Backdoor attacks pose a critical threat to machine learning models, causing them to behave normally on clean data but misclassify poisoned data into a poisoned class. Existing defenses often attempt to identify and remove backdoor neurons based on Trigger-Activated Changes (TAC) which is the activation differences between clean and poisoned data. These methods suffer from low precision in identifying true backdoor neurons due to inaccurate estimation of TAC values. In this work, we propose a novel backdoor removal method by accurately reconstructing TAC values in the latent representation. Specifically, we formulate the minimal perturbation that forces clean data to be classified into a specific class as a convex quadratic optimization problem, whose optimal solution serves as a surrogate for TAC. We then identify the poisoned class by detecting statistically small $L^2$ norms of perturbations and leverage the perturbation of the poisoned class in fine-tuning to remove backdoors. Experiments on CIFAR-10, GTSRB, and TinyImageNet demonstrated that our approach consistently achieves superior backdoor suppression with high clean accuracy across different attack types, datasets, and architectures, outperforming existing defense methods.