Adaptive Imaginary Gabor Wavelet: Per-Pixel Kernel Modulation via Local Gradient Activity for Edge-Based Boundary Detection

Edge-based boundary detection, which is frequently enhanced by recent methods like adaptive thresholding [1] and multi-scale selection [2], locates pixels where image intensity changes abruptly. This fundamental process supports medical image inspection, industrial material classification, and agricultural image analysis in which objects appear at varying local densities [3]. Classical detectors such as Canny [4], Sobel and Feldman [5], and the Laplacian of Gaussian [6], together with standard Gabor Wavelets [7, 8], apply fixed-parameter filters uniformly across an image. Even when recent works extend these detectors with the aforementioned adaptive thresholding [1, 9] or multi-scale selection [2, 10], the underlying convolution kernel remains spatially invariant. A single kernel cannot resolve fine boundaries in cluttered regions and capture broad contours in homogeneous regions at the same time, which produces spatially non-uniform responses that lower precision, recall, and the F-measure.

Adaptive approaches act on the threshold [1, 9, 11], on the choice of filter scale at the block level [2, 10], or on Gabor parameters learned inside a convolutional network [12-16]. A related line embeds Gabor filters in deep networks as learnable or fixed front ends [17-21]. None of these sets the kernel parameters of a handcrafted detector at each pixel from a local estimate of image content, and the learned variants require labelled data and a graphics processing unit (GPU). Section 2 reviews these strategies, and Table 1 positions the present formulation against them.

The imaginary-only Gabor Wavelet (GWi) halves the convolution count of the complex Gabor Wavelet (GWC) while preserving the edge-sensitive antisymmetric response, and it inherits the same spatial-invariance limitation. This paper asks whether the kernel parameters of that formulation can be set per pixel in closed form, and what such a mechanism contributes once its effect is separated from the effect of the operating point it selects. The proposed method, the Adaptive Imaginary Gabor Wavelet (A-GWi), modulates the centre frequency and the Gaussian scale at each pixel through a sigmoid mapping of a Sobel-derived local gradient activity estimate. Table 1 places the mechanism against prior adaptation strategies along four dimensions.

This work makes three contributions. First, it proposes a handcrafted spatially varying Gabor formulation in which the kernel frequency and scale are computed at each pixel in closed form, without training. Second, it reports an ablation that separates the contribution of the per-pixel mechanism from the contribution of the operating point, together with two negative controls on the driving signal. Third, it reports an evaluation under the Berkeley protocol on two datasets against five handcrafted baselines, with bootstrap confidence intervals and paired significance tests for every comparison.

Table 1. Adaptation strategies in prior edge detectors and in the proposed method, positioned along four dimensions

Figure 1 first compares the static GWC and GWi pipelines that motivate the imaginary-only design. Specifically, GWC convolves each orientation with both the real cosine kernel and the imaginary sine kernel and combines them by energy, which requires 2N convolutions for N orientations. In contrast, GWi convolves each orientation with the imaginary kernel only and takes the absolute magnitude, which requires N convolutions. By discarding the symmetric response, which mainly captures ridge-like features, GWi retains the antisymmetric response aligned with step edges at half the filtering cost [26]. The final cross-orientation maximum is identical in both pipelines, so the reduction is concentrated entirely in the per-orientation filtering stage. Based on this foundation, the proposed A-GWi method consists of six stages, as shown in Figure 2.

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Figure 2 shows the six processing stages: preprocessing; local gradient activity estimation; adaptive parameter computation; per-pixel imaginary kernel construction; spatially varying multi-orientation convolution; and magnitude extraction with cross-orientation maximum pooling. The adaptive parameters are computed before convolution and vary per pixel, which is the mechanism that turns a static filter bank into a spatially varying convolution. The stages are described next with their governing equations.

3.1 Stage 1: Preprocessing

The colour input is converted to grayscale with ITU-R BT.601 coefficients, as given in Eq. (1), where R, G, and B denote the red, green, and blue channels.

$I_{\text {gray }}(u, v)=0.299 R+0.587 G+0.114 B$              (1)

The grayscale image is then contrast-enhanced by histogram equalization and divided by 255, as in Eq. (2), which places the intensity values in the floating-point range from 0 to 1 and simplifies the subsequent computation.

$I_{\text {norm }}(u, v)=$ HistEq $\left(I_{\text {gray }}\right) / 255$                 (2)

The same preprocessed image is supplied to every method compared in Section 5, so that the filtering stage is the only difference between conditions.

