LoDD

We propose a robust and efficient boundary point detection method based on Local Direction Dispersion (LoDD). The core of boundary point detection lies in measuring the difference between boundary points and internal points. It is a common observation that an internal point is surrounded by its neighbors in all directions, while the neighbors of a boundary point tend to be distributed only in a certain directional range. By considering this observation, we adopt density-independent K-Nearest Neighbors (KNN) method to determine neighboring points and design a centrality metric LoDD using the eigenvalues of the covariance matrix to depict the distribution uniformity of KNN. This paper has been published in IEEE Transactions on Circuits and Systems for Video Technology, and more details can be seen https://ieeexplore.ieee.org/document/10906636.

Datasets

Type	Dataset	Samples	Features	Classes
Synthetic	DS1	459	2	6
Synthetic	DS2	1177	2	4
Synthetic	DS3	1310	3	4
UCI	Breast-Cancer	683	10	2
UCI	PenDigits	10,992	16	10
UCI	Control	600	60	6
UCI	Mice	1,077	77	8
Digit Images	Digits	5,620	8×8	10
Digit Images	MNIST10k	10,000	28×28	10
Face Images	Yale	165	64×64	15
Face Images	FEI-1	700	480×640	50
scRNA-seq	Segerstolpe	2,133	22,757	13
scRNA-seq	Xin	1,449	33,889	4
Text Documents	TDT2-5	2,173	36,771	5
Text Documents	TDT2-10	4,315	36,771	10
Point Cloud	Sphere	46,299	3	1
Point Cloud	Surf	47,046	3	1

How To Run

MATLAB

MATLAB code of LoDD is in the 'lodd_mat' file, where the 'lodd' function provides two parameters for user configuration as follows

function [int_id, bou_id] = LoDD(X, varargin)
%   This function returns the id of internal and boundary points of the N by D matrix X. Each row in X represents an observation.
% 
%   Parameters are:
% 
%   'k_num'      - A non-negative integer specifying the number of nearest neighbors.
%                  Default: 20
%   'ratio'      - A positive scalar specifying the ratio of boundary points.
%                  Default: 0.1

The 'main1.m' file provides an example for detecting boundary points

% Input data
data = textread('Datasets/DS1.txt');

% Obtain data size and true annotations
n = size(data, 1);
X = data(:, 1:end-1);
ref = data(:, end);

% Perform the LoDD algorithm
addpath Functions\
start_time = clock;
true_ratio = sum(ref)/length(ref);
[int_pts, bou_pts] = LoDD(X, 'k_num', 8, 'ratio', true_ratio);
end_time = clock;
disp(['Elapsed time:', num2str(etime(end_time,start_time)), 's']);

% Evaluate the accuracy
res = zeros(n,1);
res(bou_pts) = 1;
ACC = getACC(ref, res);
disp(['Accuracy:', num2str(ACC)]);

% Visualize the result
plot(X(int_pts,1),X(int_pts,2),'ro');
hold on;
plot(X(bou_pts,1),X(bou_pts,2),'bo');
hold on;

The 'main2.m' file provides an example for clustering with K-means

% Input data
data = csvread('Datasets/Control.csv');

% Obtain data size and true annotations
m = size(data, 2);
X = data(:,1:m-1);
X = mapminmax(X',0,1)';
ref = data(:,m);
clus_num = length(unique(ref));

% Perform the KM+LoDD algorithm
clus = ModKmeans(X,'k_num',20,'ratio',0.2,'NumClus',clus_num,'Method','lodd');
ACC = getACC(ref,clus);
NMI = getNMI(ref,clus);
disp(['ACC:', num2str(ACC), ' NMI:', num2str(NMI)]);

Python

Python code of LoDD is in the 'lodd_py' file, where the 'lodd' function provides two parameters for user configuration as follows

def lodd(
        X,
        k_num=20,
        ratio=0.1,
):
    """
        This function returns the id of internal and boundary points of the N by D matrix X. Each row in X
        represents an observation.

        Parameters are:

        'k_num'      - A non-negative integer specifying the number of nearest neighbors.
                       Default: 20
        'ratio'      - A positive scalar specifying the ratio of boundary points.
                       Default: 0.1
    """

The 'main1.py' file provides an example for detecting boundary points

import numpy as np
from Functions.lodd import lodd
import time
import matplotlib.pyplot as plt
from sklearn.metrics import accuracy_score

# Input data
data = np.loadtxt('Datasets/DS1.txt')

# Obtain data size and true annotations
m = data.shape[1]
X = data[:, :m - 1]
ref = data[:, m - 1]

# Perform the LoDD algorithm
start_time = time.time()
true_ratio = sum(ref)/len(ref)
[int_pts, bou_pts] = lodd(X, k_num=8, ratio=true_ratio)
end_time = time.time()
print("Elapsed time:", end_time - start_time, 's')

# Evaluate the accuracy
res = np.zeros(len(ref))
res[bou_pts] = 1
ACC = accuracy_score(ref, res)
print("Accuracy:", ACC)

# Visualize the result
plt.scatter(X[int_pts, 0], X[int_pts, 1], c='red', s=10, marker='o')
plt.scatter(X[bou_pts, 0], X[bou_pts, 1], c='blue', s=10, marker='o')
plt.show()

The 'main2.py' file provides an example for clustering with K-means

from sklearn.preprocessing import MinMaxScaler
from Functions.ModKmeans import mod_kmeans
import numpy as np
import pandas as pd
from sklearn.metrics import normalized_mutual_info_score
from Functions.getACC import acc

# Input data
raw_data = pd.read_csv('Datasets/Control.csv', header=None)
data = np.array(raw_data)

# Obtain data size and true annotations
m = data.shape[1]
X = data[:, :m - 1]
X = MinMaxScaler().fit(X).transform(X)
ref = data[:, m - 1]
clus_num = len(np.unique(ref))

# Perform the KM+LoDD algorithm
clus = mod_kmeans(X, k_num=20, ratio=0.2, c=clus_num, method='lodd')
ACC = acc(ref, clus)

# Citation Request
NMI = normalized_mutual_info_score(ref, clus)
print("Accuracy:", ACC, "NMI:", NMI)

Citation Request

D. Peng, Z. Gui, J. Gui and H. Wu, "A Robust and Efficient Boundary Point Detection Method by Measuring Local Direction Dispersion," in IEEE Transactions on Circuits and Systems for Video Technology, vol. 35, no. 8, pp. 7409-7422, Aug. 2025, doi: 10.1109/TCSVT.2025.3546401.