysis of specificity and generalization ability.

Wideband tympanometry patterns in relation to intracranial pressure

Wideband tympanometry performs a more thorough analysis of middle-ear mechanics than the conventional single-frequency method with a 226-Hz probe tone. The present work examines the sensitivity of wideband tympanometry to the stiffness of the stapes-annular ligament system in relation to intracranial pressure (ICP) and labyrinthine fluid pressure. Here, body tilt allowed ICP to be set at different values. Sixty-eight ears of volunteers were tested sequentially in upright, supine, head-down (-30°) and upright postures. Energy absorbance of the ear was measured in these postures with a commercially available wideband-tympanometry device between 0.25 and 3 kHz, at ear-canal pressures between -600 and 300 daPa. In each posture, it was possible to find a single (posture-dependent) pressure in the ear canal at which a tympanometric peak occurred at all frequencies below about 1.1 kHz. The average across ears of tympanometric-peak pressure (TPP), close to 0 in upright posture, got increasingly positive, +19 daPa in supine and +27 daPa in head-down positions. The three-dimensional plot of energy absorbance against frequency and pressure displayed an invariant shape, merely shifting with TPP along the pressure axis. Thus, a properly adjusted ear-canal pressure neutralized the effects of ICP on the ear's energy absorbance. Comparisons to published invasive assessments of ICP in the different tested body positions led to the proposed relationship ICP ≈ 15 TPP, likely describing the transformer effect between tympanic membrane and stapes-annular ligament system at quasi-static pressures. With wideband tympanometry, the middle ear may serve as a precision scales for noninvasive ICP measurements.

The ghost-box-ball system: A unified perspective on soliton cellular automata, the RSK algorithm and phase shifts

In this paper, we introduce the ghost-box-ball system, which is an extended version of the classical soliton cellular automaton. It is initially motivated as a mechanism for making precise a connection between the Schensted insertion (of the Robinson–Schensted–Knuth correspondence) and the dynamical process of the box-ball system. In addition to this motivation, we explore generalisations of classical notions of the box-ball system, including the solitonic phenomenon, the asymptotic sorting property, and the invariant shape construction.We analyse the ghost-box-ball system beyond its initial relevance to the Robinson–Schensted–Knuth correspondence, unpacking its relationship to its underlying dynamical evolution on a coordinatisation and using a mechanism for augmenting a regular box-ball configuration to study the classical ultradiscrete phase shift phenomenon.

Analysis of waveform and amplitude of mouse rod and cone flash responses.

Most vertebrate eyes have rod and cone photoreceptors, which use a signal transduction pathway consisting of many biological processes to transform light into an electrical response. We dissect and quantify the contribution of each of these processes to the photoreceptor light response by using a novel method of analysis that provides an analytical solution for the entire time course of the dim-flash light response. We find that the shape of the light response is exclusively controlled by deactivation parameters. Activation parameters scale this shape and alter the response amplitude. We show that the rising phase of the response depends on Ca2+ feedback, and we identify the deactivation parameters that control the recovery phase of the response. We devise new methods to extract values for deactivation and activation parameters from a separate analysis of response shape and response amplitude. Vertebrate eyes have rod and cone photoreceptors, which use a complex transduction pathway comprising many biological processes to transform the absorption of light into an electrical response. A fundamental question in sensory transduction is how these processes contribute to the response. To study this question, we use a well-accepted phototransduction model, which we analyse with a novel method based on the log transform of the current. We derive an analytical solution that describes the entire time course of the photoreceptor response to dim flashes of light. We use this solution to dissect and quantify the contribution of each process to the response. We find that the entire dim-flash response is proportional to the flash intensity. By normalizing responses to unit amplitude, we define a waveform that is independent of the light intensity and characterizes the invariant shape of dim-flash responses. We show that this waveform is exclusively determined by deactivation rates; activation rates only scale the waveform and affect the amplitude. This analysis corrects a previous assumption that the rising phase is determined entirely by activation rates. We further show that the rising phase depends on Ca2+ feedback to the cyclase, contrary to current belief. We identify the deactivation rates that control the recovery phase of the response, and we devise new methods to extract activation and deactivation rates from an analysis of response shape and response amplitude. In summary, we provide a comprehensive understanding of how the various transduction processes produce the cellular response.