3.2 Stage 2: Local gradient activity estimation

The local gradient activity is estimated from Sobel gradient energy and then smoothed. Eq. (3) computes the spatial derivative along the horizontal axis with a 5 × 5 Sobel aperture, and Eq. (4) computes the derivative along the vertical axis with the same aperture.

$G_x=\operatorname{Sobel}\left(I_{\text {norm }}, d x=1\right.$, ksize $\left.=5\right)$                 (3)

$G_y=\operatorname{Sobel}\left(I_{\text {norm }}, d y=1\right.$, ksize $\left.=5\right)$                 (4)

Eq. (5) gives the gradient energy at each pixel as the Euclidean norm of the two components, that is, the square root of the sum of their squares. Regions containing many objects or dense texture produce high energy values.

$E_{\text {grad }}(u, v)=\sqrt{ }\left(G x^2+G y^2\right)$                  (5)

Eq. (6) gives the final estimate. The energy map is normalised by a minimum-maximum rescaling and then smoothed by a 5 × 5 Gaussian filter, so that the estimate is smooth and free of isolated gradient spikes.

$\rho L(u, v)=$ $GaussianBlur$$\left({minmax}\left(E_{\text {grad}}\right), 5 \times 5\right.$            (6)

High-activity regions map to a high value and homogeneous regions to a low value. Section 5.2 reports how closely the estimate tracks the ground-truth boundary density, and how closely it tracks local intensity contrast.

3.3 Stage 3: Adaptive kernel parameter computation

3.3.1 Adaptive frequency

Eq. (7) maps the activity estimate to a centre frequency through a sigmoid, which gives a bounded transition between the two frequency limits.

$\begin{gathered}f_{0, {adapt}}(u, v)= f_{\min }+\left(f_{\max }-f_{\min }\right) S\left(k_s\left(\rho L(u, v)-\rho_c\right)\right)\end{gathered}$               (7)

Here S(z) = 1 / (1 + exp(−z)), with steepness k_s = 25, centre ρ_c = 0.5, and bounds f_min = 0.05 and f_max = 0.45 cycles per pixel. High-activity regions are intended to receive high frequencies and low-activity regions low frequencies. Because the median activity on the Berkeley Segmentation Dataset 500 (BSDS500) is 0.091, the sigmoid argument at the median is −10.2, so in practice most pixels receive a frequency close to f_min; Section 5.3 quantifies this.

3.3.2 Adaptive scale

Eq. (8) couples the Gaussian scale to the frequency through an inverse relation, so that a high frequency gives a narrow envelope and a low frequency a wide one.

$\sigma_{\text {adapt }}(u, v)=c / f_{0, \text { adapt }}(u, v), c=\sqrt{ }(\ln 2 / 2) / \pi$                 (8)

The constant is c = 0.1874. This is proportional to the standard Gabor bandwidth relation but omits its factor of 3 at one octave, and the fixed 7 × 7 window truncates the envelope, retaining 88% of its mass at f_max and 11% at f_min. The coupling is therefore a scale-frequency relation rather than a constant-bandwidth constraint. Eq. (9) sets the two envelope widths, with the second axis scaled by the aspect ratio γ = 0.5 to sharpen orientation selectivity. Eq. (9) sets the two envelope widths. The width along the second axis is scaled by the aspect ratio γ = 0.5, which elongates the envelope and sharpens orientation selectivity.

$\sigma_x=\sigma_{ {adapt }}, \quad \sigma_y=\sigma_{ {adapt }} / \gamma$                 (9)

3.4 Stage 4: Imaginary-only kernel construction

Eq. (10) constructs the imaginary-only Gabor kernel at each pixel for each orientation. The kernel is a Gaussian envelope multiplied by a sine carrier, which is the imaginary part of the complex Gabor kernel. The indices m and n run from −3 to 3 over the fixed 7 × 7 window, and the rotated indices are $m^{\prime}=m \cos \theta^{\circ}+n \sin \theta^{\circ}$ and $n^{\prime}=-m \sin \theta^o+n \cos \theta^o$. Removing the cosine component halves the number of convolutions without discarding edge information. The parameters are those computed in Stage 3, which makes the kernel unique at every location.

$\begin{gathered}\psi_{ {imag }}(m, n)= \exp \left[-0.5\left(\frac{m^{\prime 2}}{\sigma_x^2}+\frac{n^{\prime 2}}{\sigma_y^2}\right)\right] \sin \left(2 \pi f_{0, {adapt}} m^{\prime}\right)\end{gathered}$                (10)

3.5 Stage 5: Spatially varying multi-orientation convolution

Eq. (11) computes the response at orientation θ^o. Because the kernel parameters vary with position, the operation is a spatially varying convolution. The expression is written as a correlation; for the antisymmetric kernel used here, the convolution differs only in sign, so the two are equivalent after the absolute value is taken in Eq. (12). Pixels outside the image are supplied by reflection padding, which is applied identically to every method compared in Section 5.