Coupled time resolved and high frequency modulated photoluminescence probing surface passivation of highly doped n-type InP samples

Electronic passivation of III–V surfaces is essential for applications in optoelectronic devices. A key aspect is the measurement of the surface recombination properties, which can be done by various techniques including transient photoluminescence (TRPL) or luminescence quantum yield. These measurement techniques are always indirect since they suppose postulating a mathematical model for the data interpretation. Most common models use the notion of surface recombination velocity to quantify the surface recombination. In this paper, we demonstrate on the InP substrate that this notion is not always sufficient to represent the injection dependence of the surface phenomena. The study of power dependence of TRPL decays coupled with modulated photoluminescence (MPL) spectra on four samples from the same wafer with different surface treatments (epi-ready, freshly cleaned, after air exposure, and with poly-phosphazen passivation) allows us to discriminate between bulk and surface properties. We introduce surface defect trapping as an alternative to explain TRPL decays and MPL phase excitation power dependences of the three non-passivated samples. Surface trap parameters such as capture cross section and defect density are extracted. The passivated sample exhibits an invariant response shape on eight orders of magnitude of illumination. The stability of the PL response at high flux is in agreement with the perfect stability of the passivation layer, which is able to protect the InP surface without chemical changes over more than one year. They are linked to the nature of the passivation layer/InP interface. Other surface treatments were found to have an injection dependent response at high flux corresponding to different surface defect distributions but also possibly to surface chemical changes for the freshly cleaned sample.

InShaDe: Invariant Shape Descriptors for visual 2D and 3D cellular and nuclear shape analysis and classification

We present a shape processing framework for visual exploration of cellular nuclear envelopes extracted from microscopic images arising in histology and neuroscience. The framework is based on a novel shape descriptor of closed contours in 2D and 3D. In 2D, it relies on a geodesically uniform resampling of discrete curves to compute unsigned curvatures at vertices and edges based on discrete differential geometry. Our descriptor is, by design, invariant under translation, rotation, and parameterization. We achieve the latter invariance under parameterization shifts by using elliptic Fourier analysis on the resulting curvature vectors. Uniform scale-invariance is optional and is a result of scaling curvature features to z-scores. We further augment the proposed descriptor with feature coefficients obtained through sparse coding of the extracted cellular structures using K-sparse autoencoders. For the analysis of 3D shapes, we compute mean curvatures based on the Laplace-Beltrami operator on triangular meshes, followed by computing a spherical parameterization through mean curvature flow. Finally, we compute the Spherical Harmonics decomposition to obtain invariant energy coefficients. Our invariant descriptors provide an embedding into a fixed-dimensional feature space that can be used for various applications, e.g., as input features for deep and shallow learning techniques or as input for dimension reduction schemes to provide a visual reference for clustering shape collections. We demonstrate the capabilities of our framework in the context of visual analysis and unsupervised classification of 2D histology images and 3D nuclear envelopes extracted from serial section electron microscopy stacks.

Deep Predictive Learning in Neocortex and Pulvinar.

How do humans learn from raw sensory experience? Throughout life, but most obviously in infancy, we learn without explicit instruction. We propose a detailed biological mechanism for the widely embraced idea that learning is driven by the differences between predictions and actual outcomes (i.e., predictive error-driven learning). Specifically, numerous weak projections into the pulvinar nucleus of the thalamus generate top-down predictions, and sparse driver inputs from lower areas supply the actual outcome, originating in Layer 5 intrinsic bursting neurons. Thus, the outcome representation is only briefly activated, roughly every 100 msec (i.e., 10 Hz, alpha), resulting in a temporal difference error signal, which drives local synaptic changes throughout the neocortex. This results in a biologically plausible form of error backpropagation learning. We implemented these mechanisms in a large-scale model of the visual system and found that the simulated inferotemporal pathway learns to systematically categorize 3-D objects according to invariant shape properties, based solely on predictive learning from raw visual inputs. These categories match human judgments on the same stimuli and are consistent with neural representations in inferotemporal cortex in primates.

Relaxation dynamics of Pd–Ni–P metallic glass: decoupling of anelastic and viscous processes

The stress relaxation dynamics of metallic glass Pd40Ni40P20 was studied in both supercooled liquid and glassy states. Time-temperature superposition was found in the metastable liquid, implying an invariant shape of the distribution of times involved in the relaxation. Once in the glass state, the distribution of relaxation times broadens as temperature and fictive temperature decrease, eventually leading to a decoupling of the relaxation in two processes. While the slow one keeps a viscous behavior, the fast one shows an anelastic nature and a time scale similar to that of the collective atomic motion measured by x-ray photon correlation spectroscopy (XPCS). These results suggest that the atomic dynamics of metallic glasses, as determined by XPCS at low temperatures in the glass state, can be related to the rearrangements of particles responsible of the macroscopically reversible anelastic behavior.

Posture-Invariant Three Dimensional Human Hand Statistical Shape Model

Abstract A high-fidelity digital representation of (part of) the human body is a key enabler for integrating humans in a digital twin. Among different parts of human body, building the model of the hand can be a challenging task due to the posture deviations among collected scans. In this article, we proposed a posture invariant statistical shape model (SSM) of the human hand based on 59 3D scans of human hands. First, the 3D scans were spatially aligned using a Möbius sphere-based algorithm. An articulated skeleton, which contains 20 bone segments and 16 joints, was embedded for each 3D scan. Then, all scans were aligned to the same posture using the skeleton and the linear blend skinning (LBS) algorithm. Three methods, i.e., principal component analysis (PCA), kernel-PCA (KPCA) with different kernel functions, and independent component analysis (ICA), were evaluated in the construction of the SSMs regarding the compactness, the generalization ability, and the specificity. The PCA-based SSM was selected, where 20 principal components were used as parameters for the model. Results of the leave-one-out validation indicate that the proposed model was able to fit a given 3D scan of the human hand at an accuracy of 1.21 ± 0.14 mm. Experiment results also indicated that the proposed SSM outperforms the SSM that was built on the scans without posture correction. It is concluded that the proposed posture correction approach can effectively improve the accuracy of the hand SSM and therefore enables its wide usage in human-integrated digital twin applications.

Studies of emission regions of the X-ray pulsar Hercules X-1 with pulse-phase-resolved spectra observed with Suzaku

Abstract Pulse-phase-resolved X-ray spectra of the X-ray pulsar Her X-1 observed with Suzaku are analyzed. The analysis is based on a hypothesis that the observed variations of the continuum spectra associated with the neutron star spin originate in the changes of the projection areas of the emission regions viewed from the observer, and that the spectral shape observed from each of the emission regions does not change with the viewing angle variation. We calculate the ratios of the fluxes in each energy bin between arbitrary pairs of the pulse-phase-resolved spectra over the entire energy range. We identify flat portions in these ratio spectra, which are likely to indicate the intrinsic emission components that have invariant spectral shapes, in the three energy bands of below ∼1 keV, 2–6 keV, and above ∼18 keV. Then, we approximately solve the simultaneous equations for the intrinsic spectral components from a pair of the phase-resolved spectra that show flat portions in the ratio spectrum. As a result, spectra of the three components are obtained as the low-, medium-, and high-energy bands corresponding to those for the flat portions. They are found to be well fitted with a soft blackbody, a cut-off power law, and a hard blackbody model, respectively. Finally, we fit all the phase-resolved spectra simultaneously with a composite model of three continuum components together with a cyclotron absorption feature and an iron line, and obtain an acceptable result. The origins of the cut-off power-law and hard blackbody components are discussed in relation to recently developed theoretical models.