$\begin{gathered}R_o(u, v)= \sum_m \sum_n I_{ {norm}}(u+m, v+n) \psi_{ {imag }}(m, n)\end{gathered}$                  (11)

The kernel is not normalised, so its L2 norm varies with the centre frequency across the adaptive range, from 3.010 at f_min to 0.030 at f_max. This is reported here because it means that the response magnitude carries an amplitude scaling that is correlated with the activity estimate.

3.6 Stage 6: Magnitude extraction and maximum pooling

Eq. (12) takes the absolute value of each orientation response, and Eq. (13) selects the dominant orientation at each pixel.

$M_o(u, v)=\left|R_o(u, v)\right|$                 (12)

$E_{A G W i}(u, v)=\max _o M_o(u, v)$                  (13)

Eq. (14) rescales the output to the range from 0 to 1 by a minimum-maximum mapping computed per image. This step is stated explicitly because it determines what a single dataset-wide threshold means in the evaluation protocol of Section 4.2.

$\begin{gathered}E_{ {norm }}= \left(E_{ {AGWi }}-\min E_{{AGWi}}\right) /\left(\max E_{{AGWi }}-\min E_{ {AGWi}}\right)\end{gathered}$               (14)

3.7 Implementation settings

Table 2 lists every parameter used to produce the results in Section 5. The values are those in the released implementation, and Section 4.4 reports how the three sigmoid parameters were verified.

Table 2. Implementation settings of the Adaptive Imaginary Gabor Wavelet (A-GWi)

3.8 Algorithm and complexity

Algorithm 1 states the procedure. The adaptive parameters on lines 4 to 6 are recomputed at every pixel before the orientation loop, which is the operational difference from a static Gabor bank. For an image of height H and width W with N orientations and a fixed k × k window, the cost is O(HWNk²) with k = 7. The inner loops are compiled to machine code with a just-in-time compiler, and all timing in Section 5.7 is single-threaded.

5.1 Quantitative results

Table 5 reports the three metrics on the BSDS500 test set with 95% confidence intervals obtained by bootstrapping over images with 2,000 resamples. A-GWi attains the highest ODS at 0.534, above Canny at 0.533, Phase Congruency at 0.524, Sobel at 0.464, GWi at 0.417, and GWC at 0.397. The ordering is consistent with the design of the methods: the static Gabor banks sit below Sobel because their fixed kernel parameters do not match the per-pixel statistics of natural scenes, which produces over-detection in homogeneous regions and lowers precision. Table 6 reports the same metrics on UDED, where Canny attains the highest ODS at 0.670 and A-GWi follows at 0.668.

Table 5. Results on the BSD500 test set of 200 images, with 95% confidence intervals (CI) from 2,000 bootstrap resamples over images. Best value per column in bold

Table 6. Results on the Unified Dataset for Edge Detection (UDED) of 30 images, with 95% confidence intervals from 2,000 bootstrap resamples over the images. Best value per column in bold

Table 7 reports the paired comparisons. A-GWi exceeds both static Gabor baselines on both datasets with p below 10⁻⁷ and effect sizes between medium and large on the Cliff scale [41], where values below 0.147 are negligible, below 0.33 small, and below 0.474 medium. The difference against Canny is not significant on either dataset, at p = 0.82 on BSDS500 and p = 0.82 on UDED, so the two are not separable on this evidence. Against Phase Congruency the effect size differs between datasets, negligible on BSDS500 at 0.137 and medium on UDED at 0.442. The confidence intervals on UDED are wide because the dataset contains 30 images, and the interval for A-GWi overlaps those of Canny and Sobel. Figure 4 plots the precision-recall curves for both datasets.

Table 7. Paired comparison against the Adaptive Imaginary Gabor Wavelet (A-GWi) using the Wilcoxon signed-rank test [42] on per-image F-measures at the optimal dataset scale (ODS) threshold. The effect size is the Cliff delta [41]

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5.2 Validation of the activity estimate

The quantity defined in Eqs. (3) to (6) is a smoothed gradient magnitude, so its relation to boundary density is not automatic and is verified here. Across the 200 BSDS500 test images, the Spearman correlation between the estimate and the local ground-truth boundary count in a 15 × 15 window has a median of 0.238, with an interquartile range from 0.111 to 0.361. The correlation with local intensity contrast in the same window has a median of 0.763, with an interquartile range from 0.705 to 0.811. The estimate therefore tracks local gradient activity rather than boundary density, and the term used throughout this paper reflects that.

Two consequences follow from the definition and are measured directly. In 2.76% of pixels, a high activity value coincides with a low boundary count, which occurs at isolated high-contrast edges. In 2.66% of pixels a high boundary count coincides with a low activity value, which occurs in dense low-contrast texture. Figure 5 shows the distribution of the estimate and Figure 6 shows the two correlations.

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Figure 5. Local gradient activity on the BSD500, (a) distribution across the test set, with the median, the mean, and the sigmoid centre of Eq. (7) marked, (b) per-image standard deviation of the estimate, which shows that individual images contain both low-activity and high-activity regions

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Figure 6. Validation of the local gradient activity estimate on the 200 test images of the BSD500, (a) per-image Spearman correlation of the estimate against the ground-truth boundary count in a 15 × 15 window and against local intensity contrast in the same window, (b) pixel-level relation between the estimate and the ground-truth boundary density

5.3 Ablation

Table 8 isolates the contribution of each element of the pipeline on the BSDS500 test set. Replacing the per-pixel computation with a static kernel at the lower frequency bound gives ODS = 0.535 against 0.534 for the full formulation, and the paired difference is not significant. Permuting the activity estimate spatially, which destroys its correspondence with image content while preserving its distribution, lowers ODS by only 0.004. Replacing the Sobel-based estimate with a local variance estimate raises ODS to 0.540. Evaluating the activity estimate directly as an edge map gives ODS = 0.508. Rank-normalising the estimate so that the sigmoid operates across its full range lowers ODS to 0.361.

Table 8. Ablation on the BSD500 test set. ODS denotes the optimal dataset scale F-measure (ODS)

Two conclusions follow. First, the improvement over the static imaginary Gabor baseline at a wavelength of 4 pixels comes from the change in effective centre frequency rather than from per-pixel modulation. At the parameter setting used here, 92.8% of pixels receive a frequency within 1% of the lower bound, and the output correlates at 0.857 with a static bank at that frequency, as Figure 7 shows. Second, when the modulation is made active across the full range, accuracy falls, so the hypothesis that high-activity regions require high-frequency kernels is not supported on this benchmark. The per-pixel mechanism reaches the same operating point as the best static configuration without a manually selected centre frequency, rather than improving on it.

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Figure 7. Behaviour of the adaptive mechanism at the setting reported in Table 2. (a) Distribution of the local gradient activity estimate against the sigmoid centre of Eq. (7). (b) Distribution of the resulting adaptive frequency on a logarithmic count axis; most pixels receive a frequency at the lower bound. (c) L2 norm of the imaginary Gabor kernel as a function of centre frequency inside the fixed 7 × 7 window

5.4 Activity-stratified analysis

The aggregate values in Tables 5 and 6 conflate performance across pixel contexts, so Tables 9 and 10 evaluate three activity strata separately. Stratum boundaries are the tertiles of the activity estimate on the BSDS500 training split, 0.053 and 0.146, applied as absolute thresholds to both evaluation sets so that no test statistic enters the analysis and the datasets remain comparable. Within each stratum, predicted and ground-truth pixels outside the mask are discarded before matching, so precision and recall are computed on the same population, Eq. (20). The per-stratum quantity F_s is not comparable with the aggregate value.

$F_s=\max _t F(t)$ restricted to $M_s$               (20)

Table 9. Activity-stratified F-measure on the BSD500

Table 10. Activity-stratified F-measure on the Unified Dataset for Edge Detection (UDED), using the same absolute stratum thresholds as Table 9. Abbreviations as in Table 9. Best value per stratum in bold

Ground-truth boundary pixels are unevenly distributed across strata: on BSDS500, 7.1%, 25.1%, and 67.8% fall in the low, middle, and high strata, with a similar pattern on UDED, which is why the low-stratum values sit far below the aggregate. A-GWi exceeds both static Gabor baselines in every stratum on both datasets. The margin over GWi is +0.028, +0.007, and +0.096 on BSDS500 and +0.080, +0.049, and +0.024 on UDED, so the two datasets place the largest margin in different strata and the stratified view does not support a single account of where the difference arises. Canny is ahead in the middle and high strata on both datasets.

5.5 Error characterisation

Table 11 reports three quantities that address the behaviour of each detector more directly than the aggregate metrics. Precision at a fixed recall (PR) of 0.5 is defined in Eq. (21) and measures over-detection at a common operating point. The localisation error in Eq. (22) is the mean Euclidean distance between matched predicted pixels and their ground-truth counterparts at the ODS threshold. The edge width in Eq. (23) is the ratio of pixels above the threshold to pixels remaining after thinning, so a value near 1 means the response is already thin.