Extraction of compact boundary normalisation based geometric descriptors for affine invariant shape retrieval

Shape recognition and retrieval is a complex task on non-rigid objects and it can be effectively performed by using a set of compact shape descriptors. This paper presents a new technique for generating normalised contour points from shape silhouettes, which involves the identification of object contour from images and subsequently the object area normalisation (OAN) method is used to partition the object into equal part area segments with respect to shape centroid. Later, these contour points are used to derive six descriptors such as compact centroid distance (CCD), central angle (ANG), normalized points distance (NPD), centroid distance ratio (CDR), angular pattern descriptor (APD) and multi-triangle area representation (MTAR). These descriptors are a 1D shape feature vector which preserve contour and region information of the shapes. The performance of the proposed descriptors is evaluated on MPEG-7 Part-A, Part-B and multi-view curve dataset images. The present experiments are aimed to check proposed shape descriptor's robustness to affine invariance property and image retrieval performance. Comparative study has also been carried out for evaluating our approach with other state of the art approaches. The results show that image retrieval rate in OAN approach performs significantly better than that in other existing shape descriptors.

On the robustness of skeleton detection against adversarial attacks

Human perception of an object’s skeletal structure is particularly robust to diverse perturbations of shape. This skeleton representation possesses substantial advantages for parts-based and invariant shape encoding, which is essential for object recognition. Multiple deep learning-based skeleton detection models have been proposed, while their robustness to adversarial attacks remains unclear. (1) This paper is the first work to study the robustness of deep learning-based skeleton detection against adversarial attacks, which are only slightly unlike the original data but still imperceptible to humans. We systematically analyze the robustness of skeleton detection models through exhaustive adversarial attacking experiments. (2) We propose a novel Frequency attack, which can directly exploit the regular and interpretable perturbations to sharply disrupt skeleton detection models. Frequency attack consists of an excitatory-inhibition waveform with high frequency attribution, which confuses edge-sensitive convolutional filters due to the sudden contrast between crests and troughs. Our comprehensive results verify that skeleton detection models are also vulnerable to adversarial attacks. The meaningful findings will inspire researchers to explore more potential robust models by involving explicit skeleton features.

An X-Ray Luminosity-dependent “Changing-look” Phenomenon in UGC 3223

Abstract The nature of the rare “Changing-look” (CL) phenomenon in active galactic nuclei (AGNs) is still under debate. We present Swift/XRT and UVOT follow-up observations of UGC 3223, a newly discovered repeat CL-AGN with type transitions of S1.5 → S2 → S1.8 occurring in a period of about 30 yr. By comparing the values previously reported in the ROSAT All-sky Survey and in the second Swift-XRT Point Source catalog, we clearly find that the X-ray flux tightly follows the optical spectral transition, in which a spectral type closer to a Seyfert 1 type is associated with a higher X-ray flux. An invariable X-ray spectral shape is, however, found in the CL phenomenon of the object. An extremely low Eddington ratio of ∼2 × 10−4 can be obtained from the X-ray luminosity for its Seyfert 2 state, which suggests the disk-wind broad-line region model best explains the CL phenomenon. A variation of the total UV emission is not revealed when compared to the previous Galaxy Evolution Explorer NUV observation, since the UVOT images indicate that ∼90% UV emission comes from intensive star formation in the host galaxy.

Climate as a possible driver of gall morphology in the chestnut pest Dryocosmus kuriphilus across Spanish invaded areas.

The alien cynipid wasp Dryocosmus kuriphilus Yasumatsu, 1951 is a serious pest of chestnuts (Castanea spp.) in Japan, North America and Europe, causing fruit losses while inducing galls in buds. While D. kuriphilus galls have a recognizable and roughly invariable globular shape, their size varies, reaching up to 4 cm in diameter. Among other factors, such variation may depend on different climatic conditions in different attacked areas. Here, we sampled and measured 375 D. kuriphilus galls from 25 localities throughout the Iberian Peninsula, including both cold and rainy northern (Eurosiberian) areas and warm and dry central-southern (Mediterranean) areas, to test the effects of climate and geographical location on gall morphology. The analyses indicate that gall mass and volume follow a pattern that can be associated with a climatic cline. In particular, the Eurosiberian galls were smaller than the Mediterranean galls according to differences in climatic conditions. In the southern areas, the greater insolation regime does not allow the chestnut trees to be distributed at lower altitudes, but the high rainfall and humidity regime of the mountain enclaves allow their presence. These conditions of insolation and precipitation seem to influence the morphological characteristics of the galls of D. kuriphilus.