Table 11. Error characteristics on the BSD500

$P R=P\left(t^*\right), R\left(t^*\right)=0.5$               (21)

$L_{e r r}=(1 / \mid$ $Match$ $\mid) \sum\|p-g\|$                (22)

$W_{{edge }}=\left|\left\{(u, v): E_{ {norm}} \geq t_{ {ODS}}\right\}\right| /\left|P\left(t_{ {ODS}}\right)\right|$                (23)

Precision at fixed recall (PR) rises from 0.405 for GWi to 0.564 for A-GWi, which quantifies the reduction in spurious response that the qualitative comparison in Section 5.6 describes. The localisation error lies between 1.30 and 1.49 pixels for all four methods, well inside the matching tolerance of 4.3 pixels for a 481 × 321 image, so localisation accuracy is not a source of difference here. The edge width is 2.33 for GWi and 2.80 for A-GWi, which reflects the wider Gaussian envelope of the kernel that the mechanism selects at most pixels. The difference between the two methods therefore comes from response selectivity rather than from response width.

5.6 Qualitative comparison

Figures 8 and 9 show the six detectors on representative images, and Figure 10 shows the per-pixel parameters with the resulting edge maps. A-GWi produces fewer spurious responses in homogeneous regions than either static Gabor baseline, which Table 11 quantifies as precision at fixed recall, but its response is not narrower than that of GWi, so the difference lies in selectivity rather than in the spatial extent of the response. The frequency map in Figure 10 is close to uniform at the lower bound over most of each image, which is the visual counterpart of the ablation in Table 8; the activity map itself concentrates along object boundaries, so the driving signal is informative even though the mapping in Eq. (7) compresses it to a narrow frequency range at this setting.

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Figure 8. Qualitative comparison on the BSD500 for image identifiers 3063, 29030, 128035, and 35049 Edge maps are shown as dark edges on a light background, (a) Rows: original image, ground truth, Canny, Sobel (b) Phase Congruency, Gabor Wavelet Complex (GWC), imaginary-only Gabor Wavelet (GWi), and Adaptive Imaginary Gabor Wavelet (A-GWi)

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5.7 Runtime

Table 12 reports the mean per-image runtime on the 200 BSDS500 test images, measured single-threaded with the just-in-time compiler warmed before timing. Each image is timed three times and the minimum is retained, which limits the influence of background system load; the standard deviation for A-GWi is 5.82 ms, under 1% of the mean. The gradient operators are fastest, at 1.66 ms for Sobel and 2.28 ms for Canny. The static Gabor banks are an order of magnitude slower because they filter eight orientations: GWi at 15.54 ms and GWC at 33.70 ms. The ratio of 2.17 between them matches the ratio of 2N to N convolutions and confirms that the imaginary-only formulation halves the filtering cost. Phase Congruency runs at 402.16 ms. A-GWi is the slowest at 604.41 ms, because it builds a kernel at every pixel rather than reusing a precomputed bank.

Table 12. Mean per-image runtime on the 200 test images of the BSD500, single-threaded on an Intel Core i5-14400

Within the multi-scale family, A-GWi is 1.5 times slower than Phase Congruency while reaching a higher ODS on BSDS500. Against Canny it is 265 times slower for an accuracy difference that is not significant on either dataset, so the gradient operator is the more efficient choice where a single edge magnitude map is the required output; the per-pixel formulation is of interest where the multi-orientation response is used downstream or where kernel parameters must be set without manual tuning.

5.8 Limitations

Three limitations follow from the results. The per-pixel construction prevents the separable optimisation available to static kernels, and at 604 ms per image it is 265 times slower than a gradient baseline of comparable accuracy. The mechanism does not improve accuracy over the best static configuration on these benchmarks, and making the modulation fully active lowers accuracy, so the hypothesis relating activity to frequency is not confirmed; the supported claim is that the mechanism reaches that configuration without manual tuning. The evaluation is limited to natural-scene benchmarks, and validation on dense agricultural grain imagery is left for future work.

Conceptualization, K. Yudhistiro, R.S. Basuki, and N. Rijati; methodology and software, K. Yudhistiro; validation, all authors; formal analysis, data curation, and visualization, K. Yudhistiro; writing, original draft, K. Yudhistiro; writing, review and editing, R.S. Basuki and N. Rijati; supervision, R.S. Basuki and N. Rijati.

All evaluation code, ablation scripts, runtime logs, and figure-generation scripts are released at https://github.com/kukuhyudhistiro/localgradientactivity- to enable reproduction.