Digital holographic interferometry investigation of liquid hydrocarbon vapor cloud above a circular well.

The current study investigates evaporation of liquid hydrocarbons from a circular well cavity of small depth. Gravimetric analysis is performed to measure the evaporation rate and digital holographic interferometry is used for the measurement of normalized mole fraction profile inside the vapor cloud above the well. Phase unwrapping has been implemented to obtain continuous phase distribution in the image plane. The Fourier-Hankel tomographic inversion algorithm is implemented to obtain the refractive index change distribution inside the object plane, i.e., vapor cloud. Four liquid hydrocarbons, i.e., pentane, hexane, cyclohexane, and heptane, are studied. The radius of circular well cavities is varied in the range of 1.5 to 12.5 mm. Results using a quasi-steady, diffusion-controlled model are compared with the experimental evaporation rate. Measured evaporation rates are higher than the diffusion-limited model calculation for all working fluids and well sizes. This difference is attributed to natural convection occurring inside the vapor cloud due to the density difference between the gas-vapor mixture and the surrounding air. Holographic analysis confirms the presence of natural convection by revealing the formation of a flat disk-shaped vapor cloud above the well surface. Experimentally obtained vapor cloud shape is different from the hemispherical vapor cloud obtained using the pure diffusion-limited evaporation model. The gradient of vapor mole fraction at the liquid-vapor interface is higher compared to that of the diffusion-limited model because of the additional transport mechanism due to natural convection. Transient analysis of the vapor cloud reveals time invariant overall shape of the vapor cloud with a reduction in average magnitude of vapor concentration inside the vapor cloud during evaporation. The existing correlation for sessile droplet cannot successfully predict the evaporation rate from a liquid well. A new correlation is proposed for evaporation rate prediction, which can predict the evaporation rate within a root mean square error of 5.6% for a broad size range of well cavity.

Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,2</sub> (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ). Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n,s</sub> (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ). We define the ordered flip of a matrix A ∈ M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k,ns</sub> (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> ) and present some constructions of self-dual NRT codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> . We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two.

Isomorphic contact resonance force microscopy and piezoresponse force microscopy of an AlN thin film: demonstration of a new contact resonance technique

We present a new contact resonance force microscopy (CRFM) imaging technique, isomorphic contact resonance (iso-CR), that acquires data at a constant contact resonance (CR) frequency, and hence constant tip-sample contact stiffness across the scan area. Constant CR frequency is obtained by performing force versus distance measurements to vary the applied force at each pixel (i.e. force-volume mapping mode). The CR frequency increases with increasing applied force; thus, a carefully selected target frequency will be reached for most pixels at some point in the force versus distance curve. In the iso-CR mode, the cantilever maintains an invariant vibrational shape and a constant environmental damping, thus simplifying interpretation of amplitude and quality factor contrast compared to conventional CRFM. Iso-CR imaging of a piezoelectric AlN thin film sample is demonstrated. Iso-CRFM images were obtained by mechanically driving the base of the cantilever, and iso-CR piezoresponse force microscopy (iso-CR-PFM) images were obtained by electrically biasing the tip. The PFM phase images reveal that the sample contains nanoscale Al-polar (or ‘up’) and N-polar (or ‘down’) domains, with ≈180° phase contrast between oppositely polarized domains. The PFM amplitude and Q-factor images also show ‘up’ vs. ‘down’ domain contrast, which decreases with increasing CR frequency. The frequency-dependent amplitude and Q contrast is ascribed to a frequency-dependent electrostatic contribution to the signal. Domain contrast is not observed in the CRFM (mechanically driven) images. To summarize, the iso-CR capability to control the resonance frequency across multiple excitation schemes helps elucidate the origin of the electromechanical and nanomechanical image contrast.